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@@ -0,0 +1,2583 @@
+1
+Online Fusion of Multi-resolution Multispectral Images with Weakly
+Supervised Temporal Dynamics
+Haoqing Lia, Bhavya Duvvuria, Ricardo Borsoib, Tales Imbiribaa, Edward Beighleya,
+Deniz Erdo˘gmu¸sa, Pau Closasa
+aNortheastern University, Boston, 02215, MA, USA
+bUniversity of Lorraine, CNRS, CRAN, Nancy, F-54000, France
+Abstract
+Real-time satellite imaging has a central role in monitoring, detecting and estimating the
+intensity of key natural phenomena such as floods, earthquakes, etc. One important con-
+straint of satellite imaging is the trade-off between spatial/spectral resolution and their
+revisiting time, a consequence of design and physical constraints imposed by satellite or-
+bit among other technical limitations. In this paper, we focus on fusing multi-temporal,
+multi-spectral images where data acquired from different instruments with different spatial
+resolutions is used. We leverage the spatial relationship between images at multiple modal-
+ities to generate high-resolution image sequences at higher revisiting rates. To achieve this
+goal, we formulate the fusion method as a recursive state estimation problem and study
+its performance in filtering and smoothing contexts. Furthermore, a calibration strategy is
+proposed to estimate the time-varying temporal dynamics of the image sequence using only
+a small amount of historical image data. Differently from the training process in traditional
+machine learning algorithms, which usually require large datasets and computation times,
+the parameters of the temporal dynamical model are calibrated based on an analytical ex-
+pression that uses only two of the images in the historical dataset. A distributed version
+of the Bayesian filtering and smoothing strategies is also proposed to reduce its compu-
+tational complexity. To evaluate the proposed methodology we consider a water mapping
+task where real data acquired by the Landsat and MODIS instruments are fused generating
+high spatial-temporal resolution image estimates. Our experiments show that the proposed
+methodology outperforms the competing methods in both estimation accuracy and water
+mapping tasks.
+Keywords:
+Multimodal image fusion, Online Fusion, Bayesian Filtering, Water mapping,
+Super-resolution
+1. Introduction
+High spatial resolution satellite image data is a fundamental tool for remote sensing
+applications such as the monitoring of land cover changes [1, 2], deforestation [3, 4] or wa-
+ter mapping [5, 6] and water quality [7]. Moreover, to adequately deal with the variability
+Preprint submitted to ISPRS Journal of Photogrammetry and Remote Sensing
+June 2022
+arXiv:2301.02598v1  [eess.IV]  6 Jan 2023
+
+of such events over time it is important to have short time spans between different image
+acquisitions of the same scene (i.e., a high temporal resolution, or low revisit times). How-
+ever, fundamental limitations of multiband imaging instruments and large sensor-to-target
+distances impose a trade-off between spatial and temporal resolutions of satellite image
+sequences.
+This means that instruments providing high spatial resolution have long revisit times,
+while the converse holds for instruments with short revisit times. This can be illustrated, for
+instance, by considering Landsat 8 and MODIS instruments (with 30 and 250/500 meters
+spatial resolution, respectively). While MODIS is able to provide daily images at coarse
+resolution, Landsat-8 only revisits the same site once every 16 days [8].
+Considering these limitations, many works proposed multimodal image fusion techniques
+to generate high (spatial, spectral or temporal) resolution remote sensing images. Multi-
+modal image fusion aims to combine multiple observed images, each of which having high
+resolution in a given dimension – spatial, temporal, or spectral – to generate high reso-
+lution image sequences.
+Several instances of image fusion have been considered, some
+works aim to directly supply classification maps from multiple satellite image and surface
+elevation data at each time instant [9], integrating optical and radar data for time-series
+crop classification [10, 11], or fusing spatio-temporal optical and elevation data to obtain
+high-resolution land temperature maps [12].
+In particular, classification or mapping tasks based on time-series remote sensing data
+is receiving increasing interest in the literature [13, 14, 10, 11]. Thus, to overcome the limi-
+tations of existing instruments, fusing images with different spectral and spatial resolutions
+has been extensively studied to generate images with high spatial and spectral resolutions,
+which are critical for accurately distinguishing different materials in a pixel [15, 16, 17].
+Recently, an increasing interest has been observed in applying multimodal image fusion to
+generate image sequences with high spatial and temporal resolutions [18], with particular
+interest dedicated to fusing data from multiple satellites to obtain daily images with high
+(e.g., 30 m) resolution [19].
+This has already had an important impact in applications
+such as the generation of daily snow cover maps [20] and the study of drought-induced
+tree mortality [21]. Existing spatiotemporal image fusion methods are usually divided in
+weighted fusion, umixing-based, learning-based and Bayesian approaches [22]. There also
+exist hybrid techniques, which leverage ideas from more than one family of approaches.
+Weighted fusion methods assume that the temporal changes occurring between two time
+instants are consistent between the high and low spatial resolution images for low resolution
+pixels which are composed of only a single material [23]. However, coarse resolution pixels
+are often mixtures of different materials.
+The predicted high resolution pixels are then
+computed as a weighted linear combination of the previous high resolution pixels and of
+the changes occurring at low resolution pixels in a given neighborhood [24, 25]. Different
+works have designed various weighting functions, which aim to select neighboring pixels
+that are homogeneous and spatially/spectrally similar to the pixel whose change is being
+predicted [24, 22, 26].
+Other works have extended such framework account for sudden
+changes [27] or to use different weighting functions [25].
+2
+
+Figure 1: Overview of the proposed method. Multimodal (e.g., Landsat and MODIS over time) images time
+series are fused by the Distributed Multimodal Bayesian Fusion algorithm resulting in a high spatial-temporal
+resolution estimated sequence. Covariance estimates for the dynamical model are estimated through a weakly
+supervised strategy based on local high-resolution historical data. We highlight that the Bayesian fusion
+methodology employed here is agnostic to the multimodal measurement model making the strategy easily
+generalizable to different data scenarios.
+Unmixing-based methods make use of the linear mixing model (LMM), which assumes
+that each pixel in the low resolution image can be represented as a convex combination of the
+reflectance of a small number of pure spectral signatures, called endmembers [28, 29]. The
+LMM has been used for multimodal image fusion by assuming the proportions of each mate-
+rial in a low resolution pixel to be stable/constant over time [30, 31, 32]. This way, spectral
+unmixing [28] is used to estimate the endmembers at different time instants from low reso-
+lution images, while using different strategies to mitigate the spectral variability of a single
+material [30, 33, 34]. However, abrupt abundance variations (originating from, e.g., land
+cover changes) are commonly found in multitemporal image streams [35, 36, 37, 38], which
+may negatively impact the performance of such methods and can be particularly challeng-
+ing to address when occurring jointly with finer endmember variations [35]. Thus, special
+care is required when fusing images which are temporally distant from one another [39],
+motivating the development of strategies using, e.g., spatially adaptive quantification of the
+reliability of the input images to guide unmixing based image fusion strategies [40].
+Learning-based approaches leverage training data and different machine learning al-
+gorithms in order to perform image fusion.
+Those approaches are varied, ranging from
+approaches such as dictionary learning [41], which are based on a sparse representation
+of image pixels and have a strong connection to the LMM, to convolutional neural net-
+works [42], which are flexible function approximations which are typically used to learn a
+mapping from the low-resolution to high resolution data.
+Bayesian methods are flexible alternatives to the previous approaches that take into ac-
+count the uncertainty present both in the imaging model and in the estimated images. The
+3
+
+Bayesian framework is based on the definition of probabilistic models to describe the rela-
+tionship between images of different spatial, spectral and temporal resolutions acquired by
+different instruments. This allows image fusion to be formulated as a maximum a posteriori
+estimation problem [43]. Although Bayesian methods usually consider Gaussian distribu-
+tions for mathematical tractability, different variations have been proposed depending on
+how the image acquisition process is modelled and on how the mean and covariance matri-
+ces are estimated. This included assuming them diagonal [44], estimating image covariance
+matrices based on an initial estimate of the high resolution image [43], or based on the low
+resolution image pixels [45].
+A recent work considered a Kalman filter-based approach to estimate a high resolution
+image sequence based on mixed resolution observations from the Landsat and MODIS in-
+struments [46]. However, to define the model for the Kalman filter, two Landsat+MODIS
+image pairs at times t0 and tN are considered, as well as a time series of MODIS images
+at instants tk P rt0, tNs, making it unsuitable for online operation. Moreover, changes be-
+tween each pair of images were assumed to be constant/uniform over predefined groups of
+high resolution image pixels, which can be restrictive (due to the large resolution difference
+between the measured images, the groups must contain many pixels in order to make the
+model well-posed). It also does not benefit from auxiliary information that could aid the
+estimation of the high resolution images. Another work used the Kalman filter to esti-
+mate normalized difference vegetation indices (NDVI) time series images from Landsat and
+MODIS observations, using an affine model for the dynamics of the states whose coeffi-
+cients are selected based on the seasonality, and another affine model to relate the NVDI
+estimate obtained from MODIS and Landsat measurements [47]. The Kalman filter was
+also recently applied to estimate land surface temperature by fusing thermal infrared and
+microwave data [48].
+In this paper, we propose a weakly supervised Kalman filter and smoother framework
+for spatio-temporal fusion of multispectral images. The proposed framework relies on ex-
+plicit modeling assumptions about the image acquisition and temporal evolution processes,
+under which the proposed solution is statistically optimal. The Kalman filter-based meth-
+ods can operate in a fully online setting, where high-resolution images are only available
+as past data. We also develop a smoother-based method to optimally exploit information
+contained in future high-resolution observed images when processing images in a time win-
+dow. However, the quality of the reconstruction of Kalman filter and smoother strategies
+depend directly on the quality of the dynamical image evolution model. Thus, to overcome
+this limitation, a weakly supervised strategy is proposed to learn the temporal dynamics
+of the high-resolution images from a small amount of past data. More precisely, instead of
+considering the changes to be constant over areas comprising large amounts of image pixels,
+we propose an analytical calibration strategy to estimate a more informative time-varying
+dynamical image model by leveraging historical data. This allows for a better localiza-
+tion of changes in the high resolution image even in intervals where only coarse resolution
+observations (e.g., MODIS) are available. Moreover, to mitigate the high computational
+complexity of the Kalman filter and smoother, we propose a distributed implementation
+4
+
+by exploiting different independence assumptions about the high-resolution state space,
+allowing the proposed methods to be applied to large datasets and geographical areas. Fig-
+ure 1 depicts the proposed methodology where high-resolution (spatially and temporally)
+estimates are generated by fusing different data modalities. We illustrate the application
+of the proposed framework by fusing images from the Landsat and MODIS instruments.
+Experimental results indicate that the proposed method can lead to considerable improve-
+ments compared to using a non-informative dynamical model and to widely used image
+fusion algorithms, both in image reconstruction and in downstream water classification
+and hydrograph estimation tasks. A software package containing an implementation of the
+proposed method and the image dataset is available at https://github.com/HaoqingLi/
+Multi-resolution-Multispectral-image-fusion-based-weakly-supervised-constrained-Kalman-filter.
+This paper is organized as follows. In Section 2, we present the paper notation and the
+proposed imaging model. Section 3 presents the Kalman filter and smoother approaches
+for multimodal image fusion. Section 5 contains simulation experiments that illustrate the
+performance of the proposed method. Finally, Section 6 concludes the paper.
+2. Dynamical Imaging Model
+2.1. Definitions and notation
+Let us denote the the ℓ-th band of the k-th acquired image reflectances from modality
+m P Ω by ym
+k,ℓ P RNm,ℓ, with Nm,ℓ pixels for each of the bands ℓ “ 1, . . . , Lm, and Ω denoting
+the set of image modalities. As a practical example, we consider Ω “ tL, Mu to contain
+the Landsat-8, and MODIS image modalities, without loss of generality. We also denote by
+ΩH the highest resolution image modality, e.g., ΩH “ tLu. We denote the corresponding
+high resolution latent reflectances by Sk P RNHˆLH, with NH pixels and LH bands, with
+LH ě Lm and NH ě Nm,ℓ, @ℓ, m. Subindex k P N˚ denotes the acquisition time index. We
+also denote by vecp¨q, colt¨u, diagt¨u and by blkdiagt¨u the vectorization, vector stacking,
+diagonal and block diagonal matrix operators, respectively. The notation xa:b for a, b P N˚
+represents the set txa, xa`1, . . . , xbu. We use Npµ, Σq to denote a Gaussian distribution
+with mean µ and covariance matrix Σ.
+2.2. Measurement model
+To formulate our measurement model we assume that the acquired image at time index
+k, for any imaging modality, is a spatially degraded and spectrally transformed version of
+the high resolution latent reflectance image Sk. Following this assumption our measurement
+model for the m-th modality becomes:
+ym
+k,ℓ “ Hm
+ℓ pSkqcm
+ℓ ` rm
+k,ℓ ,
+ℓ “ 1, . . . , Lm ,
+(1)
+where cm
+ℓ P RLH denotes a spectral transformation vector, mapping all bands in Sk to the
+ℓ-th measured band at modality m; Hm
+ℓ
+is a linear operator representing the band-wise
+spatial degradation, modeling blurring and downsampling effects of each high resolution
+band, and rm
+k,ℓ represents the measurement noise. Note that, while we consider the spatial
+5
+
+resolution of the high resolution bands in Sk to be the same, different bands from the
+same modality can have different resolutions. We also assume the measurement noise to
+be Gaussian and uncorrelated among bands, that is, rm
+k,ℓ „ Np0, Rm
+ℓ q with time-invariant
+covariance matrix given by Rm
+ℓ P RNm,ℓˆNm,ℓ, and covprm
+k,j, rm
+k,ℓq “ 0 for all j ‰ ℓ.
+Note that satellite images may be corrupted by several effects, including dead pixels
+in the sensor, incorrect atmospheric compensation, and the presence of heavy cloud cover.
+Such pixels cannot be reliably used in the image fusion process as they may degrade the
+performance of the method.
+Directly addressing these effects using a statistical model
+would require the choice of a non-Gaussian distribution for the noise vector rm
+k,ℓ, which
+could make the computational complexity of the fusion procedure prohibitive. Thus, we
+consider a matrix Dm
+k P R r
+NmˆNm, which eliminates outlier pixels from the image, leading
+to the following transformed measurement model:
+rym
+k,ℓ “ Dm
+k Hm
+ℓ pSkqcm
+ℓ ` rrm
+k,ℓ ,
+(2)
+where rym
+k,ℓ “ Dm
+k ym
+k,ℓ and rrm
+k,ℓ “ Dm
+k rm
+k,ℓ denotes the measured image band and the mea-
+surement noise in which the outlier values have been removed.
+Using (2) and the properties of the vectorization operator, we can write this model
+equivalently as
+rym
+k,ℓ “
+“
+pcm
+ℓ qJ b Dm
+k
+‰
+vec
+`
+Hm
+ℓ pSkq
+˘
+` rrm
+k,ℓ
+“
+“
+pcm
+ℓ qJ b Dm
+k
+‰
+Hm
+ℓ sk ` rrm
+k,ℓ
+(3)
+where b denotes the Kronecker product.
+The variable sk P RLHNH denotes a vector-
+ordering of the high-resolution image Sk which is obtained by grouping all pixels such that
+the bands of a single HR pixel are adjacent to each other, and the pixels that are contained
+within a single “lowest-resolution” pixel are also adjacent to each other, that is:
+sk “
+»
+—————————–
+»
+————————–
+sk,1,ιp1,1q
+...
+sk,LH,ιp1,1q
+sk,1,ιp2,1q
+...
+sk,LH,ιpd,1q
+fi
+ffiffiffiffiffiffiffiffifl
+J
+, . . . ,
+»
+—————————–
+sk,1,ιp1,Nm1,ℓ1q
+...
+sk,LH,ιp1,Nm1,ℓ1q
+sk,1,ιp2,Nm1,ℓ1q
+...
+sk,LH,ιpd,Nm1,ℓ1q
+fi
+ffiffiffiffiffiffiffiffiffifl
+Jfi
+ffiffiffiffiffiffiffiffiffifl
+J
+,
+(4)
+where sk,i,j is the pi, jq-th position of Sk, m1 and ℓ1 are the modality and spectral band with
+the lowest spatial resolution (i.e., for which Nm,ℓ is smallest), d “ NH{Nm1,ℓ1 is the number of
+HR pixels inside each low resolution pixel of band ℓ1 and modality m1, and ι : N˚ ˆN˚ Ñ N˚
+is a function such that ιpi, jq returns the index (in Sk) of the of the i-th HR pixel contained
+inside the j-th low resolution pixel (where i P t1, . . . , du) for modality m1 and band ℓ1. Hm
+ℓ
+is a matrix form representation of the operator Hm
+ℓ , such that vecpHm
+ℓ pSkqq “ Hm
+ℓ sk.
+6
+
+We can now represent all bands from each modality in the form of a single vector
+rym
+k P R r
+NmLm as
+rym
+k “
+¨
+˚
+˝
+“
+pcm
+1 qJ b Dm
+k
+‰
+Hm
+1
+...
+“
+pcm
+LmqJ b Dm
+k
+‰
+Hm
+Lm
+˛
+‹‚
+looooooooooooooomooooooooooooooon
+Ă
+H
+m
+k
+sk ` rrm
+k ,
+(5)
+where rrm
+k „ Np0, rR
+m
+k q, and
+rym
+k “ col
+␣
+rym
+k,1, . . . , rym
+k,Lm
+(
+,
+(6)
+rrm
+k “ col
+␣
+rrm
+k,1, . . . , rrm
+k,Lm
+(
+,
+(7)
+rR
+m
+k “ blkdiag
+␣
+Dm
+k Rm
+1 pDm
+k qJ, . . . , Dm
+k Rm
+LmpDm
+k qJ(
+(8)
+Note that at most time instants k, one or more of the modalities m P Ω is not observed. In
+this case, we set the matrix Dm
+k as an empty (zero-dimensional) matrix, which simplifies
+the problem and avoids introducing additional notation.
+2.3. Dynamical evolution model
+Defining reasonable dynamical models for image fusion requires detailed knowledge re-
+garding the scene evolution over time, which is often unattainable. In this contribution, we
+aim at a complete data driven strategy assuming very little knowledge regarding the scene
+evolution except for past data coming from the imaging modalities being used. To match
+such lack of prior knowledge we consider a simple random-walk process to model the latent
+state dynamics as:
+sk`1 “ F ksk ` qk ,
+(9)
+where F k P RLHNHˆLHNH is the state transition matrix, which is assumed to satisfy
+}F k}2 ď 1, and qk „ Np0, Qkq with Qk P RLHNHˆLHNH being the state process noise
+covariance matrix. Note that the above model plays a crucial role in the estimation results,
+as it describes both the distribution of the changes occurring in the image at time k, as well
+as the marginal distribution of the states. This means that more sophisticated dynamics
+can be introduced in the problem through the appropriate design of the process noise co-
+variance matrix Qk. Although expectation maximization (EM) can be used to estimate Qk
+in time invariant models [49], the problem becomes extremely ill-posed in the time-varying
+setting. Another issue relates to the computational complexity of EM-based strategies re-
+quiring the solution of the Kalman filter and smoother systems multiple times, becoming
+unfeasible when dealing with large images. For these reasons, we propose an alternative
+route to estimate Qk.
+7
+
+2.4. A weakly supervised approach for estimating Qk
+We consider QkpDkq as a function of the set Dk “ t˜ymPΩH
+ℓ
+uℓăk of past high resolution
+images. The set Dk represents historical data and images currently being fused up the the
+time step k. Although many strategies could be leveraged to find suitable past time windows
+to account for more relevant covariance estimation and consider full covariance matrices,
+in this preliminary work we choose a simple route to validate this type of approach. For
+this, let ymPΩH
+k´τ
+be the the most recently observed high resolution image1. We compute Qk
+by finding in our historical data the most similar image to ymPΩH
+k´τ
+and then computing the
+pixelwise variance across the following n P N˚ images in our historical data. That is, we
+compute Qk executing the following three steps for every time step k:
+1. Identify the most similar state over Dk, that is, the image that is most similar, ac-
+cording to a metric L
+ℓ˚ “ arg min
+ℓPIDk
+L
+`
+ymPΩH
+k´τ
+, rDksℓ
+˘
+,
+(10)
+with rDksℓ being the ℓ-th image in the historical set Dk, and IDk Ď Z is the set
+containing the time index of each image in Dk.
+2. select a time window rDksℓ˚:ℓ˚`n.
+3. compute the diagonal process noise covariance matrix, i.e., Qk “ diagtq2
+k,1, . . . , q2
+k,LHNHu,
+as
+q2
+k,j “ max
+ˆvar
+`
+rDkspjq
+ℓ˚:ℓ˚`n
+˘
+∆ℓ˚
+Dk
+, ε2
+˙
+ˆ ∆k ,
+(11)
+where rDkspjq
+ℓ˚:ℓ˚`n “ r˜ymPΩH
+ℓ˚,j
+, . . . , ˜ymPΩH
+ℓ˚`n,js, ε ą 0 is a small scalar allowing for changes on the
+scene that were unseen on the historical data window rDksℓ˚:ℓ˚`n, ∆k is the time interval
+(in days) between ymPΩH
+k
+and ymPΩH
+k`1
+, and ∆ℓ˚
+Dk is the time interval (in days) between rDksℓ˚
+and rDksℓ˚`n. As similarity metric we used the cosine similarity Lpy, zq “ cospy, zq.
+3. Multimodal image fusion using a weakly supervised constrained Kalman fil-
+ter
+Considering models (5) and (9), the online multimodal image fusion problem can be
+formulated as the problem of computing the posterior distribution of the high resolution
+image given all previous measurements available, i.e.,
+p
+`
+sk
+ˇˇtrym
+1:kumPΩ
+˘
+“ N
+`
+sk|k, P k|k
+˘
+.
+(12)
+Due to the choice of a linear Gaussian model, this distribution is also Gaussian. Moreover,
+its mean vector sk|k and covariance matrix P k|k can be computed recursively using the
+standard Kalman filter with a prediction and update steps [50].
+1That is, τ P Z` is the smallest integer such that a high resolution image was observed at time instant
+k ´ τ.
+8
+
+More precisely, the prediction step of the Kalman filter computes the first and second
+order moments of p
+`
+sk
+ˇˇtrym
+1:k´1umPΩ
+˘
+as:
+sk|k´1 “ F k´1sk´1|k´1
+(13)
+P k|k´1 “ F k´1P k´1|k´1F J
+k´1 ` Qk´1
+(14)
+The update step computes then computes of (12).
+Note that the update step can be
+simplified and implemented separately for each data modality by using the Markov property
+of the model and the independence between noise vectors of different modelities:
+p
+`
+sk
+ˇˇtrym
+1:kumPΩ
+˘
+9p
+`
+trym
+k umPΩ
+ˇˇsk
+˘
+p
+`
+sk
+ˇˇtrym
+1:k´1umPΩ
+˘
+“ p
+`
+sk
+ˇˇtryu
+1:k´1uuPΩ
+˘ ź
+mPΩ
+p
+`
+rym
+k
+ˇˇsk
+˘
+.
+(15)
+By computing the first product in the right hand side as:
+p
+`
+sk
+ˇˇtryu
+1:k´1uuPΩ
+˘
+p
+`
+rym
+k
+ˇˇsk
+˘
+9p
+`
+sk
+ˇˇtryu
+1:k´1uuPΩ, rym
+k
+˘
+,
+(16)
+which is an update step of the Kalman filter with image modality m to yield a new posterior
+in the r.h.s. of (16). This can be computed as:
+vm
+k “ rym
+k ´ Ă
+H
+m
+k sk|k´1
+(17)
+T m
+k “ Ă
+H
+m
+k P k|k´1
+`Ă
+H
+m
+k
+˘J ` rR
+m
+k
+(18)
+Km
+k “ P k|k´1
+`Ă
+H
+m
+k
+˘J`
+T m
+k
+˘´1
+(19)
+sk|k “ sk|k´1 ` Km
+k vm
+k
+(20)
+P k|k “ P k|k´1 ´ Km
+k T m
+k
+`
+Km
+k
+˘J
+(21)
+for m P Ω.
+By proceeding with the computation of the product in the r.h.s.
+of (15)
+recursively, the Kalman update can then be performed separately for each of the modalities
+observed at time instant k. Note that after the first modality is processed, the update
+equations above are used again for the subsequent modalities by setting sk`1|k and P k`1|k
+as equal to the posterior estimates from the previously processed modality.
+3.1. The Linear Smoother
+Given a window of K image samples, the Bayesian smoothing problem consists of com-
+puting the posterior distribution of the high resolution image given all available measure-
+ments available, i.e.,
+p
+`
+sk
+ˇˇtrym
+1:KumPΩ
+˘
+“ N
+`
+sk|K, P k|K
+˘
+,
+(22)
+9
+
+which is also a Gaussian. Just like in the filtering problem, the linear and Gaussian model
+allows this solution to be computed efficiently using the Rauch-Tung-Striebel (RTS) smooth-
+ing equations [50], which consist of a forward pass of the Kalman filter (as described before),
+followed by a backwards recursion that updates the previously computed mean and covari-
+ances matrices of the state with information from future time instants.
+We note that the smoothing can also be performed efficiently for the case when multiple
+image modalities are available. Let us consider the Bayesian smoothing equations as defined
+in [51, 50], which is performed in two steps. Starting from the Kalman state estimate at
+time K, given by p
+`
+sK
+ˇˇtrym
+1:KumPΩ
+˘
+, the smoothing distribution is computed recursively for
+k “ k ´ 1, . . . , 1, according to the following relation:
+p
+`
+sk
+ˇˇtrym
+1:KumPΩ
+˘
+“ p
+`
+sk
+ˇˇtrym
+1:kumPΩ
+˘
+ˆ
+ż ppsk`1|skqp
+`
+sk`1
+ˇˇtrym
+1:KumPΩ
+˘
+p
+`
+sk`1
+ˇˇtrym
+1:kumPΩ
+˘
+dsk`1 ,
+(23)
+where p
+`
+sk
+ˇˇtrym
+1:kumPΩ
+˘
+“ Npsk|k, P k|kq is the Kalman estimate of the state PDF at time k,
+ppsk`1|skqq is the state transition PDF, computed according to (9), p
+`
+sk`1
+ˇˇtrym
+1:KumPΩ
+˘
+“
+Npsk`1|K, P k`1|Kq is the smoothing distribution obtained at the previous iteration, and
+p
+`
+sk`1
+ˇˇtrym
+1:kumPΩ
+˘
+is the predictive state distribution, which is computed exactly as in the
+prediction step of the Kalman filter.
+In the linear and Gaussian case this translates into the following closed form solution [50],
+with
+sk`1|k “ F ksk|k
+(24)
+P k`1|k “ F kP k|kF J
+k ` Qk
+(25)
+being used to compute the predictive state distribution, and
+Gk “ P k|kF J
+k P ´1
+k`1|k
+(26)
+sk|K “ sk|k ` Gkpsk`1|K ´ sk`1|kq
+(27)
+P k|K “ P k ` GkpP k`1|K ´ P k`1|kqGJ
+k
+(28)
+to update the covariances. It should be noted that the mean and covariance sk|k and P k|k
+used in the Smoothing equations are the final result obtained from the Kalman update after
+processing all image modalities that were available at instant k.
+Thus, while in the Kalman filtering the update equations must be computed sequentially
+at each time step w.r.t. the different image modalities, smoothing only needs only the final
+state estimates at each instant, no matter how many modalities are present.
+3.2. Constraining the estimates
+Although the Kalman filter provides closed-form solutions to the estimation of the high-
+resolution image sequence, it relies on a Gaussian assumption on the states and observations
+10
+
+which does not correspond to the physics of the problem. In fact, represented in reflectance
+values, each pixel and band of a high-resolution images sk is actually constrained to an
+interval sk,i,j P r0, smaxs, where smax is the maximum reflectance values of the scene. Since
+this information can potentially improve the accuracy of the estimated states, we propose
+to incorporate this information by considering the linearly constrained Kalman filter [52],
+in which the final constrained state s`
+k|k is obtained as the solution to a constrained opti-
+mization problem:
+s`
+k|k “ arg min
+s
+`
+s ´ sk|k
+˘JP ´1
+k|k
+`
+s ´ sk|k
+˘
+subject to s P r0, smaxsNHLH
+.
+(29)
+Problem (29) consists in a constrained quadratic program, which can be costly to solve due
+to the high dimensionality of the variables. Thus, we propose a simple solution consisting
+of truncating the result of the traditional Kalman update:
+s`
+k|k “ max
+`
+min
+`
+sk|k, smax
+˘
+, 0
+˘
+,
+(30)
+where functions maxp¨, ¨q and minp¨, ¨q compute the elementwise maximum and minimum
+value between a vector and a scalar. Note that this truncation provides the exact solution
+when P k|k is diagonal. The same truncation strategy was also applied to the results of the
+linear smoother sk|K. We generally observed that this gave good results in practice. smax
+can be estimated as the maximum value of the observed images in a time window, or from
+the historical data.
+4. A distributed implementation
+A problem with the Kalman filter is the need to compute and store the state covariance
+matrix, P k|k. This incurs in storage and operations asymptotic complexity in the order of
+OpN2
+HL2
+Hq and OpN3
+HL3
+Hq, respectively. This can make the method intractable for images
+with a large number of pixels.
+Thus, to reduce the complexity of the filter and of the
+smoother, we consider splitting the pixels in the estimated state sk into multiple groups
+which are assumed to be statistically independent [53, 54, 55]. To this end, we divide the
+state space into G groups as:
+sk “ vec
+`
+rsp1q
+k , . . . , spGq
+k
+s
+˘
+,
+(31)
+where the variables within each block spgq
+k
+are correlated, but different blocks spg1q
+k
+and spg2q
+k
+are assumed to be independent for g1 ‰ g2. This leads to the following approximation for
+the predictive and posterior covariance matrices P k|k´1 and P k|k as block diagonal matrices:
+P k|k´1 “ blkdiag
+!
+P p1q
+k|k´1, . . . , P pGq
+k|k´1
+)
+(32)
+P k|k “ blkdiag
+!
+P p1q
+k|k, . . . , P pGq
+k|k
+)
+(33)
+We consider different splitting possibilities, with different trade-offs between approxi-
+mation accuracy with respect to the full-state-covariance Kalman filter and complexity:
+11
+
+iq A fully diagonal model (with G “ NHLH blocks).
+iiq A block diagonal model where each block consists of all bands of one single high-
+resolution pixel (with G “ NH blocks).
+iiiq A block diagonal model, with blocks corresponding to the high-resolution pixels which
+reside inside a single MODIS pixel (with G “ NHLH{NMODIS blocks).
+Following [54], the Kalman equations for the prediction step (13)–(14) can be written
+for each block as:
+spgq
+k`1|k “
+“
+F k
+‰
+pgq,:sk
+(34)
+P pgq
+k`1|k “
+“
+F k
+‰
+pgq,:P k
+`“
+F k
+‰
+pgq,:
+˘J ` Qpgq
+k
+(35)
+where
+“
+F k
+‰
+pgq,: means the matrix formed by taking from F k the rows which correspond to
+the indices in the group of states g, and all columns. Matrices Qpgq
+k
+are defined as:
+Qk “ blkdiag
+␣
+Qp1q
+k , . . . , QpGq
+k
+(
+.
+(36)
+Similarly, the Kalman update equations (17)–(21) are performed separately for each block
+of variables, and are given by:
+spgq
+k
+“ spgq
+k|k´1 ` Kpgq
+k vm
+k
+(37)
+P pgq
+k
+“ P pgq
+k|k´1 ´ Kpgq
+k T m
+k
+`
+Kpgq
+k
+˘J
+(38)
+with:
+Kpgq
+k
+“ Σpgq
+xy,k|k´1
+`
+T m
+k
+˘´1
+(39)
+vm
+k “ rym
+k ´ Ă
+H
+m
+k sk|k´1
+(40)
+T m
+k “ Ă
+H
+m
+k P k|k´1
+`Ă
+H
+m
+k
+˘J ` rR
+m
+k
+(41)
+Σpgq
+xy,k|k´1 “
+“
+P k|k´1
+`Ă
+H
+m
+k
+˘J‰
+pgq,:
+“
+“
+P k|k´1
+‰
+pgq,:
+`Ă
+H
+m
+k
+˘J
+(42)
+where
+“
+P k|k´1
+‰
+pgq,: means the matrix formed by taking from P k|k´1 the rows which corre-
+spond to the indices in the group of states g, and all columns. Note that the block diagonal
+structure of P k|k´1 and P k|k can be explored to perform the above operations efficiently,
+since these matrices are very sparse.
+Following the same approach, the linear smoother can also be approximated in blockwise
+fashion as in [55], for the predictive equations (24)–(25):
+spgq
+k`1|k “
+“
+F k
+‰
+pgq,:sk
+(43)
+P pgq
+k`1|k “
+“
+F k
+‰
+pgq,:P k
+`“
+F k
+‰
+pgq,:
+˘J ` Qpgq
+k
+(44)
+12
+
+and for the smoothing equations (26)–(28):
+Gpgq
+k
+“
+“
+P kF J
+k
+‰
+pgq,pgq
+`
+P pgq
+k`1|k
+˘´1
+“ rP kspgq,pgq
+`“
+F k
+‰
+pgq,pgq
+˘J`
+P pgq
+k`1|k
+˘´1
+(45)
+spgq
+k|K “ spgq
+k
+` Gpgq
+k
+`
+spgq
+k`1|K ´ spgq
+k`1|k
+˘
+(46)
+P pgq
+k|K “ P pgq
+k
+` Gpgq
+k pP pgq
+k`1|K ´ P pgq
+k`1|kqpGpgq
+k qJ
+(47)
+where
+Gk “ blkdiag
+␣
+Gp1q
+k , . . . , GpGq
+k
+(
+.
+(48)
+One last issue is that the innovation covariance matrix T m
+k can also be large for big
+images (e.g., Landsat measurements), as it has pLm
+śLm
+ℓ“1 Nm,ℓq2 elements. Fortunately, the
+model implicitly imposes a simple structure for this matrix. To show this, let us consider a
+permutation of the pixels Πm, such that Πmrym
+k reorders rym
+k by making different bands of
+each LR pixel contiguous:
+Πmrym
+k “
+»
+—–
+»
+—–
+rym
+k,1,1
+...
+rym
+k,Lm,1
+fi
+ffifl
+J
+, . . . ,
+»
+—–
+rym
+k,1,Nm
+...
+rym
+k,Lm,Nm
+fi
+ffifl
+Jfi
+ffifl
+J
+,
+(49)
+where rym
+k,ℓ,n is the n-th pixel of the ℓ-th band of ryk.
+If we assume that Hm
+ℓ
+is a local filter, i.e., each pixel in the low-resolution image is
+generated according to a fixed linear combination of a distinct subset of HR pixels, this
+allows us to express the row-permuted version of Ă
+H
+m
+k equivalently as:
+ΠmĂ
+H
+m
+k “ blkdiag
+␣
+H, H, . . . , H
+looooooomooooooon
+Nm times
+(
+,
+(50)
+where matrix H P RLmˆd2LH is given by:
+H “ hm b Cm ,
+(51)
+where Cm “
+“
+pcm
+1 qJ, . . . , pcm
+LmqJ‰J is the spectral response function for all bands, hm P
+R1ˆd is the local spatial response filter, which defined how the HI pixels inside each LR
+pixels are combined, and d is the number of HR pixel in each LR pixel.
+Using this permutation, the innovation covariance matrix can be written as:
+ΠmT m
+k ΠJ
+m “ ΠmĂ
+H
+m
+k P k|k´1
+`Ă
+H
+m
+k
+˘JΠJ
+m ` Πm rR
+m
+k ΠJ
+m
+“ blkdiagtH, . . . , HuP k|k´1 blkdiagtHJ, . . . , HJu
+` Πm rR
+m
+k ΠJ
+m
+“ blkdiagtHP p1q
+k|k´1HJ, . . . , HP pGq
+k|k´1HJu
+` Πm rR
+m
+k ΠJ
+m .
+(52)
+13
+
+Algorithm 1: Weakly supervised online image fusion
+Input
+: Measured multimodal images ym
+k , for all time instants k “ 1, . . . , K and
+modalities m, historical datasets of high-resolution images Dk, parameters smax.
+Output: Estimated image sequence sk|K
+1 Initialize P 0|0 and s0|0;
+2 // Filter ;
+3 for k “ 1, 2, . . . , K do
+4
+Compute innovation covariance matrix Qk using sk´1 and Dk according to Section 2.4 ;
+5
+Compute sk|k´1 and P k|k´1 using equations (31), (32), (34) and (35) ; // Prediction
+6
+Compute sk|k and P k|k using equation (33) and equations (37)–(42) ;
+// Update
+7
+Constrain sk|k using (30) ;
+8 end
+9 // Smoother ;
+10 for k “ K, K ´ 1, . . . , 1 do
+11
+Compute sk`1|k and P k`1|k using equations (31), (32), (43) and (44) ; // Prediction
+12
+Compute sk|K and P k|K using equations (45)–(47) and equation (48) ;
+// Backwards
+update
+13 end
+14 return Estimated images sk|K
+Thus, as long as the noise is independent among different pixels (i.e., rR
+m
+k is block diago-
+nal), it is possible to express the innovation covariance matrix in block diagonal form by
+adequately permuting the LR image pixels. This shows that each pixel from the lowest
+resolution image modality can be processed independently when Qk and P 0|0 also have a
+block diagonal structure. The proposed image fusion method is summarized in Algorithm 1.
+5. Experiments
+In this section, we use the proposed methodology to fuse Landsat and MODIS image
+over time. The Kalman filter and smoother are built under the three different assumptions
+for the state covariance matrices regarding the distributed implementation discussed in
+Section 4: iq diagonal state covariance (denoted by KF-D and SM-D); iiq block-diagonal
+state covariance with one block per Landsat multispectral pixel (denoted by KF-B and SM-
+B); and iiiq block-diagonal with blocks for all Landsat multispectral pixels corresponding to
+the same coarse pixel in a MODIS image being correlated (denoted by KF-F and SM-F). A
+filter in which Landsat multispectral pixels corresponding to more than one coarse pixel in
+a MODIS image being all correlated could not be implemented due to computational and
+memory limitations.
+Although in our experiments we consider only two modalities the proposed methodology
+admits multiple different modalities provided that enough computational power is available.
+As benchmark, we compare the performance of Kalman filter and smoother under all three
+assumptions to that of the Enhanced Spatial and Temporal Adaptive ReFlectancefusion
+Model (ESTARFM) algorithm [25], and the Prediction Smooth Reflectance Fusion Model
+14
+
+(PSRFM) algorithm [56, 57]. The ESTARFM algorithm requires two high-resolution (e.g.,
+Landsat) images at the beginning of the image sequence, and can generate high-resolution
+reconstructions at later time instants based on MODIS measurements. Thus, it is a good
+candidate for comparison with the Kalman filtering based strategies, which also do not
+require future data. The PSRFM method, on the other hand, uses two high-resolution (e.g.,
+Landsat) images (one at the beginning and one at the end of the sequence), and provides
+high-resolution reconstruction for the intermediate MODIS images. Thus, it consists in an
+adequate comparison to the smoother algorithms, which also require future high-resolution
+images. In the following, we describe the data and simulation setup, followed by the results
+and the discussions.
+5.1. Study region
+For the experiments, we consider two sites. The first is the Oroville dam (Figure 2, left
+panel), located on the Feather River, in the Sierra Nevada Foothills (38° 35.3’ North and
+122° 27.8’ W) is the tallest dam in USA and is major water storage facility in California
+State Water Project.
+The reservoir has a maximum storage capacity of 1.54 ˆ 1011 ft3
+or 4.36 ˆ 109 m3, which fills during heavy rains or large spring snow melts and water is
+carefully released to prevent flooding in downstream areas, mainly to prevent large flooding
+in Butte County and area along the Feather River. The reservoir water storage change in
+between 07/03 and 09/21 of 2018 is as shown as the hydrograph curve in Figure 8. Another
+unique characteristic is that it has three power plants at this reservoir. The water released
+downstream is used to maintain the Feather and Sacramento Rivers and the San Francisco-
+San Joaquin delta. Lake Oroville is at an elevation of 935 feet (285 meters) above sea level.
+We focus at a particular area of the Oroville dam delimited by the red box in Figure 2.
+The second site is the Elephant Butte reservoir (Figure 2, right panel), located in the
+southern part of the Rio Grande river, in New Maxico, USA (33° 19.4’ N and 107° 26.2’ W).
+It is the largest reservoir in New Mexico, providing power and irrigation to southern New
+Mexico and Texas. Elephant Butte reservoir is at an elevation of 4,414 ft (1,345 meters),
+and has a surface area of 36,500 acres (14,800 ha).
+Table 1: Spectral angle mapper between the estimated high-resolution image and the Landsat measurement
+for the Oroville Dam example (note that the Landsat images at dates 07/19, 08/20, and 09/05 were not
+supplied to the algorithms and only used for evaluation purposes). However, the Landsat image at 09/21
+was available to all algorithms. Note that the spectral angle is not reported for PSRFM at 09/21. This is so
+since PSRFM uses the last pair (MODIS-Landsat) of images and directly sets its estimations at this dates
+to the ground-truth.
+Method
+KF-F
+SM-F
+KF-B
+SM-B
+KF-D
+SM-D
+ESTARFM
+PSRFM
+Image (07/19)
+7.1240
+10.8537
+4.2356
+6.1515
+4.9304
+5.9064
+6.0810
+6.8837
+Image (08/20)
+27.6343
+26.2786
+26.1229
+25.1520
+27.1928
+26.1758
+29.0892
+27.7802
+Image (09/05)
+8.5741
+6.0366
+6.6246
+3.6838
+7.4482
+4.4135
+11.4553
+6.0354
+Image (09/21)
+8.0588
+3.6385
+6.4042
+0.5471
+6.9754
+0.6960
+11.9584
+–
+Average
+12.8478
+11.7019
+10.8468
+8.8836
+11.6367
+9.2979
+14.6460
+10.1748
+15
+
+National Geographic, Esri, Garmin, HERE, UNEP-WCMC, USGS, NASA,
+ESA, METI, NRCAN, GEBCO, NOAA, increment P Corp.
+0
+0.55
+1.1
+0.28
+mi
+0
+0.9
+1.8
+0.45
+km
+1:44,418
+National Geographic, Esri, Garmin, HERE, UNEP-WCMC, USGS, NASA,
+ESA, METI, NRCAN, GEBCO, NOAA, increment P Corp.
+0
+5
+10
+2.5
+mi
+0
+8.5
+17
+4.25
+km
+1:368,824
+Figure 2: (Left) Oroville dam site. (Right) Elephant Butte site. The red boxes delimit the specific study
+areas used in our experiments.
+Table 2: Percentage of misclassified pixels for the Oroville Dam example (the Landsat image at 09/21 was
+available to all algorithms). Note that the misclassification percentage is not reported for PSRFM at 09/21.
+This is so since PSRFM uses the last pair (MODIS-Landsat) of images and directly sets its estimations at
+this dates to the ground-truth.
+Method
+KF-F
+SM-F
+KF-B
+SM-B
+KF-D
+SM-D
+ESTARFM
+PSRFM
+Image (07/19)
+9.5412
+7.6360
+6.4472
+8.2914
+6.1119
+8.0171
+5.4870
+5.2431
+Image (08/20)
+14.9215
+10.4405
+7.8647
+4.1000
+7.2245
+3.7799
+18.2899
+17.9851
+Image (09/05)
+13.4888
+8.2152
+9.6632
+4.7859
+9.4345
+4.5877
+22.7404
+20.8962
+Image (09/21)
+11.7360
+3.8409
+9.3583
+0.2439
+9.2974
+0.2591
+26.3374
+–
+Average
+12.4219
+7.5332
+8.3333
+4.3553
+8.0171
+4.1610
+18.2137
+11.0311
+5.2. Remote Sensed data
+For our simulations with the Oroville Dam site, we collected MODIS and Landsat data
+acquired from the region marked with a red square on Figure 2, and on a interval ranging
+from 2018{07{03 to 2018{09{21. This interval was selected since the hydrograph analysis
+indicates high variation in the water level of the reservoir, see, the hydrograph curve in
+Figure 8. Such variation in the water levels result in large changes in the acquired images,
+exposing flooded areas. In this experiment we will focus on the red and near-infrared (NIR)
+bands since they are often used to distinguish water from other landcover elements in the
+image [58]. We also collected 5 Landsat data from 2017{08{01 to 2017{12{07 to serve as a
+past historical dataset Dk.
+The study region marked in the left panel of Figure 2 corresponds to Landsat and
+MODIS images with 81 ˆ 81 and 9 ˆ 9 pixels, respectively2. After filtering for heavy cloud
+cover during the designated time periods, a set of 6 Landsat and 16 MODIS images were
+obtained. We used the first MODIS and Landsat images for initialization of all methods
+leading to 5 and 15 images used in the remaining fusion process.
+2The Landsat images were also upsampled to a spatial resolution of 27.77 meters to make its resolution
+exactly 9 times that of MODIS.
+16
+
+Lake Oroville
+Lake Oroville
+State
+Recreation AreaNOSA
+SPRINGDRAW
+Elephant
+Butte
+Reservoi
+R
+52
+M
+Conseqgences
+Truhort
+MuniAirport
+V
+A
+1991m
+GARCIA
+PEAKS
+TruthOr
+Z
+Consequences.
+R
+o-Grande
+JOHNSON
+IEMESA
+MCCLENFrom the set of 5 Landsat images of the Oroville Dam site that were available for
+testing, three of them were set aside and not processed by any of the the algorithms.
+These images were acquired at dates 07/19, 08/20 and 09/05, when MODIS observations
+were also available, and will be used in the form of a reference for the evaluation of the
+algorithms’ capability of estimating the high resolution images at these dates solely from
+the low resolution MODIS measurements.
+For the simulations with the Elephant Butte site, shown in the right panel of Figure 2, we
+aim to evaluate the performance of the algorithms when processing a larger geographical
+area, with an area of approximately 9km ˆ 9km.
+The setup is similar to the Oroville
+Dam example. We focus on the red and near-infrared bands of the Landsat and MODIS
+instruments, and collect 47 Landsat images from 2014/01/16 to 2017/11/24 to serve as the
+past historical dataset Dk.
+The study region corresponds to Landsat and MODIS images with 324ˆ324 and 36ˆ36
+pixels, respectively. After removing images with significant cloud cover, we obtained a set
+of 5 Landsat and 7 MODIS images to process. We used the first MODIS and Landsat
+image pair to initialize the algorithms, leading to 4 Landsat and 6 MODIS images to be
+used in the remaining fusion process. From the set of 4 Landsat images that were available
+for testing, 2 of them were set aside as ground truth to evaluate the algorithms. Theses
+images are acquired at dates 06/07 and 06/23. However, the MODIS measurements at those
+dates contained significant cloud cover, and had to be discarded. Therefore, we evaluate
+the performance of the algorithms through the estimation results obtained dates 06/14 and
+06/27 (in which the MODIS observations were available).
+5.3. Algorithm setup
+We initialized the proposed Kalman filter and smoother using a high resolution Landsat
+observation as the state, i.e., s0|0 “ ryL
+0, and set P 0|0 “ 10´10P 0. The structure of P 0
+varies with different assumptions: iq P 0 “ I if the state covariance is diagonal; iiq P 0 “
+blkdiagtP 0,1, P 0,2, ¨ ¨ ¨ , P 0,NHu, where P 0,i “ 1
+21 ` 1
+2I, with 1 being an all ones matrix,
+if the state covariance matrix has a block-diagonal structure with one block per Landsat
+multispectral pixel; iiiq P 0 “ blkdiagtP 0,1, P 0,2, ¨ ¨ ¨ , P 0, ˜
+NmˆLmu, where P 0,i “ 1
+21 ` 1
+2I if
+the state covariance matrix has a block-diagonal structure with each block containing all
+Landsat multispectral pixels corresponding to the same coarse pixel in a MODIS image.
+Figure 7 shows an example of the final P k|k, k “ 13, obtained with the KF under all the
+assumptions discussed in Section 4. The noise covariance matrices were set as RL
+ℓ “ 10´10I
+and RM
+ℓ “ 10´4I, for all ℓ. The blurring and downsampling matrices were set as HL
+ℓ “ I
+for Landsat, while for MODIS HM
+ℓ consisted of a convolution by an uniform 9 ˆ 9 filter,
+defined by h “
+1
+8119ˆ9 (where 19ˆ9 is a 9 ˆ 9 matrix of ones), followed by decimation by
+a factor of 9, which represents the degradation occurring at the sensor (see, e.g., [44]). We
+also set F k “ I for all k. The vectors cm
+ℓ
+contained a positive gain in the ℓ-th position
+which compensated for scaling differences between Landsat and MODIS sensors, and zeros
+elsewhere.
+The matrices DM
+k were constructed based on the quality codes (i.e., the QA bits) released
+17
+
+by MODIS for each image pixel [59]. QA bits provides information regarding pixel quality
+and cloud cover for all pixels and all bands. In our experiments we dropped any pixel not
+classified as corrected product produced at ideal quality in the QA bits [59] by adding zeros
+at corresponding positions in DM
+k . Matrices Qk were computed following our data-driven
+strategy described in Section 2.4 where ε2 “ 10´5 and n “ 1.
+The ESTARFM algorithm was parametrized as follows [25], w “ 14 as half of the window
+size, the number of classes was set to 4, and the pixels range was set to r0, 0.5s. The PSRFM
+algorithm was parametrized as follows, CLUSTER METHOD “ KMEAN, and CLUSTER DATA “
+fine`coarse. We highlight that all methods have access only to the first (07/03) and last
+(09/21) Landsat images, which allows the algorithms to produce estimates for the MODIS
+images observed from the second (07/09, k “ 2) up to the last date (09/21, k “ 16).
+However, PSRFM uses the the last pair (MODIS-Landsat) during its inference process. For
+this reason, error metrics computed for PSRFM on (09/21) should be disregarded as the
+estimate is directly the ground-truth (i.e., the Landsat image) and, thus, are not reported
+in the experimental results.
+All algorithms are evaluated using three metrics, which are computed taking as reference
+the Landsat images, three of which are not observed by the algorithms. The first metric
+is the Spectral Angle Mapper (SAM), which attempts to measure the estimation accuracy
+directly:
+SAMpS, pSq “
+1
+NH
+NH
+ÿ
+r“1
+arccos
+´
+sJ
+r psr
+}sr}}psr}
+¯
+,
+(53)
+where S and pS denote the true and the estimated images, respectively. sr and psr denote
+the r-th pixels of different bands in S and pS, respectively. The two remaining metrics are
+related to downstream tasks of water classification and water level monitoring, which are
+performed on the reconstructed image sequence.
+We evaluate the direct benefit of the different fusion strategies in classifying water pixels
+from the estimated images. To classify water pixels we resorted to a KNN classifier whose
+centroids of water and non-water 2-band pixels were computed using K-Means algorithm.
+Finally, we evaluate the performance of the algorithms for hydrograph estimation by plot-
+ting the proportion of pixels in the image classified as water over time against the true
+hydrograph for the period, for all algorithms.
+5.4. Results for the Oroville Dam site
+As discussed, we fused the red and NIR reflectance bands of MODIS and Landsat for
+the selected study region. In Figure 3, we show the fused red (Figure 3a) and NIR (Fig-
+ure 3b) reflectances as well as the acquired red and NIR reflectance values from MODIS and
+Landsat. Acquisition dates are displayed in the top labels at each column with a character,
+M for MODIS and L for Landsat, indicating the image used in the fusion algorithms. We
+recall that only the first and last Landsat images were used in the fusion process, keep-
+ing the remaining three images as ground-truth for evaluation purposes. Analyzing the
+18
+
+results we can see that the images estimated by the proposed Kalman filter and smoother
+methods, under different assumptions, produce better visual similarity with the Landsat
+(ground-truth) images for both bands. For instance, the increase in the island and the
+expansion of other land parts are clearly visible for the proposed methods. In contrast,
+analyzing ESTARFM results we note that land parts remain mainly constant through time
+until a new Landsat image is observed. Although lighter areas on the water portions can
+be noticed, specially for k ą 8, its distribution does not resemble the ground-truth. This
+is expected since ESTARFM is not designed to acknowledge prior information or historical
+data. PSRFM results show an improvement compared with ESTARFM results, since it
+uses both the first and the last Landsat images. However, the PSRFM results does not
+resemble the ground-truth very closely, and significant blurring occurs around the edge of
+the island when k ă 8. The blurring results in PSRFM are caused by the fact that the
+reconstructions provided by this algorithm are based on a form of interpolation which does
+not consider any information about the transition of the pixel reflectance values, whereas
+in our proposed methods we use the historical data to calibrate the time-varying dynamical
+model by means of matrix Qk, which can increase the accuracy of the estimations.
+Note that the images estimated by KF-F and SM-F (which used a full state covariance
+matrix) contained more artifacts when compared to the ones obtained by KF-B, SM-B,
+KF-D and SM-D (which constrained the state covariance matrix to be diagonal or block
+diagonal). This occurs due to the high-dimensionality of the state vector (i.e., equivalent
+to a vectorized Landsat image) when compared to the MODIS measurements, as this leads
+to the amount of measurements not being sufficient to provide an accurate estimate of the
+full state vector and its covariance matrix, as shown in [60]. Thus, the extra degrees of
+freedom of KF-F and SM-F end up impacting their performance negatively. By setting the
+covariance matrix of the Kalman filter and smoother to be block diagonal or fully diagonal,
+the amount of parameters to be estimated is greatly reduced in KF-B, SM-B, KF-D and
+SM-D, leading to better results.
+The results discussed above are corroborated by the absolute error maps displayed in
+Figure 4, and SAM results shown in Table 1 for dates in which ground-truth is available.
+Analyzing Figure 4 we highlight that SM-B and SM-D clearly present the smallest errors
+(i.e., overall darker pixels) for both bands and all dates. KF-B also presents low absolute
+error except for contour regions. PSRFM is the third overall darker image, followed by
+KF-B, KF-D, SM-F, KF-F and ESTARFM with exception of the results on 07/19 (first col-
+umn), where ESTARFM is close to the ground-truth. Similar conclusions can be achieved
+by analyzing Table 1. The difference between the images estimated by the Kalman filter
+and smoother under the different approximations for the state covariance matrices (which
+are discussed in Section 4 and illustrated for this example in Figure 7) is shown in Figure 5.
+It can be seen that the approximations had a more pronounced effect on the Kalman filter
+compared to the smoother. Moreover, the differences between the filter with a diagonal
+(assumption i) and block diagonal state covariance with one block per Landsat pixel (as-
+sumption ii) was relatively small. Taking in to consideration the quantitative metrics in
+Table 1, this indicates that using a diagonal or block diagonal assumption on the state
+19
+
+covariance matrix with small blocks has a positive effect on the estimation performance,
+which likely occurs since it drastically reduces the amount of unknowns in the model that
+have to be estimated by the methods.
+The left panel in Figure 6 presents the water maps for the ground-truth (first row) and
+all studied algorithms obtained using K-means clustering, while the right panel in Figure 6
+shows the misclassification maps (i.e., the absolute error between the water maps obtained
+by each algorithm and the ground-truth). When comparing the resulting classification maps
+and the misclassification error with the ground-truth, the proposed methods present classi-
+fication maps that are semantically better than the competing methods. This conclusion is
+also reached by considering the quantitative misclassification results presented in Table 2,
+in which the Kalman filter- and smoother-based methods led to smaller misclassification
+rates for all images except the ones on 07/19 and 09/21. A closer analysis reveals that the
+SM-D and SM-B methods hold the first and second best performance on average, followed
+by SM-F, KF-D, KF-B, PSRFM, KF-F and ESTARFM. Note that the PSRFM method
+requires access to the ground-truth (Landsat image) on 09/21 in order to produce an esti-
+mation for the MODIS image observed in this same date (i.e., measurement k “ 16), which
+is why the corresponding misclassification percentage is not reported. We also remark that
+KF-D and KF-B also obtained competitive misclassification performance (i.e., better than
+PSRFM), despite using no knowledge of the Landsat image at 09/21. Moreover, comparing
+the results in Table 1 and 2, it can be seen that the higher SAM results observed for all
+methods at date 08/20 does not translates into a worse classification performance. This
+indicates that the SAM results at this date were influenced by the acquisition conditions of
+the Landsat image which was used for ground truth, making the classification performance
+more straightforward to interpret.
+Finally, we plotted the percentage of pixels classified as water over the time index k in
+Figure 8, as well as a hydrograph which serves as an indicative of the dynamical evolution
+of the true level of the reservoir over time. It can be seen that ESTARFM was not able to
+properly identify the dynamical evolution of the reservoir level, leading to an estimation that
+was almost constant for all k ă 17 and very different from the hydrograph curve. PSRFM
+led to results that, although showing relatively high day-to-day variations, were closer to
+the hydrograph curve. The Kalman filter and smoother-based algorithms, particularly those
+with the diagonal and block diagonal state covariance assumption (KF-D, KF-B, SM-B and
+SM-D) led to curves that were very close to the hydrograph.
+Thus, the Kalman filter
+methods captured the general trends of the hydrograph curves, even without having access
+to information from the Landsat image at the end of the sequence (like the smoothers and
+PSRFM). We note, however, that the connection between the hydrograph and the water
+surface area is indirect; thus, small differences between the algorithms have to be interpreted
+with proper care.
+5.5. Contribution of the temporal dynamics calibration strategy
+This subsection aims to show the impact of the proposed calibration strategy, which
+learns the temporal dynamical model parameters Qk using historical data, on the perfor-
+mance of the proposed KF and SM algorithms. To this end, we compared the proposed
+20
+
+KF-D and SM-D (which estimate Qk and use a diagonal assumption on the state covari-
+ance matrix), to a Kalman filter and smoother with a fixed Qk “ 10´2I, which we denote
+by KF-I and SM-I, respectively. In Figure 9, we show the fused red (Figure 9a) and NIR
+(Figure 9b) reflectance images, as well as the acquired red and NIR reflectance values from
+MODIS and Landsat. Acquisition dates are displayed in the top labels at each column with
+a character, M for MODIS and L for Landsat indicating the image used in the fusion algo-
+rithms. We recall that only the first and last Landsat images were used in the fusion process,
+keeping the remaining three images as ground-truth for evaluation purposes. Analyzing the
+results, we can see that the images estimated by the proposed KF-D and SM-D methods
+produce significantly better visual similarity with the Landsat (ground-truth) images for
+both bands. For instance, the increase in the island and the expansion of other land parts
+at date 08/20 are clearly visible for the proposed methods. On the other hand, analyzing
+the results of the KF-I and SM-I methods, where the temporal dynamics matrix Qk was
+kept constant and independent of past data, we observe that the results appear very blurry,
+with a resolution that is comparable to that of the MODIS images. This shows that the
+proposed weakly supervised calibration strategy is key in order for the KF- and SM-based
+strategies to obtain high quality reconstructions.
+5.6. Results for larger scale Elephant Butte site
+In this subsection, we compare the proposed strategies to ESTARFM and PSRFM in
+the Elephant Butte example, which comprises a larger geographical area. For simplicity
+and to reduce the use of space, we compare only proposed Kalman filter and smoother
+methods with the block diagonal assumption on the state covariance matrices (i.e., KF-B
+and SM-B).
+The fusion results for both bands and all algorithms are shown in Figure 10, while
+Figure 11 shows the corresponding water mapping results. To measure the performances
+of different methods in this large area, the Landsat images at dates 06/07 and 06/23 were
+chosen as a ground truth to evaluate the quality of the reconstructed images at dates 06/14
+and 06/27 (we remark that the MODIS images at dates 06/07 and 06/23 were not available
+due to the presence of cloud cover). It can be seen that the proposed KF-B, SM-B and the
+PSRFM methods provide estimates that are close to the ground truth images, whereas the
+ESTARFM method shows an inferior performance. This can be seen more clearly for the
+image at date 06/14 (k “ 5), in which the smoother method better captured the increase in
+the area of the reservoir. To evaluate the performances of different methods more clearly,
+Figure 12 shows the absolute error of water maps of images compared with the ground
+truth, and Figures 13 and 14 show a zoomed-in area of the image of the fused image and
+water mapping result, respectively. It can be seen from Figure 12 that the misclassification
+errors are concentrated at the borders of the reservoir, which is the area that undergoes the
+largest amounts of changes over time, and consequently the hardest to classify correctly.
+The SM-B algorithm shows the best results, followed by KF-B, PSRFM and ESTARFM.
+Nevertheless, PSRFM provides results that contain less artifacts compared to KF-B, despite
+the lower classification accuracy. The superior visual quality of the results of SM-B and
+21
+
+PSRFM is explained by their use of Landsat images both at the beginning and at the end of
+the image sequence, whereas KF-B and ESTARFM do not have access to the last Landsat
+image.
+Table 3 presents the SAM results, and Table 4 shows the corresponding percentage of
+misclassified pixels for the different methods. It can be seen that in terms of SAM, the SM-B
+method obtained the best results for both dates, followed by PSRFM and ESTARFM. How-
+ever, the KF-B strategy was able to obtain a better water mapping performance compared
+to PSRFM. This indicates that the artifacts seen in the (comparatively noisier) reconstruc-
+tions of KF-B impact the the classification performance in a less substantial way compared
+to the SAM. This shows that the proposed Kalman-filter based strategy can provide mean-
+ingful water mapping results in a real-time setting, in which we do not have access to future
+Landsat images, precluding smoothing-based algorithms (such as SM-B and PSRFM) to be
+used.
+Table 3: Spectral angle mapper between the estimated high-resolution image and the Land- sat measurement
+for the Elephant Butte example (note that the Landsat images at dates 06/07 and 06/23 were not supplied
+to the algorithms and only used for evaluation purposes).
+Method
+KF-B
+SM-B
+ESTARFM
+PSRFM
+Image (06/07)
+5.5416
+2.9993
+9.2678
+4.2698
+Image (06/23)
+5.7514
+1.9923
+6.2158
+4.8719
+Average
+5.6465
+2.4958
+7.7418
+4.5709
+Table 4: Percentage of misclassified pixels for the Elephant Butte example (note that the Landsat images
+at dates 06/07 and 06/23 were not supplied to the algorithms and only used for evaluation purposes).
+Method
+KF-B
+SM-B
+ESTARFM
+PSRFM
+Image (06/07)
+5.3593
+1.4289
+9.2678
+6.6606
+Image (06/23)
+5.9233
+0.8250
+10.8330
+7.8675
+Average
+5.6413
+1.1269
+10.0504
+7.2640
+5.7. Discussion
+The results presented above clearly indicate that the proposed weakly supervised smoother-
+based image fusion strategy outperforms the ESTARFM and PSRFM algorithms in terms of
+image reconstruction when an appropriate covariance structure is selected (SM-D and SM-
+B). This highlights that having less model parameters to estimate (i.e., a more constrained
+state covariance model) can lead to better results. Moreover, even the Kalman filter strate-
+gies (particularly KF-B and KF-D), which estimate high-resolution images from MODIS
+without having access to any future data, have shown very competitive performance, with
+great potential for tasks in which high-resolution estimates are required online and one can-
+not wait for another Landsat image to be available before computing the high-resolution
+reconstructions.
+The advantage of the proposed filter and smoother strategies is more clear when eval-
+uated semantically by means of the water classification performance.
+For instance, the
+22
+
+growth of the island portion over time in regions that are semantically meaningful leads
+to more meaningful results that cannot be entirely captured by one standard metric such
+as the SAM. This can be observed more clearly through the spatial distribution of the
+misclassification error maps in Figure 6, which for ESTARFM and PSRFM are signifi-
+cantly more concentrated on the borders between land and water. In general, the proposed
+filtering-based strategies clearly outperformed both the ESTARFM and PSRFM algorithms,
+a standard and a state of the art remote sensing image fusion algorithms. Moreover, the
+proposed distributed implementation, described in Section 4, is able to reduce the com-
+putational power and memory demand of the standard Kalman filter and smoother when
+applied for large images.
+6. Conclusions
+In this paper, an online Bayesian approach for fusing multi-resolution space-borne mul-
+tispectral images was proposed. By formulating the image acquisition process as a linear
+and Gaussian measurement model, the proposed method leveraged the Kalman filter and
+smoother to perform image fusion by estimating the latent high resolution image from the
+different observed modalities. Moreover, a weakly supervised strategy is also proposed to
+define an informative time-varying dynamical image model by leveraging historical data,
+which leads to a better localization of changes occurring in the high-resolution image even
+in intervals where only coarse resolution observations are available. Experimental results
+indicate that the proposed strategy can lead to considerable improvements compared to
+both classical and state-of-the-art image fusion algorithms.
+7. Acknowledgments
+The authors would like to thank the support of the National Geographic Society under
+Grant NGS-86713T-21, the National Science Foundation under Award ECCS-1845833, and
+NASA – GRACE–FO Science Team (80NSSC20K0742).
+23
+
+Landsat
+07/03L
+07/09M
+07/14M
+07/19M
+07/26M
+08/01M
+08/03M
+08/08M
+08/13M
+08/20M
+08/24M
+08/29M
+09/05M
+09/11M
+09/16M
+09/21M
+09/21L
+MODIS
+KF-F
+SM-F
+KF-B
+SM-B
+KF-D
+SM-D
+ESTARFM
+k = 1
+PSRFM
+k = 2
+k = 3
+k = 4
+k = 5
+k = 6
+k = 7
+k = 8
+k = 9
+k = 10 k = 11 k = 12 k = 13 k = 14 k = 15 k = 16 k = 17
+0.00
+0.05
+0.10
+0.15
+0.20
+(a) Fused images in band 1 (MODIS) and band 4 (LandSat)
+Landsat
+07/03L
+07/09M
+07/14M
+07/19M
+07/26M
+08/01M
+08/03M
+08/08M
+08/13M
+08/20M
+08/24M
+08/29M
+09/05M
+09/11M
+09/16M
+09/21M
+09/21L
+MODIS
+KF-F
+SM-F
+KF-B
+SM-B
+KF-D
+SM-D
+ESTARFM
+k = 1
+PSRFM
+k = 2
+k = 3
+k = 4
+k = 5
+k = 6
+k = 7
+k = 8
+k = 9
+k = 10 k = 11 k = 12 k = 13 k = 14 k = 15 k = 16 k = 17
+0.00
+0.05
+0.10
+0.15
+0.20
+0.25
+0.30
+0.35
+(b) Fused images in band 2 (MODIS) and band 5 (LandSat)
+Figure 3: Fused bands from MODIS and Landsat for the Oroville Dam example using different strategies over
+time. The first two rows of each subfigure depict MODIS and Landsat bands acquired at dates displayed on
+top labels. At each time index estimation with KF and SM under different model assumptions, ESTARFM
+and PSRFM are presented. Some Landsat images were omitted from the estimation process and used solely
+as ground-truth. Images used at each update step are indicated on top labels where “M” stands for MODIS
+and “L” for Landsat.
+24
+
+KF-F
+07/19
+08/20
+09/05
+09/21
+SM-F
+KF-B
+SM-B
+KF-D
+SM-D
+ESTARFM
+k = 4
+PSRFM
+k = 10
+k = 13
+k = 16
+0.00
+0.05
+0.10
+0.15
+0.20
+KF-F
+07/19
+08/20
+09/05
+09/21
+SM-F
+KF-B
+SM-B
+KF-D
+SM-D
+ESTARFM
+k = 4
+PSRFM
+k = 10
+k = 13
+k = 16
+0.00
+0.05
+0.10
+0.15
+0.20
+0.25
+0.30
+0.35
+Figure 4: Absolute difference between the estimated and ground truth (Landsat) images for the Oroville
+Dam example in the red (upper panel) and NIR (lower panel) bands.
+25
+
+--Landsat
+07/03L
+07/09M
+07/14M
+07/19M
+07/26M
+08/01M
+08/03M
+08/08M
+08/13M
+08/20M
+08/24M
+08/29M
+09/05M
+09/11M
+09/16M
+09/21M
+09/21L
+MODIS
+KF-BF
+KF-DF
+KF-DB
+SM-BF
+SM-DF
+k = 1
+SM-DB
+k = 2
+k = 3
+k = 4
+k = 5
+k = 6
+k = 7
+k = 8
+k = 9
+k = 10 k = 11 k = 12 k = 13 k = 14 k = 15 k = 16 k = 17
+0.0
+0.1
+0.2
+0.3
+0.4
+0.5
+Landsat
+07/03L
+07/09M
+07/14M
+07/19M
+07/26M
+08/01M
+08/03M
+08/08M
+08/13M
+08/20M
+08/24M
+08/29M
+09/05M
+09/11M
+09/16M
+09/21M
+09/21L
+MODIS
+KF-BF
+KF-DF
+KF-DB
+SM-BF
+SM-DF
+k = 1
+SM-DB
+k = 2
+k = 3
+k = 4
+k = 5
+k = 6
+k = 7
+k = 8
+k = 9
+k = 10 k = 11 k = 12 k = 13 k = 14 k = 15 k = 16 k = 17
+0.0
+0.1
+0.2
+0.3
+0.4
+0.5
+Figure 5: Absolute differences between the images estimated by the KF and Smoother under different model
+assumptions for red (upper panel) and NIR (lower panel) bands, for the Oroville Dam example. KF-BF:
+difference between the estimates of KF-B and KF-F. KF-DF: difference between the estimates of KF-D and
+KF-F. KF-DB: difference between the estimates of KF-D and KF-B. An analogous notation holds for the
+smoother (SM) estimates.
+26
+
+Landsat
+07/19
+08/20
+09/05
+09/21
+KF-F
+SM-F
+KF-B
+SM-B
+KF-D
+SM-D
+ESTARFM
+k = 4
+PSRFM
+k = 10
+k = 13
+k = 16
+0.0
+0.2
+0.4
+0.6
+0.8
+1.0
+Landsat
+08/07
+08/23
+09/08
+09/24
+KF-F
+SM-F
+KF-B
+SM-B
+KF-D
+SM_D
+ESTARFM
+k = 4
+PSRFM
+k = 7
+k = 11
+k = 14
+0.0
+0.2
+0.4
+0.6
+0.8
+1.0
+Figure 6: (Upper Panel) Water map of the reconstructed images of the Oroville Dam example based
+on K-means clustering strategy, where 1 indicates land and 0 indicates water pixels. Classification maps
+obtained from Landsat images not observed by the image fusion algorithms establish the ground-truth (first
+row). (Lower Panel) Absolute error of Water map of images based on K-means clustering strategy, where
+0 indicates correctly classified pixels and 1 indicates misclassifications. The ground-truth is shown in the
+first row.
+27
+
+-:-
+L.
+1二
+.--12
+-10
+-8
+-6
+-4
+-14
+-12
+-10
+-8
+-6
+-4
+-20
+-15
+-10
+-5
+-7
+-6
+-5
+-4
+-3
+-7
+-6
+-5
+-4
+-3
+-15
+-10
+-5
+Modis Observation in Band 1
+0
+0.1
+0.2
+0.3
+0.4
+0.5
+Modis Observation in Band 2
+0
+0.1
+0.2
+0.3
+0.4
+0.5
+Landsat Observation in Band 4
+0
+0.1
+0.2
+0.3
+0.4
+0.5
+Landsat Observation in Band 5
+0
+0.1
+0.2
+0.3
+0.4
+0.5
+Figure 7: (Top Colored Panel) Estimated state covariance structure of the Kalman filter under model
+assumptions i, ii and iii for a small image area in the Oroville Dam example and k “ 13. Top row depicts
+the whole covariance matrix with a red square indicating the zoomed part displayed on the bottom row. The
+plots indicate that correlations are present when assuming block diagonal covariance matrices. (Bottom
+Panel) Zoom of the MODIS image for bands 1 and 2 (left), and the corresponding Landsat observations for
+bands 4 and 5 (right) corresponding to the covariance matrices plotted in the right panels.
+28
+
+0
+2
+4
+6
+8
+10
+12
+14
+16
+18
+20
+Image Indices (k)
+45
+50
+55
+60
+65
+70
+75
+80
+Water pixel percentage
+1.6
+1.8
+2
+2.2
+2.4
+2.6
+2.8
+Volume [m3]
+109
+KF-F
+SM-F
+KF-B
+SM-B
+KF-D
+SM-D
+ESTARFM
+PSRFM
+Hydrograph
+Figure 8: Percentage of water pixels in the estimated images over image index (time) and the reservoir
+volume in m3 (hydrograph) for the Oroville Dam example. Classification of water was done by performing
+clustering on the estimated bands for each method and time index. High resolution Landsat images were
+observed at indices k P t1, 17u.
+29
+
+Landsat
+07/03L
+07/09M
+07/14M
+07/19M
+07/26M
+08/01M
+08/03M
+08/08M
+08/13M
+08/20M
+08/24M
+08/29M
+09/05M
+09/11M
+09/16M
+09/21M
+09/21L
+MODIS
+KF-D
+SM-D
+KF-I
+k = 1
+SM-I
+k = 2
+k = 3
+k = 4
+k = 5
+k = 6
+k = 7
+k = 8
+k = 9
+k = 10 k = 11 k = 12 k = 13 k = 14 k = 15 k = 16 k = 17
+0.00
+0.05
+0.10
+0.15
+0.20
+(a) Fused images in band 1 (MODIS) and band 4 (LandSat)
+Landsat
+07/03L
+07/09M
+07/14M
+07/19M
+07/26M
+08/01M
+08/03M
+08/08M
+08/13M
+08/20M
+08/24M
+08/29M
+09/05M
+09/11M
+09/16M
+09/21M
+09/21L
+MODIS
+KF-D
+SM-D
+KF-I
+k = 1
+SM-I
+k = 2
+k = 3
+k = 4
+k = 5
+k = 6
+k = 7
+k = 8
+k = 9
+k = 10 k = 11 k = 12 k = 13 k = 14 k = 15 k = 16 k = 17
+0.00
+0.05
+0.10
+0.15
+0.20
+0.25
+0.30
+0.35
+(b) Fused images in band 2 (MODIS) and band 5 (LandSat)
+Figure 9: Fused bands from MODIS and Landsat for the Oroville Dam example using different strategies
+over time. The first two rows of each subfigure depict MODIS and Landsat bands acquired at dates displayed
+on top labels. At each time index estimation results of the diagonal Kalman filter and smoother with the
+proposed weakly supervised calibration strategy (KF-D and SM-D) are compared to the result of a Kalman
+filter and smoother with Qk being proportional to the identity (denoted by KF-I and SM-I). Landsat images
+at dates 07/19, 08/20 and 08/29 were omitted from the estimation process and used solely as ground-truth.
+Images used at each update step are indicated on top labels where “M” stands for MODIS and “L” for
+Landsat.
+30
+
+Landsat
+03/19M
+03/19L
+04/18M
+05/18M
+06/07
+06/14M
+06/23
+06/27M
+07/09M
+07/09L
+MODIS
+KF-B
+SM-B
+ESTARFM
+k = 1
+PSRFM
+k = 2
+k = 3
+k = 4
+k = 5
+k = 6
+k = 7
+k = 8
+0.00
+0.05
+0.10
+0.15
+0.20
+0.25
+0.30
+(a) Fused images in band 1 (MODIS) and band 4 (LandSat)
+Landsat
+03/19M
+03/19L
+04/18M
+05/18M
+06/07
+06/14M
+06/23
+06/27M
+07/09M
+07/09L
+MODIS
+KF-B
+SM-B
+ESTARFM
+k = 1
+PSRFM
+k = 2
+k = 3
+k = 4
+k = 5
+k = 6
+k = 7
+k = 8
+0.0
+0.1
+0.2
+0.3
+0.4
+0.5
+(b) Fused images in band 2 (MODIS) and band 5 (LandSat)
+Figure 10: Fused bands from MODIS and Landsat for the Elephant Butte example using different strategies
+over time. The first two rows of each subfigure depict MODIS and Landsat bands acquired at dates displayed
+on top labels. At each time index estimation with KF and SM under block diagonal model assumptions,
+ESTARFM and PSRFM are presented. Some Landsat images were omitted from the estimation process and
+used solely as ground-truth. Images used at each update step are indicated on top labels where “M” stands
+for MODIS and “L” for Landsat.
+31
+
+06/14
+Landsat
+KF-B
+SM-B
+ESTARFM
+k = 5
+PSRFM
+06/27
+k = 6
+0.0
+0.2
+0.4
+0.6
+0.8
+1.0
+Figure 11: Water map of images for the Elephant Butte example based on K-means clustering strategy where
+1 indicates land and 0 indicates water pixels. Unused Landsat classification maps establish the ground-truth
+(first column).
+06/14
+Landsat
+KF-B
+SM-B
+ESTARFM
+k = 5
+PSRFM
+06/27
+k = 6
+0.0
+0.2
+0.4
+0.6
+0.8
+1.0
+Figure 12: Absolute error of Water map of images for the Elephant Butte example based on K-means
+clustering strategy. Unused Landsat classification maps establish the ground-truth (first column).
+06/27
+Landsat
+KF-B
+SM-B
+ESTARFM
+k = 6
+PSRFM
+0.0
+0.2
+0.4
+0.6
+0.8
+1.0
+Figure 13: Zoomed-in water map of images for the Elephant Butte example based on K-means clustering
+strategy where 1 indicates land and 0 indicates water pixels. Unused Landsat classification map at date
+06/23 establish the ground-truth (first column).
+32
+
+7.4B4
+Landsat
+KF-B
+SM-B
+ESTARFM
+k = 6
+PSRFM
+B5
+k = 6
+0.00
+0.05
+0.10
+0.15
+0.20
+0.25
+0.30
+Figure 14: Zoomed-in version of the fused bands from MODIS and Landsat for the Elephant Butte example
+using different strategies at date 06/27 (ground-truth at 06/23 is shown in the first column).
+33
+
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+Modified Query Expansion Through Generative Adversarial Networks
+for Information Extraction in E-Commerce
+Altan Cakir∗,1, Mert Gurkan2
+A R T I C L E I N F O
+Keywords:
+Generative Adversarial Networks
+Query Expansion
+Conditional Neural Networks
+Information Retrieval
+E-Commerce
+A B S T R A C T
+This work addresses an alternative approach for query expansion (QE) using a generative adversarial
+network (GAN) to enhance the effectiveness of information search in e-commerce. We propose a
+modified QE conditional GAN (mQE-CGAN) framework, which resolves keywords by expanding the
+query with a synthetically generated query that proposes semantic information from text input. We
+train a sequence-to-sequence transformer model as the generator to produce keywords and use a re-
+current neural network model as the discriminator to classify an adversarial output with the generator.
+With the modified CGAN framework, various forms of semantic insights gathered from the query-
+document corpus are introduced to the generation process. We leverage these insights as conditions
+for the generator model and discuss their effectiveness for the query expansion task. Our experi-
+ments demonstrate that the utilization of condition structures within the mQE-CGAN framework can
+increase the semantic similarity between generated sequences and reference documents up to nearly
+10% compared to baseline models.
+1. Introduction
+In search based business models, such as e-commerce,
+given a search query, the system needs to match it to some
+relevant keywords/categories/frequencies by business part-
+ners and then pull out the related category/product for query
+searching and ranking. The query keyword matching can
+be done by some simple matching rules like exact match,
+similarity match, and phrase match, which are all based on
+matching the similar tokens shared by query and keywords.
+On the other hand, using AI-based recent techniques for smart
+match is an important yet difficult match type that can asso-
+ciate a query to some relevant keywords even they do not
+generate many similar tokens.
+In general, it is well defined that search queries math-
+ematically follow the power law distribution (Spink, Wol-
+fram, Jansen and Saracevic, 2001). The curve formed by the
+most frequent queries constitutes the main center, while the
+rare queries with low frequency form the tail of the curve.
+Although they are few in such cases, low-frequency queries
+are excluded from the query volume traffic as a whole and
+therefore cause problems in systems as a data deficiency that
+needs to be generated synthetically.
+Because of the incoming query distribution to the search
+engine, the performance of matching rare queries with docu-
+ments existing in the database is a challenging task. It is of-
+ten the case that an additional process is required to assist the
+match between rare queries and documents. To address the
+problem, various methodologies such as relevance feedback
+methods, similarity-based methods for query-document match-
+ing, machine translation models for query transformation,
+∗Corresponding author
+ORCID(s): 0000-0002-8627-7689 (A. Cakir)
+1Physics Engineering, Faculty of Science and Letters, Istanbul Tech-
+nical University, Istanbul, Turkey and Istanbul Technical University Artifi-
+cial Intelligence, Data Science Research and Application Center, Istanbul
+Turkey
+2Insider (useinsider.com), Istanbul, Turkey
+and query expansion methods are discussed in the literature.
+Query expansion is one of the significant problems stud-
+ied in the Information Retrieval (IR) domain with various
+applications such as question answering, information filter-
+ing, or multimedia document matching tasks (Carpineto and
+Romano, 2012). The problem can be described as the at-
+tempt of the increasing performance of matching input se-
+quences and document the corpus of an IR system by refor-
+mulating given input sequences (Azad and Deepak, 2019b).
+Query expansion methodologies are often applied where the
+input queries are words or sequences originating from real
+human users, while documents to match or rank them con-
+sist of predefined items. Natural language queries that match
+to same documents can differ verbally and semantically (Fur-
+nas, Landauer, Gomez and Dumais, 1987). Because of this
+ambiguity, the complexity of query-document matching is
+often increased by the innate characteristics of the data.
+Earlier studies in the query expansion domain seem to
+focus on rule-based applications. These applications evalu-
+ate candidate expansion terms by the frequency of appear-
+ing together with the words in the original query (Carpineto,
+de Mori, Romano and Bigi, 2001). In addition to word fre-
+quency based studies, systems built upon pseudo-relevance
+feedback structures are also widely utilized in the literature.
+(Metzler and Croft, 2007) uses the Markov random fields for
+modelling dependencies to assist the query expansion pro-
+cess. (Symonds, Bruza, Sitbon and Turner, 2011) provides
+a different approach to the query expansion methods with
+pseudo-relevance feedback, where they build tensor repre-
+sentations of queries that enables obtaining relevance feed-
+back based on word meanings.
+Adoption of the deep learning applications in the nat-
+ural language domains generated word embeddings as ef-
+ficient ways to represent semantic information of text data
+(Mikolov, Chen, Corrado and Dean, 2013). The utilization
+of word embeddings made it possible to evaluate the seman-
+tic relationship between words. This capability is employed
+Cakir A., Gurkan M.: Preprint submitted to Elsevier
+Page 1 of 10
+arXiv:2301.00036v1  [cs.LG]  30 Dec 2022
+
+Modified Query Expansion Through Generative Adversarial Networks
+for query expansion problems by using various ways to eval-
+uate the similarity of words that make up the queries and
+candidate terms to expand these queries.
+The popularity of the word embedding methods for vari-
+ous problems for IR and NLP, led research efforts to increase
+the accuracy of word representations in specific cases. To
+this end, alternative ways to produce different embeddings
+of tokens for query expansions are proposed (Sordoni, Ben-
+gio and Nie, 2014). Additionally, research conducted uti-
+lization of task-specific trained word embeddings for query
+expansion (Diaz, Mitra and Craswell, 2016). This way, word
+representations are more likely to capture the context and
+semantic properties of the trained corpus. Following these
+works, (Qi, Gong, Yan, Jiao, Shao, Zhang, Li, Duan and
+Zhou, 2020; Lian, Chen, Jia, You, Tian, Hu, Zhang, Yan,
+Tong, Han et al., 2021) proposed a query expansion approach
+for search engine optimization by utilizing a prefix tree to
+serve as look ahead strategy for generating expansion terms
+for given queries.
+Recent applications of GAN methods provide alternative
+methods to approach the problem. GAN models can directly
+generate expansion terms or expanded user queries by train-
+ing over user search queries and their matching documents.
+In GANs the discriminative network can learn to distinguish
+between the synthetic data created by the generator and the
+real data examples. This way, the generation process is chal-
+lenged by the network itself to create high-quality samples.
+This approach of training has proven to be very successful
+in the computer vision domain and increasing its popular-
+ity in natural language processing problems. Additionally,
+the research focusing on establishing back-propagation be-
+tween discriminator and generator models with discrete to-
+kens in text data (Yu, Zhang, Wang and Yu, 2017; Kusner
+and Hernández-Lobato, 2016) provided highly performing
+generative models.
+With initial GAN models, the model is trained with noise
+for the generation process. With conditional structures, the
+query generation of the GAN models can be assisted with
+the chosen condition mechanism. Similar to earlier works
+in the query expansion domain, enhancing user queries with
+existing relevant information is adopted by GAN-based ar-
+chitectures too. GAN models can utilize part of text data,
+class labels present during the training, or extracted proper-
+ties of the query and documents as conditions to increase the
+likelihood of matching queries with desired documents. The
+study of (Lee, Gao and Zhang, 2018) proposes a conditional
+GAN structure with a query expansion approach for enrich-
+ing rare queries in search engines. The study of (Huang,
+Wang, Liu and Ding, 2021) employs a well-known method
+of pseudo-relevance feedback in the query expansion do-
+main as the condition for their expansion term generation.
+Studies discussed intend to create a conditional GAN-
+based framework to leverage query expansion to match key-
+words for an effective search selection. In general, a sequence-
+to-sequence model, in which the input sequence is a random
+word vector followed by a query vector, is commonly used
+for the generator. The output sequence composes of the vec-
+tors of the generated keywords. As the discriminator, the
+parallelized Recurrent Neural Network (RNN) model is used
+as a binary classifier. However, most of these studies are not
+conducted from the perspective of improving search engines
+by enhancing query-document matching performance. Our
+study aims to combine GAN architecture and existing query-
+enhancing methods by utilizing them as condition structures
+for the generator model. Proposed conditional GAN models
+aim to alleviate the performance drop of search engines, by
+increasing the query-document matching performance with
+condition-assisted query expansion mechanisms.
+To alleviate the effects of the problem described, we in-
+troduce the mQE-CGAN (Modified Query Expansion Con-
+ditional Generative Adversarial Network) framework to study
+the query expansion to enhance the performance of a search
+engine by increasing the query-document matching perfor-
+mance. The generator of the model is a sequence-to-sequence
+encoder-decoder model that takes user search queries and the
+vectors from the applied condition mechanism. The output
+of the generator, expanded queries, is evaluated by the dis-
+criminator model. We use an LSTM model for the binary
+classification task between the synthetic and real samples.
+During adversarial learning, the evaluation of the discrimi-
+nator guides the performance of the generator model. With
+the mQE-CGAN framework, our contributions can be listed
+below;
+• Model: We propose a novel conditional generative
+adversarial network model that takes the semantic re-
+lationship between the query and document pairs as
+conditions. The generator of the model is a sequence-
+to-sequence encoder decoder model, while the discrim-
+inator is an LSTM-based binary classifier. We provide
+details of the model framework and the evaluation of
+the training process with a conditional approach.
+• Conditional Query Expansion: We provide alterna-
+tive methods for condition structures with generative
+adversarial networks. Condition structures discussed
+in this paper aim to capture semantic relationships be-
+tween query-document pairs.
+• Datasets: We test our generative model with the user
+query and document pairs from the customers of In-
+sider1. By testing the proposed models with differ-
+ent customer datasets, we evaluate our models against
+data with different characteristics.
+The primary aspect that mQE-CGAN framework differs
+from the existing conditional GAN frameworks is that the
+models of the framework are conditioned on the semantic
+and statistical relationships between the query-document data.
+Employed conditions are not limited to the individual re-
+lationships between the query and the matching document
+pairs. They are rather constructed with the consideration of
+the entire corpus. Hence, the generation process of the GAN
+framework utilizes conditions produced after the semantic
+analysis of the entire corpus.
+1https://useinsider.com
+Cakir A., Gurkan M.: Preprint submitted to Elsevier
+Page 2 of 10
+
+Modified Query Expansion Through Generative Adversarial Networks
+Figure 1: Diagram of the mQE-CGAN framework.
+2. System Architecture
+2.1. mQE-CGAN Framework
+The proposed framework for adversarial training with
+the mQE-CGAN framework can be observed in the Figure
+2 below. The generator model of the framework takes input
+queries and the selected condition vectors assigned for input
+queries. With a sequence-to-sequence structure, it generates
+expansion terms from the given queries. The discriminator
+model of the adversarial schema performs binary classifica-
+tion on the expanded synthetic queries and documents that
+match to original queries of users.
+Condition generation mechanisms discussed in the study
+aim to take advantage of the data used. As the query-document
+pairs in datasets denote user searches and matching docu-
+ments, condition approaches focus on the semantic and simi-
+larity metrics of given queries and their matching documents
+by the search engine.
+2.1.1. Generator Model
+The generator model of the architecture is an encoder-
+decoder sequence-to-sequence model that takes FastText (Bo-
+janowski, Grave, Joulin and Mikolov, 2016) word embed-
+ding representations of the user search queries and their cor-
+responding condition vectors as input. To be able to achieve
+back-propagation with the discrete input sequences, similar
+to the existing studies (Yu et al., 2017; Lee et al., 2018)
+Monte Carlo rollouts are used in the decoder of the gener-
+ator. With this method, rewards produced by the discrimi-
+nator can be transferred to the generator for each generation
+step.
+2.1.2. Condition Structures
+GAN models can be extended into conditional models
+if the adversarial learning process is performed with addi-
+tional information (Mirza and Osindero, 2014). With the
+introduction of conditions, the models can be inclined to
+generate samples with the desired qualities (Sohn, Lee and
+Yan, 2015). Conditions are introduced to guide the generator
+model during the sequence generation process. Condition
+structures utilized in this study are generated before training
+the model. Condition vectors of queries are concatenated
+with the word embedding representation of the user queries
+during training. To retrieve them, Ball Tree-based look-up
+tables are used.
+To this end, four different condition structures are ap-
+plied with the following expected priorities; (1) It should
+enrich the user query with other similar queries, and (2) it
+should provide information that will assist in distinction be-
+tween similar documents that can be mapped with the given
+query. To address these requirements, various condition vec-
+tor generation strategies displayed are implemented. Uti-
+lized methods are considered to be addressing the shortcom-
+ings of the encoder-decoder generator model. These condi-
+tion generation strategies are described in the list below.
+1. Query Weighting with TF-IDF Scores: Condition vec-
+tors are generated with CBOW representations of the
+TF-IDF weighed input word embeddings
+2. Search Tree Based Document Similarity: Condition
+vectors are generated with CBOW representations of
+the most similar documents to the given input query.
+3. Search Tree Based Word Similarity: Condition vec-
+tors are generated with CBOW representations of the
+most similar words in the corpus of documents to the
+given input query.
+Although these methods are commonly utilized in query
+expansion approaches (Azad and Deepak, 2019a), their in-
+tegration as condition mechanisms is not adequately experi-
+mented with generative models.
+2.1.3. Discriminator Model
+The discriminator model of the mQE-CGAN framework
+is built with the same pre-trained Fasttext word embeddings
+and LSTM layers processing embedded representations of
+generated and real document sequences. Unlike the gener-
+ator model of the framework, the discriminator model does
+not utilize the condition structures for its pre-training and ad-
+versarial learning processes. The model is designed for the
+binary classification task between real documents in corpus
+and sequences formed by the generator as the synthetic data.
+Cakir A., Gurkan M.: Preprint submitted to Elsevier
+Page 3 of 10
+
+P1: Classification as
+p2
+Real Data
+p2: Classification as
+Generated Sequences
+Generated Data
+Linear Layer
+Query 1
+ Condition 1
+Query 2
+Condition 2
+Expanded
+User Query
+ Matched Document
+queries
+Condition
+Generation
+W1
+W1
+Query n
+Condition n
+Generator model input
+Search Engine
+Discriminator Model InputModified Query Expansion Through Generative Adversarial Networks
+Figure 2: Diagram of the Monte Carlo rollouts. At each step, a batch of sequences are generated by the decoder of the
+network. These batches are evaluated by the discriminator to guide the generation process of the generator model.
+Figure 3: LSTM based discriminator model of the mQE-
+CGAN framework.
+2.2. Implementation Details
+We conducted the implementation with the PyTorch li-
+brary (Paszke, Gross, Massa, Lerer, Bradbury, Chanan, Killeen,
+Lin, Gimelshein, Antiga, Desmaison, Kopf, Yang, DeVito,
+Raison, Tejani, Chilamkurthy, Steiner, Fang, Bai and Chin-
+tala, 2019) in this study. The encoder-decoder generator model
+is implemented by using the TransformerEncoder and Trans-
+formerDecoder classes in PyTorch. The generator model
+uses 2 layers for both the encoder and the decoder parts.
+Initially, the input user queries are transformed to FastText
+word embedding representations with each word being rep-
+resented with a tensor of size 100. Originally, FastText word
+embeddings are available for Turkish with a size of 300. To
+reduce the amount of GPU RAM required, we transformed
+these embedding representations to vectors with size 100
+with the reduce_model implementation of the FastText li-
+brary. It is followed by applying positional embedding to
+assign the order context to tokens in sequences with the help
+of the attention heads (Vaswani, Shazeer, Parmar, Uszko-
+reit, Jones, Gomez, Kaiser and Polosukhin, 2017). For the
+forward pass, the given query and its paired condition vector
+are concatenated. The encoder and decoder of the generator
+take an input size of 200 from the concatenated tensors, and
+they have a hidden size of 512.
+Pre-training of the generator is performed by training the
+generator model with the learning rate 10−3 and the Adam
+(Kingma and Ba, 2014) optimizer. During pre-training, the
+generator uses a softmax layer of size 푁, where 푁 is the
+total vocabulary size of the query and document corpus. Se-
+quence generation is performed iteratively by predicting an
+expansion term at each step until the generator predicts the
+next token as < 퐸푂푆 > (end of the sequence) token. For
+many cases, it was observed that after training the genera-
+tor 16 epochs the Cross-Entropy Loss of the model does not
+improve.
+The discriminator model of the framework is intention-
+ally kept simpler than the generator. For the discriminator,
+we used a 1 layer LSTM model. To decide on the hyper-
+parameters of the discriminator, a grid search is applied to
+hyper-parameters by training discriminator models with com-
+bined datasets of synthetic data from the generator and the
+samples from the document corpus. The discriminator model
+where the loss is optimized was obtained with the number of
+epochs as 24, the learning rate as 10−2, dropout as 0.1, and
+the batch size as 256.
+3. Experiments
+3.1. Datasets
+The datasets utilized in the study are generated by the
+analysis of user behavior in a search engine product of In-
+sider. More specifically, these datasets consist of user search
+queries and the first-ranked resulting products in the plat-
+forms of Insider customers. It should be noted that the datasets
+utilized in this study do not include any specific user infor-
+mation. During the data collection step, any information that
+Cakir A., Gurkan M.: Preprint submitted to Elsevier
+Page 4 of 10
+
+Backpropagation of the
+average reward collected
+from the discriminator
+Rewards from the
+Discriminator Model
+Generated
+sequences
+current
+with Monte
+Discriminator
+t2
+Batch of
+word
+Carlo
+expanded
+Rollouts
+sequences
+t1
+t2
+Encoded
+Queryp1: Classification as Real Data
+p1
+p2
+P2: Classification as Generated Data
+Linear Layer
+h21
+h22
+h23
+h24
+h25
+LSTM
+Layers
+h11
+h12
+h13
+h14
+h15
+Word
+Embeddings
+W1
+W2
+W3
+W4
+W5
+Synthetic data
+Real data source
+source
+The
+Generator
+Model
+User queries and
+matching productsModified Query Expansion Through Generative Adversarial Networks
+can be exploited to identify the user information is discarded.
+As the general user behavior in search engines is to en-
+ter fewer words to match the desired documents (Pal, Mitra
+and Bhattacharya, 2015), queries in search engines tend to
+compose fewer words compared to the documents. This gen-
+eral observation is also present in the datasets utilized in our
+study. The average number of words in queries and docu-
+ments in datasets used in the study can be observed in Figure
+4 below.
+Figure 4: Statistics of the query and the document datasets
+utilized in the study. For each dataset, bars at the top display
+the maximum, average, and minimum number of words in
+queries. Similarly, bottom bars display statistics of the doc-
+ument corpus. For all datasets, the average number of words
+in user searches are almost four times less than their match-
+ing product equivalents, suggesting further ways to employ
+semantic information to be extracted from document data.
+The difference between the number of words in queries
+and documents introduces various challenges for search en-
+gines. In the case of rare query inputs of users, similar to
+recommendation systems search engines are more prone to
+the cold start problem (Camacho and Alves-Souza, 2018).
+Datasets generated in the study aim to challenge the mQE-
+CGAN framework in this regard.
+3.2. Experimented Evaluation Metrics
+Both the generator and the discriminator of the mQE-
+CGAN framework are trained with cross-entropy loss during
+pre-training processes. For model comparisons, changes in
+the perplexity metric were analyzed for the generator. For
+the discriminator, the accuracy of the trained models was
+tracked.
+In addition to these metrics, we track the language diver-
+sity of the expanded queries. To this end, a new evaluation
+metric, the Word Coverage (WC), is defined. Word Cover-
+age metric checks the ratio of the number of unique words
+selected as expansion terms by the generator to the number
+of unique words in the document corpus. For a successful
+model, we expect this metric to be close to 1. Obtaining a
+Word Coverage metric lower than one suggests that the gen-
+erator model was not able to cover words in the tested set in
+the query expansion process. On the other hand, obtaining
+a Word Coverage metric higher than one indicates that the
+word selection process during query expansion utilized more
+unique words from the training corpus than it should have.
+The formula of the metric can be observed below. In the
+formula, 푠푄퐸 denotes words that are selected as expansion
+terms by the generator, 푠퐶 denotes the words in the tested
+corpus.
+푊 퐶 =
+∑ 푢푛푖푞(푠푄퐸)
+∑ 푢푛푖푞(푠퐶)
+In addition to analyzing the expansion term diversity in
+generated sequences, models are also evaluated by the se-
+mantic similarity between generated sequences and refer-
+ence sequences. To this end, we utilized average cosine simi-
+larity between the generated sequences obtained with expan-
+sion terms and their corresponding references in the docu-
+ment corpus. To assess the similarity, the average CBOW
+representations of both sets are compared. CBOW represen-
+tations are obtained by averaging the embedding represen-
+tations of the words that make generated and reference se-
+quences. The formula below summarizes the Semantic Sim-
+ilarity (SS) analysis between generated and reference docu-
+ments.
+푆푆 =
+푁
+∑
+푖
+̂푤푖 ⋅ 푤푖
+‖‖ ̂푤푖‖‖2 ‖‖푤푖‖‖2
+These metrics allow us to assess the success of gener-
+ated sequences without penalizing the n-gram matching per-
+formance of the generator. As the significance of n-gram
+matching and the word order are less crucial for matching
+user queries and products, the metric provides significant in-
+sights into the generation performance with different datasets.
+3.3. Generator Evaluation Metrics
+Resulting evaluation metrics after integrating the con-
+dition generation strategies to the generator model can be
+found from the table below.
+Dataset
+Condition
+CE Loss
+Perplexity
+WC
+SS (휇, 휖)
+C1
+Baseline Generator
+1.266
+3.650
+1.07
+(0.602, 0.173)
+Word Sim.
+1.328
+3.792
+1.02
+(0.696, 0.169)
+Document Sim.
+1.258
+3.536
+0.99
+(0.659, 0.178)
+TF-IDF
+1.288
+3.644
+1.15
+(0.606, 0.176)
+C2
+Baseline Generator
+0.267
+1.307
+0.46
+(0.898, 0.144)
+Word Sim.
+0.27
+1.311
+0.45
+(0.911, 0.14)
+Document Sim.
+0.272
+1.313
+0.46
+(0.902, 0.1412)
+TF-IDF
+0.267
+1.307
+0.46
+(0.894, 0.146)
+C3
+Baseline Generator
+0.34
+1.405
+1.07
+(0.662, 0.173)
+Word Sim.
+0.337
+1.401
+0.84
+(0.81, 0.169)
+Document Sim.
+0.344
+1.411
+0.98
+(0.809, 0.171)
+TF-IDF
+0.33
+1.391
+0.74
+(0.819, 0.162)
+C4
+Baseline Generator
+1.292
+3.650
+1.26
+(0.709, 0.217)
+Word Sim.
+1.285
+3.626
+1.02
+(0.736, 0.209)
+Document Sim.
+1.28
+3.605
+1.15
+(0.721, 0.203)
+TF-IDF
+1.218
+3.39
+1.20
+(0.686, 0.272)
+Table 1: Generator evaluation metrics of the selected dataset of companies. Company
+names are replaced with placeholders as C. To provide further context; Company 1
+(C1) is a Turkey-based cosmetics company, Company 2 (C2) and 4 (C4) are fashion
+retailers originated in Turkey, and Company 3 (C3) is a worldwide technology com-
+pany.
+Cakir A., Gurkan M.: Preprint submitted to Elsevier
+Page 5 of 10
+
+Query and Document Length Statistics
+Avg.
+Query Length
+Avg. Document Length
+Company 1
+Company 2
+Average Length
+Company 3 -
+Company 4 -
+2
+10
+11
+12
+0
+3
+5
+8
+9
+13
+14
+15
+16
+4
+6
+7
+17
+18Modified Query Expansion Through Generative Adversarial Networks
+In Table 1 above, the Baseline Generator is trained by
+self-conditioning the input user queries. This way, the ef-
+fectiveness of the condition structures is evaluated against a
+condition mechanism that will not provide further positive
+cues for the generation process. WC denotes the Word Cov-
+erage metric discussed earlier, and SS denotes the Semantic
+Similarity metrics of trained generators. The mean and stan-
+dard deviation of cosine similarities between generated se-
+quences and reference documents can be observed in the ta-
+ble. Although generators with different models yield similar
+Cross Entropy Loss values, the Semantic Similarity obtained
+from generators with word similarity as conditions result in
+more successful generation processes. The Word Coverage
+metric is higher than it should have been for baseline gen-
+erator models, compared to models trained with additional
+conditions.
+These metrics were obtained after training each genera-
+tor model 16 epochs with the Cross-Entropy Loss function.
+As our initial observations demonstrated that generator pre-
+training tends to not improve after 16 epochs, Table 1 dis-
+plays the effectiveness of condition methods before adver-
+sarial learning.
+3.4. Adversarial Learning
+For adversarial learning, we pre-trained the generator and
+the discriminator models with half the number of epochs
+mentioned in the Implementation Details section. Thus, these
+models were not optimized for the underlying dataset. The
+pre-training of the generator is performed with the train and
+validation splits, where the discriminator is trained with the
+test splits. Below, the adversarial learning algorithm we use
+with these configurations can be observed.
+Algorithm 1 Adversarial Learning with Policy Gradients
+Require: Generator pre-training policy 퐺; rollout policy
+퐺푟; Discriminator pre-training policy 퐷; query-document
+dataset 푆 = {푋1∶푁, 푌1∶푁}
+Pre-train G using Cross-Entropy Loss on 푆
+Generate synthetic examples using G for training D as 푆휃
+Pre-train D using Cross-Entropy Loss on {푆, 푆휃}
+repeat
+for e in epochs do
+for b in batches do
+Generate 푁 rollout sequences with 퐺푟 for 푆푏
+Obtain average reward from 퐷 as 푅푏
+end for
+Update 퐺푟 with 푎푣푔(푅푏)
+end for
+until G loss of mQE-CGAN does not improve
+During adversarial learning, at each expansion term gen-
+eration step the generator model samples 푁 finished sequences
+from unfinished sequences with Monte Carlo rollouts. These
+sampled sequences are evaluated by the discriminator 퐷 to
+inform the generator model 퐺푟 about the current generation
+step. The average discriminator loss 푎푣푔(푅푏) obtained for
+this operation is used for rewarding the generator model and
+updating its parameters. These operations are repeated for
+each batch in the query-document dataset. By employing
+Policy Gradients (Sutton, McAllester, Singh and Mansour,
+1999), we convert the discriminator loss to the format that
+the generator can utilize.
+4. Results & Discussion
+Table 1 demonstrates that the generator models condi-
+tioned with the Word Similarity method result in the best
+semantic evaluation metrics. Word Similarity provides pre-
+cise embedding vectors of words that are the most similar in
+meaning to the words in the query. In this manner, models
+conditioned with it receive more insights about the context of
+the given query. The approach can also be considered similar
+to the pseudo-relevance feedback methods where the query
+is enhanced with the documents that initially matched with
+them. Compared to Word Similarity conditions, Document
+Similarity and TF-IDF Weighting conditions yield slightly
+worse semantic evaluation metrics. In some cases, condition
+methods do not increase the generator model performance
+compared to the baseline model. Our analysis displayed that
+Document Similarity conditions tend to guide the generation
+process in inaccurate ways, as the most similar documents to
+given input queries were possible to be differentiating from
+the reference documents. For many cases, TF-IDF Weight-
+ing seems to omit words in a given query to incline the gener-
+ator model to narrow the space for expansion term selection.
+When model performances are compared among datasets,
+it can be seen that the models were most successful in the Se-
+mantic Similarity metric for the dataset of Company 2 (C2
+in Table 1) and least successful for the dataset of Company
+1 (C1). This result was expected after we analyzed the prop-
+erties of different datasets in the study. The C1 dataset is the
+most challenging dataset having the largest vocabulary size
+among utilized datasets. On the other hand, the C2 dataset
+can be considered more trivial among others as having the
+smallest vocabulary size and documents are composed of
+more keywords. The evaluation metrics of conditioned mod-
+els tested with the C3 dataset should also be noted. As it is a
+dataset of a technology company, product name documents
+are often composed of words that do not have much meaning
+and context when separate. Here, prioritizing the words with
+higher TF-IDF scores for the generator model can increase
+the generator performance to select more precise expansion
+terms within the same query context.
+When the extended queries produced by the models are
+examined, it is seen that the expansion terms in the generated
+sequences have a high success in being in the same context
+as the reference document, but the prediction of the terms
+in the documents is not at the same level. It should also be
+noted that the datasets used for training the models are lim-
+ited. When the system we proposed in our study works inte-
+grated with a search engine, it will be better optimized with
+real-time data flow with higher traffic. We expect the suc-
+cess of word prediction in sequences to increase even more.
+Furthermore, due to the nature of the problem we aim to ad-
+Cakir A., Gurkan M.: Preprint submitted to Elsevier
+Page 6 of 10
+
+Modified Query Expansion Through Generative Adversarial Networks
+dress, it is more important that the added words bring the
+semantic values of the extended queries closer to the docu-
+ments instead of directly matching the words added in the ex-
+panded queries with the documents tested. Therefore, eval-
+uation metrics such as BLEU (Papineni, Roukos, Ward and
+Zhu, 2002) and ROUGE (Lin, 2004) that prioritize correct
+word prediction was not prioritized in our study. It is consid-
+ered that previously discussed Word Coverage and Semantic
+Similarity metrics were better suited for evaluating the pro-
+posed framework.
+The approach taken for condition structures is similar
+to pseudo-relevance feedback approaches. The differenti-
+ating aspect here is that obtaining a document or a list of
+documents that are likely to match the user query requires
+multiple operations within the search engine. As this re-
+quirement would hinder the time performance of the query-
+document matching, we avoided utilizing pseudo-relevance
+feedback approaches directly in our studies. Applied condi-
+tion mechanisms are designed to be stored outside the search
+engine environment and memory. Hence, operations needed
+for reaching condition vectors are not reflected in the perfor-
+mance of the search engine. However, the time and space re-
+quirements of these conditions are the primary drawbacks of
+these approaches. As for all condition structures discussed
+construction of a lookup table is necessary, these lookup ta-
+bles should be generated or updated before model training
+and tuning. Thus, these structures form an additional step
+for complete model training.
+Applying the word and document similarity for condi-
+tions intends to enrich the initial user query that often con-
+sists of one to two words. However, we observed that with
+the word and document similarity condition mechanisms as-
+sistance for rare user queries may not be adequate to decrease
+the effects of the cold start problem. The reason for this is
+that the condition vectors obtained by word similarity and
+document similarity may affect the sentence production of
+the model in undesirable ways. The sequences that can be
+produced with the word and document information added
+with the conditions can be differentiated from the document
+information corresponding to the search made by the user.
+When a user query consisting of very few words is combined
+with conditions that are almost the same size and contain the
+same amount of semantic meaning, the indexes produced by
+the model can diverge from the sequences desired to be ob-
+tained.
+There are differences between the conditioned GAN ar-
+chitectures in the literature and the conditioned GAN archi-
+tectures presented in this study. It has been seen that the
+conditional GAN architectures in the literature utilize condi-
+tions for both the generator and the discriminator models. In
+these studies, it is a correct approach to feed both the genera-
+tor and the discriminator with this information, as the condi-
+tions are usually made up of class labels. In our study, since
+the condition structures consist of semantic information that
+increases the sentence generation performance of the model,
+the condition structures were used only in the generators.
+The discriminator model only performs binary classification
+between synthetic data and product information correspond-
+ing to user queries. Another differentiating issue is the train-
+ing phase of the generative model. In the mQE-CGAN archi-
+tecture, Monte Carlo simulations were not used in the pre-
+training phase of the generative models. Softmax operations
+were used in an iterative manner for the models to predict the
+next words in the sequences. Monte Carlo simulations were
+used only during adversarial learning.
+It is observed that similar semantic evaluation metric val-
+ues can be obtained with adversarial learning. For the dataset
+of Company 2, the adversarial learning phase improves the
+Semantic Similarity metric between generated and reference
+sequences from 0.911 to 0.914. Likewise, the Semantic Sim-
+ilarity metric increases from 0.736 to 0.808 for the dataset of
+Company 4. Hence, the average cosine similarity between
+generated sequences and reference documents increase by
+nearly 10% after the adversarial learning phase compared to
+the generator evaluation metrics with the baseline model.
+For other datasets, adversarial training until the generator
+loss function does not improve did not yield better semantic
+evaluation metrics. It suggests that there are further tuning
+and optimization steps for the adversarial learning process of
+the framework. Due to this, we take the 10% performance
+increase as the best improvement of the adversarial learning
+phase.
+The table below displays the sequences obtained after the
+adversarial learning phase of the mQE-CGAN framework.
+When examples in Table 2 are analyzed, it can be seen
+that the model tends to generate the company name as an
+expansion term. It is because company names are the most
+common words in the case of C2 and C4 datasets. Thus, the
+models have the bias of outputting the most occurred word
+in the dataset. For these datasets, the expansion terms seem
+to be meaningful in general. For user queries such as "mont"
+(coat) or "gömlek" (shirt), the model generates terms such as
+"regular fit" or "slim fit". On the other hand, when the initial
+user query does not have a matching document in the dataset,
+the generated expansion terms seem to be less successful.
+The most obvious example of this observation is the first ex-
+ample given for C2. The query "bandana" is expanded with
+"siyah parka" (black parka) where the query initially matches
+with the document "sarı bucket çanta" (yellow bucket bag).
+It seems that trained models are more successful for the
+expansion generation task where the relationship between
+words is more precise. If the candidate words to expand the
+given query are more limited, models seem to capture the
+semantic relationship between different words in a smaller
+space. Generation results of the C3 dataset exemplify this
+phenomenon. In the first example, the query "şarj" (charge)
+is expanded with words such as "c-type", "hızlı" (fast), and
+"seyehat" (travel). The second example adds the memory
+information that is very common to be included in product
+names to the search query of a specific device. The third
+example adds "kablolu" (wired) and "mikrofonlu" (with mi-
+crophone) to the user query of "kulaklık" (headphones).
+The presence of this phenomenon can also be seen in
+the results of the C1 dataset. The query "saçlar" (hair) is
+Cakir A., Gurkan M.: Preprint submitted to Elsevier
+Page 7 of 10
+
+Modified Query Expansion Through Generative Adversarial Networks
+Dataset
+Query
+Generated Sequence
+Reference Document
+C1
+saçlar
+saçlar nemlendirici krem 50
+water nemlendirici şampuan
+köpük
+karma köpük 150
+vitaminli 150 ml yüz köpüğü
+yıpranma krem
+yıpranmış nemlendirici krem 50
+brand name yıpranma karşıtı nemlendirici krem 50 ml
+C2
+bandana
+siyah parka
+sarı bucket çanta
+krem ceket
+kapüşonlu siyah ceket
+kapüşonlu beyaz ceket
+{company name} black jake
+jake {company name} black jean pantolon
+jake {company name} black gölgeli jean pantolon
+C3
+şarj
+{company name} {model name} c-type hızlı seyahat
+{company name} {model name} siyah
+{company name} {model name}
+{company name} {model name} 128 gb
+{company name} {model name} 128 gb
+kulaklık
+{company name} {model name} kablolu mikrofonlu
+{company name} {model name} kulaklık
+C4
+mont
+{company name} klasik regular fit
+standart fit mont
+kareli gömlek
+slim fit gömlek
+slim fit kareli gömlek
+polo yaka tisort
+{company name} polo yaka cepsiz
+regular fit polo yaka tisort
+Table 2: Randomly selected generated samples and their corresponding query and reference document pairs. Generated sequences are obtained after the adversarial learning The
+generator of the framework was selected as Word Similarity model. For each different company dataset, three examples are displayed in the table. Whenever the company name
+is included in the generated sequence, they are marked as {company name}. {brand name} is added to not reveal specific brand names in the C3 dataset. {model name} is added
+to hide specific product models in the C3 to not reveal the further information about the company.
+paired with "nemlendirici" (moisturizer) and "krem" (con-
+ditioner). In the third example, the query "yıpranma krem"
+is expanded with "nemlendirici" (moisturizer) and the cor-
+rect volume of the product. As the C1 dataset is mostly
+composed of cosmetics products, the dataset usually con-
+sists of documents that have volume information. Results
+display that the trained model is not successful at generat-
+ing sequences with correct volume information consisting of
+the volume value and its unit. We observed that our model
+tended to include a numerical value to generated sequences
+often but did not include its unit such as "ml" or "cc". It
+suggests that models can be further optimized to capture the
+relationships between individual word pairs in a better way.
+5. Conclusion
+Our work focused on bringing concepts of generative
+adversarial networks, query expansion, and condition struc-
+tures originated from query-document relationships together.
+Results from the mQE-CGAN framework demonstrate that
+given user queries with limited information can be enriched
+with query expansion to obtain sequences that are semanti-
+cally more similar to the documents in the datasets. As the
+trained models yield successful evaluation metrics for cap-
+turing the context of given query-document pairs, utilization
+of the framework can be beneficial for optimizing search en-
+gines in the e-commerce domain.
+Various aspects of the proposed GAN framework can
+be improved. Firstly, we believe that the sequence gener-
+ation process could benefit from utilizing context-specific
+word embeddings. To this end, word embeddings obtained
+from language models fine-tuned for datasets will be tested
+in the future. Secondly, alternative condition mechanisms
+can be introduced during the training process. The proposed
+framework allows the replacement of condition mechanisms
+to adapt specific cases by capturing different semantic re-
+lationships in query-document data. One of the condition
+structures to be applied is the combination of the conditions
+experimented with in this study. Lastly, we aim to experi-
+ment with the integration of the proposed GAN framework
+with the existing search engine. This way, the advantages
+and shortcomings of a search engine with an integrated GAN
+model for query expansion can be observed in high-traffic
+environments. In future works, we aim to assess the practi-
+cal evaluation metrics of the query expansion approach for
+its performance against the cold start problem.
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+Symonds, M., Bruza, P., Sitbon, L., Turner, I., 2011. Tensor
+query expansion: A cognitively motivated relevance
+model.
+Cakir A., Gurkan M.: Preprint submitted to Elsevier
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+Modified Query Expansion Through Generative Adversarial Networks
+Vaswani, A., Shazeer, N., Parmar, N., Uszkoreit, J., Jones,
+L., Gomez, A.N., Kaiser, L.u., Polosukhin, I., 2017.
+Attention is all you need, in: Guyon, I., Luxburg,
+U.V., Bengio, S., Wallach, H., Fergus, R., Vish-
+wanathan, S., Garnett, R. (Eds.), Advances in Neural
+Information Processing Systems, Curran Associates,
+Inc. URL: https://proceedings.neurips.cc/paper/2017/
+file/3f5ee243547dee91fbd053c1c4a845aa-Paper.pdf.
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+Page 10 of 10
+

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+arXiv:2301.01920v1  [math.NA]  5 Jan 2023
+Double-Exponential transformation:
+A quick review of a Japanese tradition*
+Kazuo Murota†and Takayasu Matsuo‡
+January 5, 2023
+Abstract
+This article is a short introduction to numerical methods using the double exponential
+(DE) transformation, such as tanh-sinh quadrature and DE-Sinc approximation. The
+DE-based methods for numerical computation have been developed intensively in Japan
+and the objective of this article is to describe the history in addition to the underlying
+mathematical ideas.
+Keywords: Double exponential transformation, DE integration formula, tanh-sinh quadra-
+ture, DE-Sinc method.
+1
+Introduction
+The double exponential (DE) transformation is a generic name of variable transformations
+(changes of variables) used effectively in numerical computation on analytic functions, such
+as numerical quadrature and function approximation. A typical DE transformation is a change
+of variable x to another variable t by x = φ(t) with the function
+φ(t) = tanh
+�π
+2 sinh t
+�
+.
+The term “double exponential” refers to the property that the derivative
+φ′(t) =
+π
+2 cosh t
+cosh2(π
+2 sinh t)
+decays double exponentially
+φ′(t) ≈ exp
+�
+−π
+2 exp |t|
+�
+(1)
+as |t| → ∞.
+*This is a preliminary version of an article to be included in ICIAM 2023, Tokyo Intelligencer.
+†The Institute of Statistical Mathematics, Tokyo 190-8562, Japan; and Faculty of Economics and Business
+Administration, Tokyo Metropolitan University, Tokyo 192-0397, Japan, murota@tmu.ac.jp
+‡Department of Mathematical Informatics, Graduate School of Information Science and Technology, Uni-
+versity of Tokyo, Tokyo 113-8656, Japan matsuo@mist.i.u-tokyo.ac.jp
+1
+
+This article is a short introduction to numerical methods using DE transformations such as
+the double exponential formula (tanh-sinh quadrature) for numerical integration and the DE-
+Sinc method for function approximation. The DE-based methods for numerical computation
+have been developed intensively in Japan [5, 7, 34, 38] and a workshop titled “Thirty Years of
+the Double Exponential Transforms” was held at RIMS (Research Institute for Mathematical
+Sciences, Kyoto University) on September 1–3, 2004 [14]. The objective of this article is to
+describe the history of the development of the DE-based methods in addition to the underlying
+mathematical ideas.
+This article is written to the memory of Professors Masao Iri (President of Japan SIAM,
+1996), Masatake Mori (President of Japan SIAM, 1998), and Masaaki Sugihara (Vice Presi-
+dent of Japan SIAM, 2008).
+2
+DE formula for numerical integration
+The DE formula for numerical integration invented by Hidetosi Takahasi and Masatake Mori
+[37] was first presented at the RIMS workshop “Studies on Numerical Algorithms,” held on
+October 31–November 2, 1972. The celebrated term of “double exponential formula” was
+proposed there, as we can see in the proceedings paper [36].
+2.1
+Quadrature formula
+The DE formula was motivated by the fact that the trapezoidal rule is highly effective for
+integrals over the infinite interval (−∞, +∞). For an integral
+I =
+� 1
+−1
+f (x)dx,
+for example, we employ a change of variable x = φ(t) using some function φ(t) satisfying
+φ(−∞) = −1 and φ(+∞) = 1, and apply the trapezoidal rule to the transformed integral
+I =
+� +∞
+−∞
+f (φ(t))φ′(t)dt,
+to obtain an infinite sum of discretization
+Ih = h
+∞
+�
+k=−∞
+f (φ(kh))φ′(kh).
+(2)
+A finite-term approximation to this infinite sum results in an integration formula
+I(N)
+h
+= h
+N
+�
+k=−N
+f (φ(kh))φ′(kh).
+(3)
+Such combination of the trapezoidal rule with a change of variables was conceived by several
+authors [2, 24, 25, 35] around 1970.
+The error I − I(N)
+h
+of the formula (3) consists of two parts, the error ED ≡ I − Ih incurred
+by discretization (2) and the error ET ≡ Ih − I(N)
+h
+caused by truncation of an infinite sum Ih to
+a finite sum I(N)
+h .
+2
+
+The major findings of Takahasi and Mori consisted of two ingredients. The first was that
+the double exponential decay of the transformed integrand f (φ(t))φ′(t) achieves the optimal
+balance (or trade-off) between the discretization error ED and the truncation error ET. The
+second finding was that a concrete choice of
+φ(t) = tanh
+�π
+2 sinh t
+�
+(4)
+is suitable for this purpose thanks to the double exponential decay shown in (1). With this
+particular function φ(t) the formula (3) reads
+I(N)
+h
+= h
+N
+�
+k=−N
+f
+�
+tanh
+�π
+2 sinh(kh)
+��
+(π/2) cosh(kh)
+cosh2((π/2) sinh(kh))
+,
+which is sometimes called “tanh-sinh quadrature.” The error of this formula is estimated
+roughly as
+���I − I(N)
+h
+��� ≈ exp(−CN/ log N)
+(5)
+with some C > 0. The DE formula has an additional feature that it is robust against end-point
+singularities of integrands.
+The idea of the DE formula can be applied to integrals over other types of intervals of
+integration. For example,
+I =
+� +∞
+0
+f (x)dx, x = exp
+�π
+2 sinh t
+�
+,
+(6)
+I =
+� +∞
+−∞
+f (x)dx, x = sinh
+�π
+2 sinh t
+�
+.
+(7)
+Such formulas are also referred to as the double exponential formula. The DE formula is
+available in Mathematica (NIntegrate), Python library SymPy, Python library mpmath, C++
+library Boost, Haskell package integration, etc.
+2.2
+Optimality
+Optimality of the DE transformation (4) was discussed already by Takahasi and Mori [37].
+Numerical examples also support its optimality. Figure 1 (taken from [5]) shows the compar-
+ison of the DE transformation (4) against other transformations
+φ(t) = tanh t,
+φ(t) = tanh
+�π
+2 sinh t3�
+,
+φ(t) = erf(t) =
+2√π
+� t
+0
+exp(−s2)ds
+for
+� 1
+−1
+1
+(x − 2)(1 − x)1/4(1 + x)3/4 dx
+that has integrable singularities at both ends of the interval of integration. The DE formula
+converges much faster than others. It is known that the tanh-rule (using φ(t) = tanh t) has
+3
+
+Discovery of the DE Transformation
+915
+Figure 4. Comparison of the efficiency of several variable transformations for
+the integral
+dx/
+2)(1
+(1 +
+uations and the ordinate is the absolute error
+in logarithmic scale
+actually computed. The number attached to each curve in the figure is the
+mesh size
+used for actual computation. Transformation c gives the DE for-
+mula. From this figure we see that the efficiency becomes higher as the decay
+of
+is faster, and it attains the highest when the DE transformation is ap-
+plied. Then, as the decay becomes faster than double exponential the efficiency
+turns to be lower.
+Thus, Takahasi and Mori were convinced of the optimality of the DE trans-
+formation and presented the result orally at a RIMS symposium in 1973 [79]
+and published as a paper in Publ. RIMS in 1974 [80].
+4.3.
+Application of the DE transformation to
+other types of integrals
+The idea of the DE transformation can be applied to various kinds of
+integrals.
+Takahasi and Mori gave some examples other than (4.8) in their
+paper in 1974 [80]. We list here typical types of integrals and corresponding
+Figure 1: Comparison of the efficiency of several variable transformations for the integral
+� 1
+−1 dx/{(x − 2)(1 − x)1/4(1 + x)3/4}; taken from Mori [5, Fig. 4] with permission from the
+European Mathematical Society; u and N in the figure correspond, respectively, to t and
+2N + 1 in the present notation.
+the (rough) convergence rate exp(−C
+√
+N), in contrast to exp(−CN/ log N) in (5) of the DE
+formula.
+The optimality argument of [37], based on complex function theory, was convincing
+enough for the majority of scientists and engineers, but not perfectly satisfactory for theo-
+reticians. Rigorous mathematical argument for optimality of the DE formula was addressed
+by Masaaki Sugihara [28, 29, 30] in the 1980–1990s in a manner comparable to Stenger’s
+framework [26] for optimality of the tanh rule. It is shown in [30] (also [42]) that the DE
+formula is optimal with respect to a certain class (Hardy space) of integrand functions.
+In principle, for each class of integrand functions we may be able to find an optimal
+quadrature formula, and the optimal formula naturally depends on our choice of the admissi-
+ble class of integrands. Thus the optimality of a quadrature formula is only relative. However,
+it was shown by Sugihara that no nontrivial class of integrand functions exists that admits a
+quadrature formula with smaller errors than the DE formula. We can interpret this fact as the
+absolute optimality of the DE formula.
+2.3
+Fourier-type integrals
+For Fourier-type integrals like
+I =
+� +∞
+0
+f1(x) sin x dx,
+the DE formula like (6) is not very successful. To cope with Fourier-type integrals, a novel
+technique, in the spirit of DE transformation, was proposed by Ooura and Mori [22, 23]. In
+[22] they proposed to use
+φ(t) =
+t
+1 − exp(−K sinh t)
+4
+
+h
+口
+Q8
+0.8
+10--5
+W0.1
+0.6=h
+tanh u
+04
+0.075
+K
+0.5
+045
+aok
+0.3
+0425
+10-10
+0.05
+0.25
+0.3
+?0.04
+0.2
+0.25
+erf
+10-15
+u
+0.03
+T
+TT
+tanh
+tanh
+sinh u
+0.1$
+2
+2
+0.2
+10-20
+150
+250
+100
+200
+N
+0
+50(K > 0), which maps (−∞, +∞) to (0, +∞) in such a way that (i) φ′(t) → 0 double exponen-
+tially as t → −∞ and (ii) φ(t) → t double exponentially as t → +∞. The proposed formula
+changes the variable by x = Mφ(t) to obtain
+I = M
+� +∞
+−∞
+f1(Mφ(t)) sin(Mφ(t))φ′(t)dt,
+to which the trapezoidal rule with equal mesh h is applied, where M and h are chosen to
+satisfy Mh = π. The transformed integrand decays double-exponentially toward t → −∞
+because of the factor φ′(t) and also toward t → +∞ because Mφ(t) for t = kh (sample point
+of the trapezoidal rule) tends double-exponentially to Mt = Mkh = kπ, at which sine function
+vanishes. Another (improved) transformation function
+φ(t) =
+t
+1 − exp(−2t − α(1 − e−t) − β(et − 1)),
+is given in [23], where β = 1/4 and α = β/
+�
+1 + M log(1 + M)/(4π).
+2.4
+IMT rule
+In 1969, prior to the DE formula, a remarkable quadrature formula was proposed by Masao
+Iri, Sigeiti Moriguti, and Yoshimitsu Takasawa [2]. The formula is known today as the “IMT
+rule,” which name was introduced in [35] and used in [1].
+For an integral
+I =
+� 1
+0
+f (x)dx
+over [0, 1], the IMT rule applies the trapezoidal rule to the integral
+I =
+� 1
+0
+f (φ(t))φ′(t)dt
+resulting from the transformation by
+φ(t) = 1
+Q
+� t
+0
+exp
+�
+−
+�1
+τ +
+1
+1 − τ
+��
+dτ,
+where
+Q =
+� 1
+0
+exp
+�
+−
+�1
+τ +
+1
+1 − τ
+��
+dτ
+is a normalizing constant to render φ(1) = 1.
+The transformed integrand g(t) = f (φ(t))φ′(t) has the property that all the derivatives
+g(j)(t) (j = 1, 2, . . .) vanish at t = 0, 1. By the Euler–Maclaurin formula, this indicates that
+the IMT rule should be highly accurate. Indeed, it was shown in [2] via a complex analytic
+method that the error of the IMT rule can be estimated roughly as exp(−C
+√
+N), which is
+much better than N−4 of the Simpson rule, say, but not as good as exp(−CN/ log N) of the DE
+formula. Variants of the IMT rule have been proposed for possible improvement [4, 10, 21,
+29], but it turned out that an IMT-type rule, transforming
+� 1
+0 dx to
+� 1
+0 dt rather than to
+� +∞
+−∞ dt,
+cannot outperform the DE formula.
+5
+
+3
+DE-Sinc methods
+Changing variables is also useful in the Sinc numerical methods. The book [27] of Stenger
+in 1993 describes this methodology to the full extent, focusing on single exponential (SE)
+transformations like φ(t) = tanh(t/2). Use of the double exponential transformation in the
+Sinc numerical methods was initiated by Sugihara [31, 33] around 2000, with subsequent
+development mainly in Japan. Such numerical methods are often called the DE-Sinc methods.
+The subsequent results obtained in the first half of 2000s are described in [5, 7, 34].
+3.1
+Sinc approximation
+The Sinc approximation of a function f (x) over (−∞, ∞) is given by
+f (x) ≈
+N
+�
+k=−N
+f (kh)S (k, h)(x),
+(8)
+where S (k, h)(x) is the so-called Sinc function defined by
+S (k, h)(x) = sin[(π/h)(x − kh)]
+(π/h)(x − kh)
+and the step size h is chosen appropriately, depending on N. The technique of variable trans-
+formation x = φ(t) is also effective in this context. By applying the formula (8) to f (φ(t)) we
+obtain
+f (φ(t)) ≈
+N
+�
+k=−N
+f (φ(kh))S (k, h)(t),
+or equivalently,
+f (x) ≈
+N
+�
+k=−N
+f (φ(kh))S (k, h)(φ−1(x)).
+To approximate f (x) over [0, 1], for example, we choose
+φ(t) = 1
+2 tanh t
+2 + 1
+2,
+(9)
+φ(t) = 1
+2 tanh
+�π
+2 sinh t
+�
++ 1
+2,
+(10)
+etc. The methods using (9) and (10) are often called the SE- and DE-Sinc approximations,
+respectively. The error of the SE-Sinc approximation is roughly exp(−C
+√
+N) and that of the
+DE-Sinc approximation is exp(−CN/ log N).
+These approximation schemes are compared in Fig. 2 (taken from [34]) for function
+f (x) = x1/2(1 − x)3/4
+over [0, 1]. In Fig. 2, “Ordinary-Sinc” means the SE-Sinc approximation using (9), and the
+polynomial interpolation with the Chebyshev nodes is included for comparison.
+Detailed theoretical analyses on the DE-Sinc method can be found in Sugihara [33] as
+well as Tanaka et al. [41] and Okayama et al. [16, 20]. An optimization technique is used to
+improve the DE-Sinc method in Tanaka and Sugihara [39].
+6
+
+M. Sugihara, T. Matsuo / Journal of Computational and Applied Mathematics 164–165 (2004) 673–689
+10−14
+10−12
+10−10
+10−8
+10−6
+10−4
+10−2
+100
+0
+10 20 30 40 50 60 70 80 90 100 110 120
+|ERROR|
+n
+Chebyshev
+Ordinary-Sinc
+DE-Sinc
+3. Errors in the Sinc approximation for the function
+(1
+) and (13
+in the polynomial interpolation with the Chebyshev nodes is also displayed).
+An
+n=
+))
+of the Sinc-collocation method
+We here consider the Sinc-collocation method for the problem whose solution decays double ex-
+on the real line. We can prove the following theorem, which shows that the convergence
+of the Sinc-collocation method is given by O(exp(
+n=
+18
+a unique solution
+),
+is analytic on the real line. Furthermore assume
+A; B; ; ; 
+; 
+in the strip region
+on the real line are
+as follows
+y
+to
+);
+on the real line
+Re
+to
+on the real line
+is
+))
+to
+on the real line
+is
+))
+we have
+−∞¡x¡
++3
+Figure 2:
+Errors in the Sinc approximations for x1/2(1 − x)3/4 using (9) and (10) and the
+Chebyshev interpolation; taken from Sugihara and Matsuo [34, Fig. 3] with permission from
+Elsevier; n in the figure corresponds to N in (8).
+3.2
+Application to other problems
+Once a function approximation scheme is at hand, we can apply it to a variety of numerical
+problems. Indeed this is also the case with the DE-Sinc approximation as follows.
+• Indefinite integration by Muhammad and Mori [8], Tanaka et al. [40], and Okayama
+and Tanaka [19].
+• Initial value problem of differential equations by Nurmuhammad et al. [11] and Okayama
+[15].
+• Boundary value problem of differential equations by Sugihara [32], followed by Nur-
+muhammad et al. [12, 13] and Mori et al. [6].
+• Volterra integral equation by Muhammad et al. [9] and Okayama et al. [18].
+• Fredholm integral equation by Kobayashi et al. [3], Muhammad et al. [9], and Okayama
+et al. [17].
+Acknowledgement. The authors are thankful to Ken’ichiro Tanaka and Tomoaki Okayama
+for their support in writing this article.
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+exponential sinc method, Math. Comput., 74 (2004), 655–679.
+[41] K. Tanaka, M. Sugihara, and K. Murota: Function classes for successful DE-Sinc ap-
+proximations, Math. Comput., 78 (2009), 1553–1571.
+[42] K. Tanaka, M. Sugihara, K. Murota, and M. Mori: Function classes for double expo-
+nential integration formulas, Numer. Math., 111 (2009), 631–655.
+10
+

39AzT4oBgHgl3EQf9f54/content/tmp_files/load_file.txt

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+filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AzT4oBgHgl3EQf9f54/content/2301.01920v1.pdf,len=387
+page_content='arXiv:2301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AzT4oBgHgl3EQf9f54/content/2301.01920v1.pdf'}
+page_content='01920v1  [math.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AzT4oBgHgl3EQf9f54/content/2301.01920v1.pdf'}
+page_content='NA]  5 Jan 2023  Double-Exponential transformation:  A quick review of a Japanese tradition*  Kazuo Murota†and Takayasu Matsuo‡  January 5, 2023  Abstract  This article is a short introduction to numerical methods using the double exponential  (DE) transformation, such as tanh-sinh quadrature and DE-Sinc approximation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AzT4oBgHgl3EQf9f54/content/2301.01920v1.pdf'}
+page_content=' The  DE-based methods for numerical computation have been developed intensively in Japan  and the objective of this article is to describe the history in addition to the underlying  mathematical ideas.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AzT4oBgHgl3EQf9f54/content/2301.01920v1.pdf'}
+page_content='  Keywords: Double exponential transformation, DE integration formula, tanh-sinh quadra-  ture, DE-Sinc method.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AzT4oBgHgl3EQf9f54/content/2301.01920v1.pdf'}
+page_content='  1  Introduction  The double exponential (DE) transformation is a generic name of variable transformations  (changes of variables) used effectively in numerical computation on analytic functions, such  as numerical quadrature and function approximation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AzT4oBgHgl3EQf9f54/content/2301.01920v1.pdf'}
+page_content=' A typical DE transformation is a change  of variable x to another variable t by x = φ(t) with the function  φ(t) = tanh  �π  2 sinh t  �  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AzT4oBgHgl3EQf9f54/content/2301.01920v1.pdf'}
+page_content='  The term “double exponential” refers to the property that the derivative  φ′(t) =  π  2 cosh t  cosh2(π  2 sinh t)  decays double exponentially  φ′(t) ≈ exp  �  −π  2 exp |t|  �  (1)  as |t| → ∞.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AzT4oBgHgl3EQf9f54/content/2301.01920v1.pdf'}
+page_content='  This is a preliminary version of an article to be included in ICIAM 2023, Tokyo Intelligencer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AzT4oBgHgl3EQf9f54/content/2301.01920v1.pdf'}
+page_content='  †The Institute of Statistical Mathematics, Tokyo 190-8562, Japan;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AzT4oBgHgl3EQf9f54/content/2301.01920v1.pdf'}
+page_content=' and Faculty of Economics and Business  Administration, Tokyo Metropolitan University, Tokyo 192-0397, Japan, murota@tmu.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AzT4oBgHgl3EQf9f54/content/2301.01920v1.pdf'}
+page_content='ac.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AzT4oBgHgl3EQf9f54/content/2301.01920v1.pdf'}
+page_content='jp  ‡Department of Mathematical Informatics, Graduate School of Information Science and Technology, Uni-  versity of Tokyo, Tokyo 113-8656, Japan matsuo@mist.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AzT4oBgHgl3EQf9f54/content/2301.01920v1.pdf'}
+page_content='i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AzT4oBgHgl3EQf9f54/content/2301.01920v1.pdf'}
+page_content='u-tokyo.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AzT4oBgHgl3EQf9f54/content/2301.01920v1.pdf'}
+page_content='ac.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AzT4oBgHgl3EQf9f54/content/2301.01920v1.pdf'}
+page_content='jp  1  This article is a short introduction to numerical methods using DE transformations such as  the double exponential formula (tanh-sinh quadrature) for numerical integration and the DE-  Sinc method for function approximation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AzT4oBgHgl3EQf9f54/content/2301.01920v1.pdf'}
+page_content=' The DE-based methods for numerical computation  have been developed intensively in Japan [5, 7, 34, 38] and a workshop titled “Thirty Years of  the Double Exponential Transforms” was held at RIMS (Research Institute for Mathematical  Sciences, Kyoto University) on September 1–3, 2004 [14].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AzT4oBgHgl3EQf9f54/content/2301.01920v1.pdf'}
+page_content=' The objective of this article is to  describe the history of the development of the DE-based methods in addition to the underlying  mathematical ideas.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AzT4oBgHgl3EQf9f54/content/2301.01920v1.pdf'}
+page_content='  This article is written to the memory of Professors Masao Iri (President of Japan SIAM,  1996), Masatake Mori (President of Japan SIAM, 1998), and Masaaki Sugihara (Vice Presi-  dent of Japan SIAM, 2008).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AzT4oBgHgl3EQf9f54/content/2301.01920v1.pdf'}
+page_content='  2  DE formula for numerical integration  The DE formula for numerical integration invented by Hidetosi Takahasi and Masatake Mori  [37] was first presented at the RIMS workshop “Studies on Numerical Algorithms,” held on  October 31–November 2, 1972.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AzT4oBgHgl3EQf9f54/content/2301.01920v1.pdf'}
+page_content=' The celebrated term of “double exponential formula” was  proposed there, as we can see in the proceedings paper [36].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AzT4oBgHgl3EQf9f54/content/2301.01920v1.pdf'}
+page_content='  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AzT4oBgHgl3EQf9f54/content/2301.01920v1.pdf'}
+page_content='1  Quadrature formula  The DE formula was motivated by the fact that the trapezoidal rule is highly effective for  integrals over the infinite interval (−∞, +∞).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AzT4oBgHgl3EQf9f54/content/2301.01920v1.pdf'}
+page_content=' For an integral  I =  � 1  −1  f (x)dx,  for example, we employ a change of variable x = φ(t) using some function φ(t) satisfying  φ(−∞) = −1 and φ(+∞) = 1, and apply the trapezoidal rule to the transformed integral  I =  � +∞  −∞  f (φ(t))φ′(t)dt,  to obtain an infinite sum of discretization  Ih = h  ∞  �  k=−∞  f (φ(kh))φ′(kh).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AzT4oBgHgl3EQf9f54/content/2301.01920v1.pdf'}
+page_content='  (2)  A finite-term approximation to this infinite sum results in an integration formula  I(N)  h  = h  N  �  k=−N  f (φ(kh))φ′(kh).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AzT4oBgHgl3EQf9f54/content/2301.01920v1.pdf'}
+page_content='  (3)  Such combination of the trapezoidal rule with a change of variables was conceived by several  authors [2, 24, 25, 35] around 1970.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AzT4oBgHgl3EQf9f54/content/2301.01920v1.pdf'}
+page_content='  The error I − I(N)  h  of the formula (3) consists of two parts, the error ED ≡ I − Ih incurred  by discretization (2) and the error ET ≡ Ih − I(N)  h  caused by truncation of an infinite sum Ih to  a finite sum I(N)  h .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AzT4oBgHgl3EQf9f54/content/2301.01920v1.pdf'}
+page_content='  2  The major findings of Takahasi and Mori consisted of two ingredients.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AzT4oBgHgl3EQf9f54/content/2301.01920v1.pdf'}
+page_content=' The first was that  the double exponential decay of the transformed integrand f (φ(t))φ′(t) achieves the optimal  balance (or trade-off) between the discretization error ED and the truncation error ET.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AzT4oBgHgl3EQf9f54/content/2301.01920v1.pdf'}
+page_content=' The  second finding was that a concrete choice of  φ(t) = tanh  �π  2 sinh t  �  (4)  is suitable for this purpose thanks to the double exponential decay shown in (1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AzT4oBgHgl3EQf9f54/content/2301.01920v1.pdf'}
+page_content=' With this  particular function φ(t) the formula (3) reads  I(N)  h  = h  N  �  k=−N  f  �  tanh  �π  2 sinh(kh)  ��  (π/2) cosh(kh)  cosh2((π/2) sinh(kh))  ,  which is sometimes called “tanh-sinh quadrature.” The error of this formula is estimated  roughly as  ���I − I(N)  h  ��� ≈ exp(−CN/ log N)  (5)  with some C > 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AzT4oBgHgl3EQf9f54/content/2301.01920v1.pdf'}
+page_content=' The DE formula has an additional feature that it is robust against end-point  singularities of integrands.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AzT4oBgHgl3EQf9f54/content/2301.01920v1.pdf'}
+page_content='  The idea of the DE formula can be applied to integrals over other types of intervals of  integration.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AzT4oBgHgl3EQf9f54/content/2301.01920v1.pdf'}
+page_content=' For example,  I =  � +∞  0  f (x)dx, x = exp  �π  2 sinh t  �  ,  (6)  I =  � +∞  −∞  f (x)dx, x = sinh  �π  2 sinh t  �  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AzT4oBgHgl3EQf9f54/content/2301.01920v1.pdf'}
+page_content='  (7)  Such formulas are also referred to as the double exponential formula.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AzT4oBgHgl3EQf9f54/content/2301.01920v1.pdf'}
+page_content=' The DE formula is  available in Mathematica (NIntegrate), Python library SymPy, Python library mpmath, C++  library Boost, Haskell package integration, etc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AzT4oBgHgl3EQf9f54/content/2301.01920v1.pdf'}
+page_content='  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AzT4oBgHgl3EQf9f54/content/2301.01920v1.pdf'}
+page_content='2  Optimality  Optimality of the DE transformation (4) was discussed already by Takahasi and Mori [37].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AzT4oBgHgl3EQf9f54/content/2301.01920v1.pdf'}
+page_content='  Numerical examples also support its optimality.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AzT4oBgHgl3EQf9f54/content/2301.01920v1.pdf'}
+page_content=' Figure 1 (taken from [5]) shows the compar-  ison of the DE transformation (4) against other transformations  φ(t) = tanh t,  φ(t) = tanh  �π  2 sinh t3�  ,  φ(t) = erf(t) =  2√π  � t  0  exp(−s2)ds  for  � 1  −1  1  (x − 2)(1 − x)1/4(1 + x)3/4 dx  that has integrable singularities at both ends of the interval of integration.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AzT4oBgHgl3EQf9f54/content/2301.01920v1.pdf'}
+page_content=' The DE formula  converges much faster than others.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AzT4oBgHgl3EQf9f54/content/2301.01920v1.pdf'}
+page_content=' It is known that the tanh-rule (using φ(t) = tanh t) has  3  Discovery of the DE Transformation  915  Figure 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AzT4oBgHgl3EQf9f54/content/2301.01920v1.pdf'}
+page_content=' Comparison of the efficiency of several variable transformations for  the integral  dx/  2)(1  (1 +  uations and the ordinate is the absolute error  in logarithmic scale  actually computed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AzT4oBgHgl3EQf9f54/content/2301.01920v1.pdf'}
+page_content=' The number attached to each curve in the figure is the  mesh size  used for actual computation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AzT4oBgHgl3EQf9f54/content/2301.01920v1.pdf'}
+page_content=' Transformation c gives the DE for-  mula.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AzT4oBgHgl3EQf9f54/content/2301.01920v1.pdf'}
+page_content=' From this figure we see that the efficiency becomes higher as the decay  of  is faster, and it attains the highest when the DE transformation is ap-  plied.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AzT4oBgHgl3EQf9f54/content/2301.01920v1.pdf'}
+page_content=' Then, as the decay becomes faster than double exponential the efficiency  turns to be lower.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AzT4oBgHgl3EQf9f54/content/2301.01920v1.pdf'}
+page_content='  Thus, Takahasi and Mori were convinced of the optimality of the DE trans-  formation and presented the result orally at a RIMS symposium in 1973 [79]  and published as a paper in Publ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AzT4oBgHgl3EQf9f54/content/2301.01920v1.pdf'}
+page_content=' RIMS in 1974 [80].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AzT4oBgHgl3EQf9f54/content/2301.01920v1.pdf'}
+page_content='  4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AzT4oBgHgl3EQf9f54/content/2301.01920v1.pdf'}
+page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AzT4oBgHgl3EQf9f54/content/2301.01920v1.pdf'}
+page_content='  Application of the DE transformation to  other types of integrals  The idea of the DE transformation can be applied to various kinds of  integrals.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AzT4oBgHgl3EQf9f54/content/2301.01920v1.pdf'}
+page_content='  Takahasi and Mori gave some examples other than (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AzT4oBgHgl3EQf9f54/content/2301.01920v1.pdf'}
+page_content='8) in their  paper in 1974 [80].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AzT4oBgHgl3EQf9f54/content/2301.01920v1.pdf'}
+page_content=' We list here typical types of integrals and corresponding  Figure 1: Comparison of the efficiency of several variable transformations for the integral  � 1  −1 dx/{(x − 2)(1 − x)1/4(1 + x)3/4};' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AzT4oBgHgl3EQf9f54/content/2301.01920v1.pdf'}
+page_content=' taken from Mori [5, Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AzT4oBgHgl3EQf9f54/content/2301.01920v1.pdf'}
+page_content=' 4] with permission from the  European Mathematical Society;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AzT4oBgHgl3EQf9f54/content/2301.01920v1.pdf'}
+page_content=' u and N in the figure correspond, respectively, to t and  2N + 1 in the present notation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AzT4oBgHgl3EQf9f54/content/2301.01920v1.pdf'}
+page_content='  the (rough) convergence rate exp(−C  √  N), in contrast to exp(−CN/ log N) in (5) of the DE  formula.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AzT4oBgHgl3EQf9f54/content/2301.01920v1.pdf'}
+page_content='  The optimality argument of [37], based on complex function theory, was convincing  enough for the majority of scientists and engineers, but not perfectly satisfactory for theo-  reticians.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AzT4oBgHgl3EQf9f54/content/2301.01920v1.pdf'}
+page_content=' Rigorous mathematical argument for optimality of the DE formula was addressed  by Masaaki Sugihara [28, 29, 30] in the 1980–1990s in a manner comparable to Stenger’s  framework [26] for optimality of the tanh rule.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AzT4oBgHgl3EQf9f54/content/2301.01920v1.pdf'}
+page_content=' It is shown in [30] (also [42]) that the DE  formula is optimal with respect to a certain class (Hardy space) of integrand functions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AzT4oBgHgl3EQf9f54/content/2301.01920v1.pdf'}
+page_content='  In principle, for each class of integrand functions we may be able to find an optimal  quadrature formula, and the optimal formula naturally depends on our choice of the admissi-  ble class of integrands.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AzT4oBgHgl3EQf9f54/content/2301.01920v1.pdf'}
+page_content=' Thus the optimality of a quadrature formula is only relative.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AzT4oBgHgl3EQf9f54/content/2301.01920v1.pdf'}
+page_content=' However,  it was shown by Sugihara that no nontrivial class of integrand functions exists that admits a  quadrature formula with smaller errors than the DE formula.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AzT4oBgHgl3EQf9f54/content/2301.01920v1.pdf'}
+page_content=' We can interpret this fact as the  absolute optimality of the DE formula.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AzT4oBgHgl3EQf9f54/content/2301.01920v1.pdf'}
+page_content='  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AzT4oBgHgl3EQf9f54/content/2301.01920v1.pdf'}
+page_content='3  Fourier-type integrals  For Fourier-type integrals like  I =  � +∞  0  f1(x) sin x dx,  the DE formula like (6) is not very successful.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AzT4oBgHgl3EQf9f54/content/2301.01920v1.pdf'}
+page_content=' To cope with Fourier-type integrals, a novel  technique, in the spirit of DE transformation, was proposed by Ooura and Mori [22, 23].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AzT4oBgHgl3EQf9f54/content/2301.01920v1.pdf'}
+page_content=' In  [22] they proposed to use  φ(t) =  t  1 − exp(−K sinh t)  4  h  口  Q8  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AzT4oBgHgl3EQf9f54/content/2301.01920v1.pdf'}
+page_content='8  10--5  W0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AzT4oBgHgl3EQf9f54/content/2301.01920v1.pdf'}
+page_content='1  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AzT4oBgHgl3EQf9f54/content/2301.01920v1.pdf'}
+page_content='6=h  tanh u  04  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AzT4oBgHgl3EQf9f54/content/2301.01920v1.pdf'}
+page_content='075  K  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AzT4oBgHgl3EQf9f54/content/2301.01920v1.pdf'}
+page_content='5  045  aok  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AzT4oBgHgl3EQf9f54/content/2301.01920v1.pdf'}
+page_content='3  0425  10-10  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AzT4oBgHgl3EQf9f54/content/2301.01920v1.pdf'}
+page_content='05  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AzT4oBgHgl3EQf9f54/content/2301.01920v1.pdf'}
+page_content='25  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AzT4oBgHgl3EQf9f54/content/2301.01920v1.pdf'}
+page_content='3  ?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AzT4oBgHgl3EQf9f54/content/2301.01920v1.pdf'}
+page_content='0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AzT4oBgHgl3EQf9f54/content/2301.01920v1.pdf'}
+page_content='04  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AzT4oBgHgl3EQf9f54/content/2301.01920v1.pdf'}
+page_content='2  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AzT4oBgHgl3EQf9f54/content/2301.01920v1.pdf'}
+page_content='25  erf  10-15  u  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AzT4oBgHgl3EQf9f54/content/2301.01920v1.pdf'}
+page_content='03  T  TT  tanh  tanh  sinh u  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AzT4oBgHgl3EQf9f54/content/2301.01920v1.pdf'}
+page_content='1$  2  2  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AzT4oBgHgl3EQf9f54/content/2301.01920v1.pdf'}
+page_content='2  10-20  150  250  100  200  N  0  50(K > 0), which maps (−∞, +∞) to (0, +∞) in such a way that (i) φ′(t) → 0 double exponen-  tially as t → −∞ and (ii) φ(t) → t double exponentially as t → +∞.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AzT4oBgHgl3EQf9f54/content/2301.01920v1.pdf'}
+page_content=' The proposed formula  changes the variable by x = Mφ(t) to obtain  I = M  � +∞  −∞  f1(Mφ(t)) sin(Mφ(t))φ′(t)dt,  to which the trapezoidal rule with equal mesh h is applied, where M and h are chosen to  satisfy Mh = π.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AzT4oBgHgl3EQf9f54/content/2301.01920v1.pdf'}
+page_content=' The transformed integrand decays double-exponentially toward t → −∞  because of the factor φ′(t) and also toward t → +∞ because Mφ(t) for t = kh (sample point  of the trapezoidal rule) tends double-exponentially to Mt = Mkh = kπ, at which sine function  vanishes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AzT4oBgHgl3EQf9f54/content/2301.01920v1.pdf'}
+page_content=' Another (improved) transformation function  φ(t) =  t  1 − exp(−2t − α(1 − e−t) − β(et − 1)),  is given in [23], where β = 1/4 and α = β/  �  1 + M log(1 + M)/(4π).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AzT4oBgHgl3EQf9f54/content/2301.01920v1.pdf'}
+page_content='  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AzT4oBgHgl3EQf9f54/content/2301.01920v1.pdf'}
+page_content='4  IMT rule  In 1969, prior to the DE formula, a remarkable quadrature formula was proposed by Masao  Iri, Sigeiti Moriguti, and Yoshimitsu Takasawa [2].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AzT4oBgHgl3EQf9f54/content/2301.01920v1.pdf'}
+page_content=' The formula is known today as the “IMT  rule,” which name was introduced in [35] and used in [1].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AzT4oBgHgl3EQf9f54/content/2301.01920v1.pdf'}
+page_content='  For an integral  I =  � 1  0  f (x)dx  over [0, 1], the IMT rule applies the trapezoidal rule to the integral  I =  � 1  0  f (φ(t))φ′(t)dt  resulting from the transformation by  φ(t) = 1  Q  � t  0  exp  �  −  �1  τ +  1  1 − τ  ��  dτ,  where  Q =  � 1  0  exp  �  −  �1  τ +  1  1 − τ  ��  dτ  is a normalizing constant to render φ(1) = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AzT4oBgHgl3EQf9f54/content/2301.01920v1.pdf'}
+page_content='  The transformed integrand g(t) = f (φ(t))φ′(t) has the property that all the derivatives  g(j)(t) (j = 1, 2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AzT4oBgHgl3EQf9f54/content/2301.01920v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AzT4oBgHgl3EQf9f54/content/2301.01920v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AzT4oBgHgl3EQf9f54/content/2301.01920v1.pdf'}
+page_content=') vanish at t = 0, 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AzT4oBgHgl3EQf9f54/content/2301.01920v1.pdf'}
+page_content=' By the Euler–Maclaurin formula, this indicates that  the IMT rule should be highly accurate.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AzT4oBgHgl3EQf9f54/content/2301.01920v1.pdf'}
+page_content=' Indeed, it was shown in [2] via a complex analytic  method that the error of the IMT rule can be estimated roughly as exp(−C  √  N), which is  much better than N−4 of the Simpson rule, say, but not as good as exp(−CN/ log N) of the DE  formula.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AzT4oBgHgl3EQf9f54/content/2301.01920v1.pdf'}
+page_content=' Variants of the IMT rule have been proposed for possible improvement [4, 10, 21,  29], but it turned out that an IMT-type rule, transforming  � 1  0 dx to  � 1  0 dt rather than to  � +∞  −∞ dt,  cannot outperform the DE formula.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AzT4oBgHgl3EQf9f54/content/2301.01920v1.pdf'}
+page_content='  5  3  DE-Sinc methods  Changing variables is also useful in the Sinc numerical methods.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AzT4oBgHgl3EQf9f54/content/2301.01920v1.pdf'}
+page_content=' The book [27] of Stenger  in 1993 describes this methodology to the full extent, focusing on single exponential (SE)  transformations like φ(t) = tanh(t/2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AzT4oBgHgl3EQf9f54/content/2301.01920v1.pdf'}
+page_content=' Use of the double exponential transformation in the  Sinc numerical methods was initiated by Sugihara [31, 33] around 2000, with subsequent  development mainly in Japan.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AzT4oBgHgl3EQf9f54/content/2301.01920v1.pdf'}
+page_content=' Such numerical methods are often called the DE-Sinc methods.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AzT4oBgHgl3EQf9f54/content/2301.01920v1.pdf'}
+page_content='  The subsequent results obtained in the first half of 2000s are described in [5, 7, 34].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AzT4oBgHgl3EQf9f54/content/2301.01920v1.pdf'}
+page_content='  3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AzT4oBgHgl3EQf9f54/content/2301.01920v1.pdf'}
+page_content='1  Sinc approximation  The Sinc approximation of a function f (x) over (−∞, ∞) is given by  f (x) ≈  N  �  k=−N  f (kh)S (k, h)(x),  (8)  where S (k, h)(x) is the so-called Sinc function defined by  S (k, h)(x) = sin[(π/h)(x − kh)]  (π/h)(x − kh)  and the step size h is chosen appropriately, depending on N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AzT4oBgHgl3EQf9f54/content/2301.01920v1.pdf'}
+page_content=' The technique of variable trans-  formation x = φ(t) is also effective in this context.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AzT4oBgHgl3EQf9f54/content/2301.01920v1.pdf'}
+page_content=' By applying the formula (8) to f (φ(t)) we  obtain  f (φ(t)) ≈  N  �  k=−N  f (φ(kh))S (k, h)(t),  or equivalently,  f (x) ≈  N  �  k=−N  f (φ(kh))S (k, h)(φ−1(x)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AzT4oBgHgl3EQf9f54/content/2301.01920v1.pdf'}
+page_content='  To approximate f (x) over [0, 1], for example, we choose  φ(t) = 1  2 tanh t  2 + 1  2,  (9)  φ(t) = 1  2 tanh  �π  2 sinh t  �  + 1  2,  (10)  etc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AzT4oBgHgl3EQf9f54/content/2301.01920v1.pdf'}
+page_content=' The methods using (9) and (10) are often called the SE- and DE-Sinc approximations,  respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AzT4oBgHgl3EQf9f54/content/2301.01920v1.pdf'}
+page_content=' The error of the SE-Sinc approximation is roughly exp(−C  √  N) and that of the  DE-Sinc approximation is exp(−CN/ log N).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AzT4oBgHgl3EQf9f54/content/2301.01920v1.pdf'}
+page_content='  These approximation schemes are compared in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AzT4oBgHgl3EQf9f54/content/2301.01920v1.pdf'}
+page_content=' 2 (taken from [34]) for function  f (x) = x1/2(1 − x)3/4  over [0, 1].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AzT4oBgHgl3EQf9f54/content/2301.01920v1.pdf'}
+page_content=' In Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AzT4oBgHgl3EQf9f54/content/2301.01920v1.pdf'}
+page_content=' 2, “Ordinary-Sinc” means the SE-Sinc approximation using (9), and the  polynomial interpolation with the Chebyshev nodes is included for comparison.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AzT4oBgHgl3EQf9f54/content/2301.01920v1.pdf'}
+page_content='  Detailed theoretical analyses on the DE-Sinc method can be found in Sugihara [33] as  well as Tanaka et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AzT4oBgHgl3EQf9f54/content/2301.01920v1.pdf'}
+page_content=' [41] and Okayama et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AzT4oBgHgl3EQf9f54/content/2301.01920v1.pdf'}
+page_content=' [16, 20].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AzT4oBgHgl3EQf9f54/content/2301.01920v1.pdf'}
+page_content=' An optimization technique is used to  improve the DE-Sinc method in Tanaka and Sugihara [39].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AzT4oBgHgl3EQf9f54/content/2301.01920v1.pdf'}
+page_content='  6  M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AzT4oBgHgl3EQf9f54/content/2301.01920v1.pdf'}
+page_content=' Sugihara, T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AzT4oBgHgl3EQf9f54/content/2301.01920v1.pdf'}
+page_content=' Matsuo / Journal of Computational and Applied Mathematics 164–165 (2004) 673–689  10−14  10−12  10−10  10−8  10−6  10−4  10−2  100  0  10 20 30 40 50 60 70 80 90 100 110 120  |ERROR|  n  Chebyshev  Ordinary-Sinc  DE-Sinc  3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AzT4oBgHgl3EQf9f54/content/2301.01920v1.pdf'}
+page_content=' Errors in the Sinc approximation for the function  (1  ) and (13  in the polynomial interpolation with the Chebyshev nodes is also displayed).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AzT4oBgHgl3EQf9f54/content/2301.01920v1.pdf'}
+page_content='  An  n=  ))  of the Sinc-collocation method  We here consider the Sinc-collocation method for the problem whose solution decays double ex-  on the real line.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AzT4oBgHgl3EQf9f54/content/2301.01920v1.pdf'}
+page_content=' We can prove the following theorem, which shows that the convergence  of the Sinc-collocation method is given by O(exp(  n=  18  a unique solution  ),  is analytic on the real line.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AzT4oBgHgl3EQf9f54/content/2301.01920v1.pdf'}
+page_content=' Furthermore assume  A;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AzT4oBgHgl3EQf9f54/content/2301.01920v1.pdf'}
+page_content=' B;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AzT4oBgHgl3EQf9f54/content/2301.01920v1.pdf'}
+page_content='  ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AzT4oBgHgl3EQf9f54/content/2301.01920v1.pdf'}
+page_content='  ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AzT4oBgHgl3EQf9f54/content/2301.01920v1.pdf'}
+page_content='  \x16;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AzT4oBgHgl3EQf9f54/content/2301.01920v1.pdf'}
+page_content=' \x16  in the strip region  on the real line are  as follows  \x17y  to  );' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AzT4oBgHgl3EQf9f54/content/2301.01920v1.pdf'}
+page_content='  on the real line  Re  to  on the real line  is  ))  to  on the real line  is  ))  we have  −∞¡x¡  +3  Figure 2:  Errors in the Sinc approximations for x1/2(1 − x)3/4 using (9) and (10) and the  Chebyshev interpolation;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AzT4oBgHgl3EQf9f54/content/2301.01920v1.pdf'}
+page_content=' taken from Sugihara and Matsuo [34, Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AzT4oBgHgl3EQf9f54/content/2301.01920v1.pdf'}
+page_content=' 3] with permission from  Elsevier;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AzT4oBgHgl3EQf9f54/content/2301.01920v1.pdf'}
+page_content=' n in the figure corresponds to N in (8).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AzT4oBgHgl3EQf9f54/content/2301.01920v1.pdf'}
+page_content='  3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AzT4oBgHgl3EQf9f54/content/2301.01920v1.pdf'}
+page_content='2  Application to other problems  Once a function approximation scheme is at hand, we can apply it to a variety of numerical  problems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AzT4oBgHgl3EQf9f54/content/2301.01920v1.pdf'}
+page_content=' Indeed this is also the case with the DE-Sinc approximation as follows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AzT4oBgHgl3EQf9f54/content/2301.01920v1.pdf'}
+page_content='  Indefinite integration by Muhammad and Mori [8], Tanaka et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AzT4oBgHgl3EQf9f54/content/2301.01920v1.pdf'}
+page_content=' [40], and Okayama  and Tanaka [19].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AzT4oBgHgl3EQf9f54/content/2301.01920v1.pdf'}
+page_content='  Initial value problem of differential equations by Nurmuhammad et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AzT4oBgHgl3EQf9f54/content/2301.01920v1.pdf'}
+page_content=' [11] and Okayama  [15].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AzT4oBgHgl3EQf9f54/content/2301.01920v1.pdf'}
+page_content='  Boundary value problem of differential equations by Sugihara [32], followed by Nur-  muhammad et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AzT4oBgHgl3EQf9f54/content/2301.01920v1.pdf'}
+page_content=' [12, 13] and Mori et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AzT4oBgHgl3EQf9f54/content/2301.01920v1.pdf'}
+page_content=' [6].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AzT4oBgHgl3EQf9f54/content/2301.01920v1.pdf'}
+page_content='  Volterra integral equation by Muhammad et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AzT4oBgHgl3EQf9f54/content/2301.01920v1.pdf'}
+page_content=' [9] and Okayama et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AzT4oBgHgl3EQf9f54/content/2301.01920v1.pdf'}
+page_content=' [18].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AzT4oBgHgl3EQf9f54/content/2301.01920v1.pdf'}
+page_content='  Fredholm integral equation by Kobayashi et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AzT4oBgHgl3EQf9f54/content/2301.01920v1.pdf'}
+page_content=' [3], Muhammad et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AzT4oBgHgl3EQf9f54/content/2301.01920v1.pdf'}
+page_content=' [9], and Okayama  et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AzT4oBgHgl3EQf9f54/content/2301.01920v1.pdf'}
+page_content=' [17].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AzT4oBgHgl3EQf9f54/content/2301.01920v1.pdf'}
+page_content='  Acknowledgement.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AzT4oBgHgl3EQf9f54/content/2301.01920v1.pdf'}
+page_content=' The authors are thankful to Ken’ichiro Tanaka and Tomoaki Okayama  for their support in writing this article.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AzT4oBgHgl3EQf9f54/content/2301.01920v1.pdf'}
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+page_content=' Kobayashi, H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AzT4oBgHgl3EQf9f54/content/2301.01920v1.pdf'}
+page_content=' Okamoto, and J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AzT4oBgHgl3EQf9f54/content/2301.01920v1.pdf'}
+page_content=' Zhu: Numerical computation of water and solitary  waves by the double exponential transform, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AzT4oBgHgl3EQf9f54/content/2301.01920v1.pdf'}
+page_content=' Comput.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AzT4oBgHgl3EQf9f54/content/2301.01920v1.pdf'}
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+page_content=' Math.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AzT4oBgHgl3EQf9f54/content/2301.01920v1.pdf'}
+page_content=', 152 (2003), 229–  241.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AzT4oBgHgl3EQf9f54/content/2301.01920v1.pdf'}
+page_content='  [4] M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AzT4oBgHgl3EQf9f54/content/2301.01920v1.pdf'}
+page_content=' Mori: An IMT-type double exponential formula for numerical integration, Publ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AzT4oBgHgl3EQf9f54/content/2301.01920v1.pdf'}
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+Pseudopotential Bethe-Salpeter calculations for shallow-core x-ray absorption
+near-edge structures: excitonic effects in α-Al2O3
+M. Laura Urquiza,1, 2 Matteo Gatti,1, 2, 3 and Francesco Sottile1, 2
+1LSI, CNRS, CEA/DRF/IRAMIS, École Polytechnique, Institut Polytechnique de Paris, F-91120 Palaiseau, France
+2European Theoretical Spectroscopy Facility (ETSF)
+3Synchrotron SOLEIL, L’Orme des Merisiers, Saint-Aubin, BP 48, F-91192 Gif-sur-Yvette, France
+(Dated: January 12, 2023)
+We present an ab initio description of optical and X-ray absorption spectroscopies, in a unified
+formalism based on the pseudopotential plane-wave method at the level of the Bethe-Salpeter Equa-
+tion (BSE) within Green’s functions theory. We show that norm-conserving pseudopotentials are
+very reliable and accurate not only for valence, but also for semi-core electron absorption spectra.
+In order to validate our approach, we compare BSE results with two codes: EXC, based on pseu-
+dopotentials, and Exciting, an all-electron full-potential code. We take corundum α-Al2O3 as an
+example, a prototypical system that presents strong electron-hole interaction in both valence and
+core electron excitations. We analyze the optical, as well as the L1 and L2,3 edges, in detail in terms
+of anisotropy, crystal local fields, interference and excitonic effects. We conclude with a thorough
+inspection of the origin and localization of bright and dark excitons.
+I.
+INTRODUCTION
+X-ray absorption spectroscopy (XAS) and optical ab-
+sorption are complementary techniques to determine ma-
+terials properties. In optical absorption, valence electrons
+are excited into unoccupied conduction states across the
+band gap (or the Fermi energy in metals). Their excita-
+tions determine the color (or the transparency) of materi-
+als and are crucial to many materials properties and func-
+tionalities, spanning from optoelectronics to solar energy
+conversion and storage. In XAS, promoted to unoccu-
+pied conduction bands are instead core electrons, tightly
+bound to the nuclei. X-ray absorption near-edge struc-
+tures (XANES), also known as near-edge X-ray absorp-
+tion fine structure (NEXAFS), being element specific,
+is a probe of the atomic environment, giving structural
+and chemical information1. In the simplest independent-
+particle picture, XANES spectra are proportional to the
+unoccupied density of states, projected on the absorbing
+atom and the angular momentum component that is se-
+lected by dipole selection rules, whereas optical spectra
+can be interpreted on the basis of the joint density of
+states of valence and conduction bands. In both spectro-
+scopies, the interaction between the excited electron and
+the hole left behind can strongly alter this independent-
+particle picture. Indeed, the electron-hole attraction can
+give rise to excitons, i.e bound electron-hole pairs, lead-
+ing to a transfer of spectral weight to lower energies in
+the spectra, including the formation of sharp peaks at
+their onset.
+Given the importance of XANES spectroscopy, sev-
+eral theoretical methods have been developed to interpret
+the measured spectra in solids, taking care of core-hole
+effects at different levels of approximation2. The most
+efficient approaches are, on one side, multiple scattering
+methods3–8, and, on the other side, multiplet models9–11.
+While the former usually neglect the electronic interac-
+tions, the latter are often semi-empirical (i.e., not entirely
+parameter-free) and generally neglect solid-state effects,
+being a many-body solution of finite-cluster models.
+Since the excitations of the core electrons are localised at
+the absorbing atoms, delta-self-consistent-field (∆SCF)
+methods can be also employed, nowadays usually within
+first-principles density-functional theory12–20. The core-
+excited atom is treated as an impurity in a supercell ap-
+proach, and the presence of the core hole is taken into ac-
+count in different ways, from the Z+1 approximation21,22
+(the absorbing atom is assumed to have one additional
+nuclear charge), to the half core-hole approximation23,24
+(also known as Slater’s transition-state method) or the
+full core-hole approximation (the electron removed from
+the core is put at lowest conduction band, or ionized).
+Alternatively, XANES excitation spectra can be directly
+obtained within linear-response theory25,26, which is the
+standard approach for valence excitations and optical
+spectra as well27. In this case, two possible options are
+time-dependent density-functional theory28–30 (TDDFT)
+and the Bethe-Salpeter equation31–35 (BSE) of Green’s
+function theory36,37. Since TDDFT lacks of efficient ap-
+proximations for describing accurately excitonic effects in
+solids38, the BSE, even though computationally more ex-
+pensive, is usually more reliable27. In the present work,
+the solution of the BSE will therefore be also our pre-
+ferred choice to simulate valence and shallow-core exci-
+tation spectra within the same formalism.
+In the simulation of core excitation spectra, the in-
+tuitive technique to represent the single-particle wave
+functions are all-electron methods. They explicitly deal
+with core electrons in extended materials by partitioning
+the space into interstitial and muffin-tin (MT) regions,
+where wave functions are described differently according
+to their localisation degree39–42. Instead, methods that
+are based on plane-wave expansions cannot deal explic-
+itly with the quickly oscillatory behavior of core elec-
+trons, tightly localised near the nuclei, which are instead
+generally taken into account effectively through the de-
+sign of suitable pseudopotentials43. Plane-wave methods
+arXiv:2301.04199v1  [cond-mat.mtrl-sci]  10 Jan 2023
+
+2
+are computationally cheaper and new theoretical devel-
+opments are easier to implement in plane-waves computer
+codes. Moreover, the separation between core electrons,
+kept frozen, and valence electrons, treated explicitly, is
+often not rigid. Between valence and deep core electrons,
+there are often also shallow core (or semicore) electrons,
+which in the pseudopotential framework can be in princi-
+ple also treated as valence electrons, although at a price of
+higher computational cost. However, in all the cases, the
+pseudopotential formalism also introduces an important
+approximation, requiring a pseudization of the valence
+wave functions near the nuclei that make them smoother
+and node free. In the recent past, much work has been de-
+voted to assess pseudopotential calculations for excited-
+state properties with respect to all-electron methods, no-
+tably for self-energy calculations of quasiparticle band
+structure energies44–51. In the present work, we directly
+address the question of the validity of the pseudopotential
+approximation for XANES spectra of shallow-core edges
+(i.e., for electron binding energies smaller than ∼180 eV),
+investigating the limits of use of pseudo wave functions
+for shallow core states in many-body BSE calculations.
+It is clear that the description of deep core levels will
+be always out of reach for plane-wave basis methods.
+However, the high plane-wave cutoff required by semi-
+core states can be now alleviated by the new generation
+of ultrasoft norm-conserving pseudopotentials52. Besides
+the promised lower computational cost for shallower core
+levels, an advantage of pseudopotential plane-wave calcu-
+lations with respect to all-electron methods is that they
+do not make any hypothesis concerning the localisation
+of the core hole inside the muffin tin53.
+In particular, here we investigate the effects of the
+electron-hole interactions on the optical absorption and
+shallow-core XANES spectra of alumina. α-Al2O3 is a
+wide-gap insulator, with many possible applications as a
+structural ceramic (e.g. as a replacement to SiO2 gate ox-
+ide technology) and optical material (also thanks to the
+high-damage threshold for UV laser applications), and
+a prototypical system to investigate core-hole effects in
+XANES spectroscopy12,54–59.
+The article is organised as follows. After a short de-
+scription of the employed methodology in Sec. II, com-
+prising a review of the theoretical background (Sec. II A)
+and a summary of the computational details (Sec. II B),
+Sec. III presents the results of the calculations together
+with their analysis. In Sec. III B pseudopotential cal-
+culations are assessed with respect to all-electron bench-
+marks for both optical and Al L2,3 XANES spectra, while
+Sec. III C contains a discussion on the issue of the core-
+hole localisation in the muffin tin for the Al L1 XANES
+spectrum.
+Sec.
+III D compares the calculated spectra
+with available experiments and analyses the effects of the
+electron-hole interactions on the spectra.
+Finally, Sec.
+IV draws the conclusions summarizing the results of the
+work.
+II.
+METHODOLOGY
+A.
+Theoretical background
+In the framework of Green’s function theory36, the
+Bethe-Salpeter equation (BSE) yields the density re-
+sponse function from the solution of a Dyson-like equa-
+tion for the two-particle correlation function60. In the
+GW approximation (GWA) to the self-energy61, with
+a statically screened Coulomb interaction W, the BSE
+takes the form of an excitonic Hamiltonian27 in the basis
+|vck⟩ of transitions between occupied vk and unoccupied
+bands ck (i.e., uncorrelated electron-hole pairs):
+⟨vck|Hexc|v′c′k′⟩ = Evckδvv′δcc′δkk′+⟨vck|¯vc−W|v′c′k′⟩.
+(1)
+Here Evck = Eck − Evk are the interband transition en-
+ergies calculated in the GWA, while ¯vc is the Coulomb
+interaction without its macroscopic component (i.e., the
+component G = 0 in reciprocal space).
+The stati-
+cally screened Coulomb interaction W = ϵ−1vc is usu-
+ally calculated adopting the random-phase approxima-
+tion (RPA) for the inverse dielectric function ϵ−1.
+The GWA-BSE is nowadays the state-of-the-art ap-
+proach for the simulation, interpretation and prediction
+of optical spectra in solids36,37,62–64, and is more and
+more used also for the simulation of core-level excita-
+tion spectra2,65–81. A great advantage of theory with re-
+spect to experiments is the possibility to separately sup-
+press (or activate) the various interactions at play in the
+materials, which allows one to single out their specific
+effect on the spectra and the materials properties. By
+setting to zero the two electron-hole interactions, ¯vc and
+−W, the excitonic Hamiltonian (1) reduces to a diago-
+nal matrix and corresponds to the independent-particle
+approximation (IPA). By switching on the electron-hole
+exchange interaction ¯vc in Eq.
+(1), one retrieves the
+RPA. With respect to the IPA, the RPA includes the
+so-called crystal local field effects. They are related to
+the inhomogeneous charge response of materials through
+the induced microscopic Hartree potentials counteract-
+ing the external perturbations. Finally, by also switch-
+ing on the electron-hole direct interaction −W, the full
+BSE (1) describes excitonic effects, which are due to the
+electron-hole attraction.82 The electron-hole interactions
+contributing to the off-diagonal matrix elements of the
+BSE (1) give rise to a mixing of the independent-particle
+transitions, which is formally obtained from the solution
+of the eigenvalue equation for the excitonic hamiltonian:
+HexcAλ = EλAλ.
+The absorption spectra, expressed both in the opti-
+cal and XANES regimes by the imaginary part of the
+macroscopic dielectric function, ImϵM(ω), in the long
+wavelength limit q → 0,in the so-called Tamm-Dancoff
+approximation can be directly written in terms of eigen-
+vectors Aλ and eigenvalues Eλ of the BSE Hamiltonian
+
+3
+(1) as:
+ImϵM(ω) = lim
+q→0
+8π2
+Ωq2
+�
+λ
+�����
+�
+vck
+Avck
+λ
+˜ρvck(q)
+�����
+2
+δ(ω − Eλ),
+(2)
+where
+Ω
+is
+the
+crystal
+volume,
+and
+˜ρvck(q)
+=
+�
+ϕ∗
+vk−q(r)e−iq·rϕck(r)dr are the independent-particle
+oscillator strengths. Here the single-particle orbitals ϕi
+are usually Kohn-Sham orbitals. If the exciton energy Eλ
+is smaller than the smallest independent-particle transi-
+tion energy Evck, the exciton λ is said to be bound: the
+difference between Evck and Eλ is its binding energy.
+The contribution of each exciton λ to the spectrum can
+be analysed by introducing the cumulative function:
+Sλ(ω) = lim
+q→0
+8π
+Ωq2
+�����
+Evck<ω
+�
+vck
+Avck
+λ
+˜ρvck(q)
+�����
+2
+.
+(3)
+Since the eigenvectors Aλ and the oscillator strengths
+˜ρ(q) are both complex quantities, the cumulative func-
+tion (3) is not a monotonic function of ω.
+The limit
+Sλ(ω → ∞) is the oscillator strength of the exciton λ
+in the absorption spectrum. If it is negligibly small, the
+exciton is said to be dark, otherwise it is called a bright
+exciton, for it contributes to the spectrum. Even in the
+q → 0, the oscillator strengths ˜ρ(q) depends on the di-
+rection of q, so each exciton λ can at the same time be a
+bright exciton in one polarization direction and dark in
+another.
+Finally, the investigation of the electron-hole correla-
+tion function for each exciton λ,
+Ψλ(rh, re) =
+�
+vck
+Avck
+λ
+φ∗
+vk(rh)φck(re),
+(4)
+gives information about the localisation in real space of
+the electron-hole pair, which results from the electron-
+hole attraction. Assuming that the hole is in a specific
+position rh = r0
+h, one can visualize the corresponding
+density distribution of the electron |Ψλ(r0
+h, re)|2.
+B.
+Computational details
+We have performed calculations using both a pseu-
+dopotential
+(PP)
+plane-wave
+method
+and
+a
+full-
+potential all-electron (AE) linearized augmented plane-
+wave method. AE calculations have been done in partic-
+ular to assess the validity of PP calculations for the core-
+level excitations (see Sec.
+III B). The converged BSE
+absorption spectra and their analysis (see Sec.
+III D)
+have been then obtained in the PP framework. In the
+pseudopotential case, we have used the Abinit code83
+for the ground-state and screening calculations, and the
+EXC code84 for the BSE calculations. In the all-electron
+case, we have used the Exciting code85 for obtaining all
+the benchmark results.
+The Kohn-Sham ground-state calculations have been
+performed within the local density approximation86
+(LDA).
+We
+have
+employed
+norm-conserving
+Troullier-
+Martins87
+(TM)
+and
+optimized
+norm-conserving
+Vanderbilt52,88
+(ONCVPSP)
+pseudopotentials.
+In
+particular, for the absorption spectra a special TM
+pseudopotential89 treating also Al 2s and 2p states as
+valence electrons has been used.
+Calculation with the
+ONCVPSP pseudopotential converged with 42 Hartree
+cutoff of the plane-wave expansion, while the hard TM
+pseudopotential required 320 Hartree.
+The statically screened Coulomb interaction W has
+been obtained (within the RPA) with the ONCVPSP
+pseudopotential (without Al 2s and 2p core levels), in-
+cluding 100 bands, and with a cutoff of 8 and 14.7 Hartree
+for the Kohn-Sham wave functions for the optical and
+shallow-core excitations, respectively.
+The size of the
+screening matrix in the plane-wave basis was 6 Hartree
+for the optical and 8 Hartree for the core spectrum. We
+have verified that, contrary to calculations of the screened
+interaction for other materials like silicon50 or simple
+metals90–92, the effect of core polarization is negligible
+in α-Al2O3.
+In the all-electron results, the ground-state calcula-
+tions were performed using a plane wave cutoff, RMT|G+
+k|max, of 18 Hartree and muffin-tin (MT) spheres RMT
+of 2a0 and 1.45a0 for aluminum and oxygen, respectively.
+The RPA screening was obtained with 100 conduction
+bands and a cutoff in the matrix size of 5 Hartree (main-
+taining the same cutoff of the ground state for the plane
+waves).
+The GW band structure has been approximated within
+a scissor correction model. The LDA conduction bands
+have been rigidly shifted upwards by 2.64 eV, which cor-
+responds to the band gap correction obtained within
+the perturbative G0W0 scheme by Marinopoulos and
+Grüning93.
+The BSE calculations for the absorption spectra have
+been performed with shifted k-point grids (i.e., not con-
+taining high-symmetry k points), which allowed for a
+quicker convergence of the spectra63.
+The optical ab-
+sorption spectrum converged with a 10×10×10 k-point
+grid, while the XANES spectra at the Al L2,3 and L1
+edges converged with a 8×8×8 k-point grid. The exciton
+analysis and plot have been instead done with a smaller
+Γ-centered 4×4×4 k-point grid.
+The BSE spectra for the optical spectrum or the
+XANES spectra at the Al L2,3 and L1 edges had a dif-
+ferent convergence rate with respect to the number of
+empty bands considered in the BSE hamiltonian. Fig. 1
+shows their convergence study (carried out here with a re-
+duced number of k points in a Γ-centered 2×2×2 k-point
+grid).
+While the optical spectrum (left panel) quickly
+converges with the number of empty bands, the XANES
+spectra (middle and right panels) require many more
+empty bands, also to converge the lowest energy peak.
+In the converged spectra, obtained with many more k
+
+4
+FIG. 1: Convergence of BSE absorption spectra with the number of unoccupied conduction bands (cb). (Left)
+Optical spectrum. (Middle) XANES at L2,3 edge. (Right) XANES at L1 edge.
+points, this slow convergence is partially attenuated by
+the fact that the spectra become smoother. The opti-
+cal absorption spectra have been thus obtained with 12
+valence bands and 12 unoccupied bands. The XANES
+spectra at the L2,3 and L1 edges included all the corre-
+sponding core levels together with 30 unoccupied bands.
+A 0.1 eV Lorentzian broadening has been applied to the
+spectra.
+In the all-electron BSE calculations, we considered the
+same parameters used in the calculation of the screen-
+ing: 9 Hartree for the wavefunction cutoff and 5 Hartree
+to describe the electron-hole terms. In the pseudopoten-
+tial BSE calculations, we have used a 30 Hartree cut-
+off for the Kohn-Sham wavefunctions expansion and 7.3
+Hartree for the plane-wave representation of the electron-
+hole interactions.
+We note that, as usual (see e.g.94),
+the plane-wave cutoffs for the BSE matrixelements can
+be significantly reduced with respect to the high cutoff
+needed for the ground-state calculation. Therefore, even
+for pseudopotential BSE calculations of shallow-core ex-
+citations, the limiting factor remains the large size of the
+BSE hamiltonian (1) in extended systems, which is given
+by the number of electron-hole transitions (i.e., the num-
+ber of occupied bands × the number of unoccupied bands
+× the number of k points in the full Brillouin zone).
+III.
+RESULTS
+A.
+Crystal and electronic structure of α-Al2O3
+The crystal structure of corundum α-Al2O3 is trigo-
+nal (see Fig. 2). In the primitive rhombohedral unit cell
+(space group R¯3c, number 167) there are two formula
+units.
+The corundum structure can also be viewed as
+a hexagonal cell containing six formula units with alter-
+nate layers of Al and O atoms in planes perpendicular
+to the hexagonal cH axis. In the α-Al2O3 structure all
+Al atoms occupy octahedral sites coordinated with 6 O
+atoms, which form two equilateral triangles located re-
+spectively slightly above and below each Al atom along
+the cH direction.
+FIG. 2: Primitive rhombohedral unit cell of the crystal
+structure of α-Al2O3. Red and grey balls represent O
+and Al atoms, respectively. The Al atoms are aligned
+along the cartesian z axis, which is the vertical direction
+in the figure, while the O atoms belong to the xy planes
+perpendicular to it.
+We adopted the experimental lattice parameters from
+Ref.95: aH = bH = 4.7589 Å and cH = 12.991 Å in the
+hexagonal unit cell, which corresponds to aR = 5.128 Å
+and α = 55.287◦ in the rhombohedral primitive cell. In
+the reference frame used in the simulations, the hexago-
+nal cH axis is aligned along the cartesian z axis, which is
+the vertical direction in Fig. 2.
+The left panel of Fig. 3 shows the Kohn-Sham LDA
+band structure along a high symmetry path in the first
+Brillouin zone, together with the projected density of
+states on the O (middle panel) and Al (right panel)
+atoms. α-Al2O3 has a direct bandgap at the Γ point,
+
+a
+C5
+FIG. 3: (Left) LDA Kohn-Sham band structure of
+α-Al2O3. The top of the valence band has been set to
+zero. Density of states projected on (middle) O and
+(right) Al atoms, resolved in the angular components: s
+(red), p (blue) and d (green).
+which amounts to 6.21 eV in the LDA. This value is
+in very good agreement with the result of Ref. 96 ob-
+tained with the same experimental lattice parameters.
+Calculations93,96–98 that adopt a crystal structure opti-
+mised within the LDA, rather than the experimental one,
+instead obtain larger band gaps. In particular, the dif-
+ference with respect to Ref. 93 is 0.51 eV. We refer to
+Ref. 98 for a detailed analysis of the dependence on the
+band gap on the lattice parameters. As usual, the Kohn-
+Sham band gap underestimates the experimental funda-
+mental gap, estimated to be 9.57 eV from temperature-
+dependent vacuum ultraviolet (VUV) spectroscopy55 and
+9.6 eV from conductivity measurements99.
+The 6 bands located between -19 eV and -15.9 eV are
+the O 2s states, while the upper 18 valence bands, start-
+ing at ∼ -7 eV, are mostly due to O 2p states, partially
+hybridised with Al states. The valence bands are quite
+flat along the entire path. The bottom conduction band
+consists of Al 3s hybridised with O 3s at the Γ point and
+also with O 2p elsewhere, showing a strong dispersion
+around the Γ point. The higher conduction bands have
+mainly Al 3p and 3d character, also hybridised with O
+states.
+This overview of the electronic properties con-
+firms the intermediate covalent-ionic nature of the chem-
+ical bond in α-Al2O3.
+Finally, the Al 2p and 2s core levels (not shown in
+Fig. 3) in LDA are located 61.7 eV and 99.4 eV below
+the top valence, which, as usual, largely underestimates
+the experimental values100 of 70.7 eV and 115.6 eV, re-
+spectively. The calculations do not include the spin-orbit
+coupling, so the 2p1/2 and 2p3/2 levels are not split. In
+all cases, we have verified that pseudopotential and all-
+electron calculations give the same band structures and
+core-level energies.
+B.
+All-electron benchmark
+One of the main goals of this work is to demonstrate
+that shallow core spectra can be calculated with high
+accuracy using the pseudopotential (PP) approximation.
+The importance of this objective is underlined by the
+many works in the same spirit101–104. However, at vari-
+ance with previous works that concern tests on ground-
+state properties, mostly related to total-energy calcula-
+tions, here we aim at a much more stringent test, which
+involves occupied (both valence and semi-core) and unoc-
+cupied states. The latter could be in particular affected
+by the presence of ghost states105, which could jeopardize
+completely the excitation spectrum, while leaving unaf-
+fected a total energy calculation. Therefore, in order to
+validate the optical and core spectra calculated with PPs,
+we benchmark the results with full-potential all-electron
+(AE) calculations, considered as a gold-standard method
+for solving DFT in extended systems85,106. In order to
+perform this comparison properly, for both optical and
+L2,3 edge absorption spectra the same choice of valence
+electrons is made in the two calculations, and the num-
+ber of plane wave was converged consistently in the two
+cases.
+The valence and L2,3 spectra obtained at different lev-
+els of approximations, IPA, RPA and BSE, are shown in
+the top and bottom panels of Fig. 4, respectively. We
+can make several observations: i) The results of the left
+panels of Fig.4 show that the pseudopotential approxi-
+mation reproduces the all-electron spectra with excellent
+accuracy within the IPA. ii) For the RPA spectrum (cen-
+tral panels) we find a similar result. This is in part related
+to the fact that local fields effects are not important in
+the energy ranges considered. iii) Finally, also the BSE
+calculations with the two approaches are in very good
+agreement.
+Recent comparisons81 between all-electron
+and projected augmented wave method approaches, for
+instance, present much bigger discrepancies than our re-
+sults appearing in the right panels of Fig.4. The origin of
+this residual difference lies in the different treatment be-
+tween the two codes of the integrable singularity of the di-
+agonal matrix elements of W in (1), calculated in recipro-
+cal space, when k − k′ = q = 0 and the reciprocal-lattice
+vectors are G = G′ = 0.
+We note that the different
+treatment of this singularity was already mentioned also
+recently in a comparison among different GW codes107.
+This singularity is, in fact, eliminated, by evaluating the
+integral
+−4π
+Ω ϵ−1
+G=0,G′=0(q → 0)
+1
+(2π)3
+�
+Ωq=0
+dq 1
+q2 ,
+where Ωq=0 = ΩBZ/Nk. In order to carry out, numer-
+ically or analytically, the integral, one has to define the
+shape for the little volume Ωq=0 around the origin of the
+Brillouin zone and, in anisotropic materials, choose the
+q → 0 direction in order to evaluate the inverse dielectric
+function ϵ−1(q → 0). The details about how this inte-
+gral is performed are in Ref.108 and Refs.109,110, for EXC
+
+6
+6
+8
+10
+12
+14
+16
+18
+20
+Energy [eV]
+0
+2
+4
+6
+8
+Im εM
+IPA - pseudopotential
+IPA - all-electron
+6
+8
+10
+12
+14
+16
+18
+20
+Energy [eV]
+0
+2
+4
+6
+8
+Im εM
+RPA - pseudopotential
+RPA - all-electron 
+6
+8
+10
+12
+14
+16
+18
+20
+Energy [eV]
+0
+2
+4
+6
+8
+Im εM
+BSE - pseudopotential
+BSE - all-electron
+68
+70
+72
+74
+76
+78
+80
+Energy [eV]
+0
+0.02
+0.04
+0.06
+0.08
+Im εM
+IPA - pseudopotential
+IPA - all-electron
+RPA - pseudopotential
+RPA - all-electron
+66
+68
+70
+72
+74
+76
+78
+80
+Energy [eV]
+0
+0.1
+0.2
+0.3
+0.4
+Im εM
+BSE - pseudopotential
+BSE - all-electron
+FIG. 4: Comparison of absorption spectra calculated with pseudopotential (red lines) and all-electron (blue lines)
+methods, using an unshifted 8 × 8 × 8 k-point grid, (left panels) in the independent particle approximation (IPA),
+(middle panels) in the random-phase approximation (RPA), (right panels) from the full Bethe-Salpeter equation
+(BSE). (Upper panels) Optical spectra (with 12 valence bands and 20 conduction bands). (Bottom panels) XANES
+spectra at Al L23 edge (with 12 core levels and 12 conduction bands).
+and Exciting, respectively. If we exclude this singular
+contribution, the two BSE results become superposed, as
+in the IPA case. In addition, this contribution vanishes
+(more or less rapidly according to the kind of exciton111)
+in the convergency with k points. Fig. 5 indeed shows
+that the differences in the spectra obtained with the two
+codes tend to vanish with increasing number of k points.
+Most importantly for the scope of the present work, we
+find that the differences between the PP and AE codes
+remain always of the same order of magnitude for both
+valence and shallow-core spectra. Therefore, in summary,
+we can safely conclude that the benchmarks with the all-
+electron approach show that pseudopotential calculations
+for optical and XANES spectroscopies (with semi-core
+states) are reliable and accurate.
+C.
+Interference effects at the L1 edge
+The comparison between all-electron and pseudopoten-
+tial approximation is more delicate for the L1 edge, since
+the electrons are treated differently in the two codes.
+While Exciting includes the 2s states of Al inside the
+muffin-tin, in EXC they are considered as valence and
+treated with plane-waves.
+One of the limitations of the linearized augmented-
+plane-wave (LAPW) method is that it could give a wrong
+description of semicore states when they are considered
+inside the muffin-tin (MT) sphere, but they overlap sig-
+nificantly with valence electrons or are too extended to be
+8
+9
+Energy [eV]
+0
+0.5
+1
+1.5
+2
+  8 kp - PP
+  8 kp - AE
+0
+0.5
+1
+1.5
+2
+Im εM
+10 kp - PP
+10 kp - AE
+0
+0.5
+1
+1.5
+2
+12 kp - PP
+12 kp - AE
+FIG. 5: Convergence of BSE absorption spectra
+calculated with pseudopotential (solid lines) and
+all-electron (dot-dashed lines) methods (with 2
+conduction and 2 valence bands), for increasing number
+of k points (Γ-centered grids with 8, 10 and 12 divisions
+for bottom, central and top panel, respectively).
+entirely contained inside the MT85,112. In order to over-
+come this problem, local orbitals are included to com-
+plement the basis.
+However, the quality of this basis
+set depends on the choice of energy parameters85,113. In
+addition, there could be some interference effects that
+
+7
+106
+108
+110
+112
+114
+116
+118
+120
+Energy [eV]
+0
+0.005
+0.01
+0.015
+0.02
+Im εM
+IPA - pseudopotential
+IPA - all-electron x 4
+106
+108
+110
+112
+114
+116
+118
+120
+Energy [eV]
+0
+0.005
+0.01
+0.015
+0.02
+Im εM
+RPA - pseudopotential
+RPA - all-electron x 4
+106
+108
+110
+112
+114
+116
+118
+120
+Energy [eV]
+0
+0.01
+0.02
+0.03
+0.04
+0.05
+Im εM
+BSE - pseudopotential
+BSE - all-electron x 4
+FIG. 6: Absorption spectra at the L1 calculated with EXC (pseudopotential code) and Exciting (all-electron code).
+All the calculations are performed using a Γ-centered 8 × 8 × 8 grid of k points and 30 unoccupied bands. In EXC we
+include the 4 2s levels corresponding to the 4 Al atoms, while in Exciting we include only one 2s level (i.e., the 2s
+state on the Al atom where the core hole is created). For this reason, the spectra of Exciting are multiplied × 4.
+play an important role, and are not obviously included
+when considering the states inside the muffin-tin80. For
+all these reasons, since we validated the pseudopotential
+approach for the valence electrons (optical and L23 edge),
+we will use it to benchmark the L1 edge.
+The absorption spectra calculated for the L1 edge using
+different levels of approximations are shown in Fig. 6.
+Notice that in EXC, the 4 bands corresponding to the 2s
+state of the 4 Al atoms need to be considered in order
+to properly represent the electronic transitions, while in
+Exciting, only one occupied level is considered, the 2s
+state of the Al atom where the core-hole is sitting. Since
+there are 4 equivalent Al atoms in the cell, the overall
+spectrum coming out of Exciting needs to be multiplied
+by 4, for a correct comparison.
+In all level of approximations, the pseudopotential and
+all-electron results differ slightly (and more than in the
+optical or L2,3 edge cases), showing that small interfer-
+ence effects among the Al atoms come to play. These
+interferences are small in the system under study, for the
+Al atoms lie in equivalent positions in the cell, but they
+are detectable. We have verified that in other systems80
+these effects can be quantitative and qualitatively more
+important. While including these effects is still feasible
+with Exciting (and all approaches that create a core-
+hole in a specific position), by doing multiple calcula-
+tions and generalizing Eq.
+(2), interferences come up
+naturally in pseudopotential approaches, for all electrons
+are treated on the same footing and belong to the whole
+system, not just to one atom.
+D.
+Optical and XANES spectra: valence and
+shallow core excitations
+1.
+Comparison with experiments
+Fig.
+7 compares the calculated absorption spectra,
+ImϵM(ω), with experiment, for both the optical absorp-
+tion corresponding to valence excitations and the XANES
+spectrum of the shallow-core excitations at the Al L2,3
+edge.
+The same figure also displays the results of the
+calculations at the Al L1 edge, where, to best of our
+knowledge, no experimental XANES spectra are avail-
+able for α-Al2O3, since this core level excitation is less
+commonly studied than the Al K edge57,58,117,118. In all
+cases, the presence of sharp and pronounced peaks at the
+onset of the BSE spectra (red lines), which are absent in
+the RPA and IPA spectra (orange and green lines), is an
+evidence of strong excitonic effects. Taking into account
+the electron-hole attraction in the BSE is the key to bring
+the calculations in agreement with experiment.
+As already discussed in Ref. 93, for the optical absorp-
+tion in the polarization direction perpendicular to the z
+axis (i.e. in the xy plane), where two VUV spectroscopy
+experiments114,115 are available, there are large discrep-
+ancies between the experimental spectra themselves [see
+Fig. 7(a)]. They agree on the position of the absorption
+onset and the presence of a sharp peak at ∼ 9.2 eV, while
+they largely differ in the intensities of the various spectral
+features. Those differences can be attributed to the fact
+that both absorption spectra have been obtained from
+measured reflectivity data using the Kramers-Kroning re-
+lations, which introduces uncertainties in the ImϵM(ω)
+spectra. The calculated optical spectra in Fig. 7(a)-(b)
+have been blueshifted by 0.7 eV. This underestimation of
+the onset of the absorption spectrum is a manifestation
+of the underestimation of the band gap by the pertur-
+bative G0W0 approach, which is a systematic error for
+large gap materials119. As a matter of fact, the 2.64 eV
+scissor correction that we have employed here, which is
+taken from the G0W0 calculation in Ref. 93, underes-
+timates the band gap correction to the LDA. The BSE
+calculation in Ref.
+93 is also in very good agreement
+with the present result: the difference in the peak posi-
+tions is actually due to the LDA band gap difference (see
+Sec. III A). The BSE spectrum in the xy polarization re-
+produces well the spectral shape measured by French et
+al.115, while there are larger differences with the experi-
+mental spectra in both polarizations measured by Tomiki
+et al.114.
+At the Al L2,3 edge, see Fig. 7(c), the calculated spec-
+
+8
+6
+7
+8
+9
+10
+11
+12
+13
+14
+15
+16
+Energy [eV]
+0
+2
+4
+6
+8
+Im εM
+IPA xy
+RPA xy
+BSE xy
+Exp Tomiki et al
+Exp French et al.
+(a)
+6
+7
+8
+9
+10
+11
+12
+13
+14
+15
+16
+Energy [eV]
+0
+1
+2
+3
+4
+5
+6
+7
+8
+Im εM
+IPA z
+RPA z
+BSE z
+Exp Tomiki et al.
+(b)
+75 76 77 78 79 80
+81 82
+83 84 85 86 87 88
+89 90
+Energy [eV]
+0
+0.1
+0.2
+0.3
+0.4
+0.5
+Im εM
+Exp Weigel et al.
+BSE xy
+BSE z
+RPA xy
+RPA z
+IPA xy
+IPA z
+(c)
+124
+126
+128
+130
+132
+134
+Energy [eV]
+0
+0.01
+0.02
+0.03
+0.04
+0.05
+Im εM
+IPA xy
+IPA z
+RPA xy
+RPA z
+BSE xy
+BSE z
+(d)
+FIG. 7: Comparison of theoretical results with experimental data from Tomiki et al.114 and French et al.115 for the
+optical absorption, and Weigel et al.116 for the XANES at the L2,3 edge. The calculated spectra are obtained in the
+independent particle approximation (IPA), green lines, in the random-phase approximation (RPA), orange lines, and
+from the solution of the Bethe-Salpeter equation (BSE), red lines. Optical absorption spectra for polarization in the
+(a) xy plane and (b) in the z direction: the calculated spectra have been blueshifted by 0.7 eV. (c) Absorption
+spectra at the L2,3 edge in the xy (solid lines) and z (dot-dashed lines) polarizations compared to the isotropic
+XANES experimental spectrum116, to which a vertical offset has been added for improved clarity. (d) Absorption
+spectra at the L1 edge in the xy (solid lines) and z (dot-dashed lines) polarizations.
+tra have been blueshifted by 9.75 eV, which matches well
+the needed correction to the LDA Al 2p core level energy
+(see Sec. III A). The calculations neglect the spin-orbit
+coupling and therefore miss the splitting of the main peak
+into a doublet separated by 0.47 eV in the high-resolution
+experimental XANES spectrum from Ref.
+116 (which
+also agrees well with previous experiments114,120,121). In
+the spectra, the first, most prominent, excitonic peak
+is followed by a series of lower intensity peaks. While
+the absolute intensity of the experimental spectrum116
+is arbitrary, the relative intensity of the first and second
+peaks gives information about the coordination number
+of Al and the nature of the chemical bond: a lower sym-
+metry enhances the intensity of the second peak. More-
+over, a lower coordination shifts the edge to lower ener-
+gies, while higher bond ionicity shifts the edge to higher
+energies59,116.
+At the Al L1 edge there is no available experiment.
+Therefore, the curves in Fig. 7(d) have been shifted by
+19.5 eV, in order for the smallest independent-particle
+transition energy, from the 2s band to the bottom-
+conduction band, to match the experimental value of
+125.2 eV, which corresponds to the sum of the fundamen-
+tal band gap plus the binding energy of the 2s states55,100
+(see Sec. III A). We find that the main prominent exci-
+tonic peak in the BSE spectra is preceded by a pre-edge
+structure, more evident in the xy direction (solid lines).
+At the Al K edge, which mainly probes the analogous
+1s → 3p transition, there has been much work to ex-
+plain the origin of a similar prepeak structure12,57,122–127,
+which has been finally interpreted in terms of atomic vi-
+brations enabling monopole transitions to unoccupied Al
+
+9
+3s states. In the present case, the calculations do not
+take into account the coupling with atomic vibrations
+and nevertheless the BSE spectra show a prepeak struc-
+ture. This finding therefore calls for a detailed compar-
+ison with other calculations including atomic vibrations
+and, possibly, experiments at the Al L1 edge.
+2.
+Anisotropy and local field effects
+The α-Al2O3 crystal is optically uniaxial. As shown
+by Fig. 7(a)-(b), at the onset of the optical spectrum
+the anisotropy is rather small, while it becomes larger
+for higher energy features. The lowest energy exciton is
+visible along the z polarization, while it is dark in the
+perpendicular xy polarization. It is separated by ∼ 25
+meV from a pair of degenerate excitons that are visible
+in the perpendicular xy direction and, conversely, dark
+in the z direction. Tomiki et al.114 experimentally deter-
+mined a similar splitting of the exciton peaks in the two
+polarization directions (35 meV at room temperature and
+86 meV at 10 K). We find that the binding energy of these
+excitons is of order of 0.3 eV, which is more than twice
+the 0.13 eV value estimated from temperature-dependent
+VUV spectroscopy55. A similar splitting of the lowest
+energy exciton occurs also at the L2,3 edge114, where its
+binding energy largely increases up to 1.6 eV. For the op-
+tical and the L2,3 cases, both the lowest energy exciton
+in the BSE spectrum and the excitation at the smallest
+independent-particle transition energy in the IPA spec-
+trum have a significant oscillator strength. Instead, at
+the L1 edge the lowest energy transitions have a 2s → 3s
+character and are dipole forbidden.
+We find that the
+binding energy of the lowest dark exciton at the L1 edge
+is 1.2 eV. The lowest bright excitons in the z and xy po-
+larization directions are located 1.6 eV and 1.8 eV above
+it, respectively. They belong to the prepeak in the spec-
+trum. In this case, we define their binding energy as the
+difference with respect to the corresponding first allowed
+transition in the IPA spectrum: it amounts to 0.6 eV.
+The splitting of the main exciton peak in the two polar-
+izations is also the largest one at the L1 edge, being more
+than 0.2 eV.
+By comparing the RPA and IPA optical spectra, or-
+ange and green lines in Fig. 7(a)-(b), respectively, we
+note that the effect of crystal local fields is quite small
+for both polarizations, in contrast to typical layered van
+der Waals materials like graphite, where local field ef-
+fects are strong for the polarization along the hexagonal
+axis128. Marinopoulos and Grüning93 also found that lo-
+cal field effects are not essential to describe satisfactorily
+the low energy part of the experimental spectra, whereas
+they become crucial for higher energies (above 16 eV, not
+shown in Fig. 7), in correspondence to the excitation of
+the more localised O 2s electrons. Indeed, the degree of
+electron localisation directly correlates with the degree
+of charge inhomogeneity, which is a key factor for the
+induced microscopic local fields. One may therefore ex-
+pect that the excitation spectra of shallow core levels,
+which are even more localised, should be more affected
+by local field effects. This phenomenon has been in fact
+observed for many shallow core levels129–133. However,
+in α-Al2O3 for both the L2,3 and L1 edges the compari-
+son of the absorption spectra calculated within the RPA
+and in the IPA shows that local field effects are actually
+negligible134 (even weaker than in the optical regime).
+We can understand this result by noticing that the in-
+tensity of the L2,3 and L1 absorption spectra is one or
+two orders of magnitude smaller than for the optical ab-
+sorption. This large intensity difference reflects the fact
+that Al 2p and 2s states are much less polarizable than
+valence states. Therefore, even though their electronic
+charge is much more localized and inhomogeneous, local
+fields associated to the excitations of Al 2p and 2s are
+small because they are weakly polarizable, which also
+leads to weak induced potentials.
+3.
+Analysis of excitonic effects
+Excitonic effects in solids can be understood as the re-
+sult of the mixing of the independent-particle, vertical
+interband transitions at various k points in the Brillouin
+zone, which are weighted by the excitonic coefficients
+Aλ
+vck, i.e., the eigenvectors of the excitonic Hamiltonian
+(1). The analysis of the excitonic coefficients therefore
+directly informs on the character of the exciton.
+Fig. 8 represents, projected on the LDA band struc-
+ture, the partial contributions
+��Aλ
+vck˜ρvck
+�� to the oscilla-
+tor strength of the lowest energy bright excitons in the
+absorption spectra of Fig. 7. Each independent-particle
+transition vk → ck is represented by a pair of circles, one
+in the occupied band v and one in the unoccupied band
+c, whose size is proportional to the value of the contri-
+bution. For the optical spectrum (left panel of Fig. 8),
+we consider the exciton giving rise to the first peak in
+the absorption spectrum in the z polarization. Our anal-
+ysis shows that the largest contribution stems from the
+top-valence bottom-conduction transition at the Γ point,
+in correspondence to the direct band gap. The next k
+points along the LΓX line in the conduction band give a
+contribution that is already 10 times smaller. The others
+are even smaller. This is due to the fact that for this exci-
+ton the top-valence bottom-conduction transition at the
+Γ point has the predominant coefficient Avck
+λ
+, together
+with a large single-particle oscillator strength ˜ρvck in the
+z direction. Instead, the same ˜ρvck is negligibly small in
+the x or y direction, explaining why the same exciton is
+dark in the xy plane.
+For the L2,3 and L1 excitation spectra, all the k points
+for the corresponding core levels are involved in the spec-
+tra, as one may expect from the fact that the core levels
+are not dispersive. Also for first exciton peak in the L2,3
+XANES spectrum (middle panel of Fig. 8), the lowest
+conduction band at the Γ point gives the largest con-
+tribution, having a large Al 3s character (see Sec. 3).
+
+10
+T
+L
+X
+10
+5
+0
+5
+10
+15
+20
+Energy [eV]
+100
+101
+102
+103
+0
+10
+20
+30
+40
+T
+L
+X
+62.0
+61.5
+10
+4
+10
+3
+10
+2
+10
+1
+FIG. 8: Contributions of independent transitions to the lowest energy bright exciton intensity in the absorption
+spectra: (left) for the optical spectrum; (middle) for the XANES at L2,3; (right) for the XANES at the L1 edge. The
+size of the circles is proportional to |˜ρvckAvck
+λ
+|.
+However, in this case the other k points of the bottom
+conduction band and the higher conduction bands signif-
+icantly contribute to the spectrum as well. This illustrate
+the deviation from a simple independent-particle picture
+of a Al 2p → 3s atomic transition, since many transitions
+are mixed together to produce the excitonic peak at the
+onset of the L2,3 XANES spectrum.
+For the L1 XANES spectrum (right panel of Fig. 8),
+we consider the first bright exciton in the z polariza-
+tion direction, which belongs to the prepeak in the spec-
+trum in Fig. 7(d). Contrary to the other two cases, the
+bottom-conduction band at the Γ point gives no contri-
+bution, consistently with the 2s → 3s character of the
+transition, which is dipole forbidden. The largest contri-
+butions are instead given by the k points along the ΓT
+line of the bottom conduction band, which have 3p char-
+acter as well. Even in this case higher conduction bands
+contribute significantly to the intensity of the excitonic
+prepeak.
+The plot in Fig.
+9 of the cumulative sums Sλ(ω),
+see Eq. (3), as a function of the number of conduction
+bands explains the different convergence behavior be-
+tween the optical and L2,3 XANES spectra shown in Fig.
+1. By increasing the number of conduction bands in the
+BSE Hamiltonian (1), the largest possible independent-
+particle transition energy progressively increases. There-
+fore, the curves for larger numbers of conduction bands
+extend to higher energies. However, in the case of the
+optical spectrum (top panel), the cumulative sum Sλ(ω)
+rapidly converges to the final result. Already considering
+transition energies within 12 eV from the smallest one
+and including 15 conduction bands in the BSE hamil-
+tonian give a result of the oscillator strength very close
+to 100%. Instead, in the case of the L2,3 edge (bottom
+panel), the range of transition energies needed to get close
+to 100% has to be much larger, of the order of 50 eV
+above the smallest transition energy. Moreover, the var-
+ious curves in the bottom panel of Fig. 9 do not overlap,
+as it is the case for the optical spectrum in the upper
+panel.
+This behavior indicates that, at the L2,3 edge,
+interband transitions to higher conduction bands in the
+BSE hamiltonian mix together with transitions to lower
+conductions bands, which affects the behavior of the cu-
+mulative sum Sλ(ω) also at lower energies. The reason
+of this strong mixing is the fact that at the L2,3 edge
+there are many interband transitions with similar weak
+intensity. This, in turns, explains why the convergence
+of the XANES spectrum with the number of conduction
+bands is slow (see Fig. 1), and requires extra care.
+0
+2
+4
+6
+8
+10
+12
+14
+16
+18
+20
+22
+Energy [eV]
+0
+0.2
+0.4
+0.6
+0.8
+1
+Sλ(ω)
+30 cb
+20 cb
+15 cb
+10 cb
+0
+10
+20
+30
+40
+50
+60
+70
+80
+Energy [eV]
+0
+0.2
+0.4
+0.6
+0.8
+1
+Sλ(ω)
+160 cb
+100 cb
+60 cb
+10 cb
+FIG. 9: Cumulative sums Sλ(ω) as a function of
+number of conduction bands (cb) in the BSE
+hamiltonian for the lowest energy bright exciton in the
+z direction for (top panel) the optical spectrum (bottom
+panel) and the XANES spectrum at the L2,3 edge. In
+each case, the zero of the energy axis has been set to
+the smallest independent-particle transition energy and
+Sλ(ω) has been normalised to its largest value.
+
+10-
+10-3
+10-4
+10-511
+The lowest-energy dark excitons, both in the opti-
+cal spectrum and the L2,3 edge, have a cumulative sum
+Sλ(ω) that is always close to zero. It means that all the
+independent-particle oscillator strengths ˜ρvck are always
+small, indicating dipole forbidden transitions. The situ-
+ation is instead different for the lowest dark exciton at
+the L1 edge. In this case, some transitions to the lowest
+conduction bands have a weak but not zero contribu-
+tion |˜ρvckAλ| to the spectrum, as shown by their repre-
+sentation on the LDA band structure in the top panel
+of Fig. 10. The corresponding cumulative sum Sλ(ω),
+bottom panel of Fig. 10, is indeed not always zero: it
+has even two distinct peaks, before progressively decreas-
+ing to zero, giving rise to a dark exciton. This suggests
+the occurrence of destructive interference of contributions
+˜ρvckAλ of different sign, involving transitions over a large
+range of energy. Moreover, it also shows that including
+not enough conduction bands in the BSE hamiltonian (1)
+would produce a weak excitonic peak in the spectrum. It
+is another indication that an independent-particle pic-
+ture is here inadequate, whereas the strong electron-hole
+interaction manifest itself as the (positive or negative)
+interference of many electron-hole pairs.
+0
+10
+20
+30
+40
+T
+L
+X
+100.0
+99.5
+10
+5
+10
+4
+10
+3
+10
+2
+0
+5
+10
+15
+20
+25
+30
+35
+Energy [eV]
+0
+0.2
+0.4
+0.6
+0.8
+1
+Sλ(ω)
+FIG. 10: Contributions of independent transitions to
+the dipole strength of the lowest energy dark exciton in
+the XANES spectrum at the L1 edge. (Top panel) The
+size of the circle is proportional to |˜ρvckAλ|. (Bottom
+panel) Corresponding cumulative sum Sλ(ω). The zero
+of the energy axis has been set to the smallest
+independent-particle transition energy and the intensity
+normalised to the largest value.
+Fig.
+11 displays the electron density distribution
+|Ψλ(r0
+h, re)|2 for a fixed position of the hole r0
+h for the
+wavefunction of the lowest bright excitons in the spec-
+tra. In the color plots, we consider a cut of the three-
+dimensional distribution in the xy plane, perpendicular
+to the z axis, containing the hole. In all cases, the hole
+position (represented by the black ball in Fig. 11) has
+been chosen slightly away from the atoms, in order to
+avoid the nodes of the orbitals. This is the reason why
+the electron distribution is not symmetrical around the
+hole.
+For an uncorrelated electron-hole pair, the elec-
+tron density would be delocalised all over the crystal,
+corresponding to a Bloch wavefunction. The effect of the
+electron-hole correlation is instead to localise the electron
+density around the hole.
+For the optical spectrum (left panel), the hole has been
+placed near an O atom, consistently with the main char-
+acter of the valence band (see Sec. III A). Here we dis-
+cover that the electron charge is also surprisingly located
+at the O atoms, and quite delocalised in the xy plane.
+This picture is indeed in contrast with the naive expec-
+tation of a charge transfer O → Al nature of the exciton,
+which is based on the largely ionic character of the elec-
+tronic properties of α-Al2O3. However, the strong Al-
+O hybridisation of the bottom conduction bands makes
+it possible for the exciton to localise entirely on the O
+atoms. The nature of the exciton in α-Al2O3 therefore
+turns out to be similar to what found135,136 in other ionic
+materials like LiF, where, analogously, for a hole fixed at
+a F atom, the electron charge is located mainly on F
+atoms as well.
+Finally, the right panel of Fig.
+11 shows the wave-
+function of the first bright exciton in the prepeak of
+the L1 edge. The hole is localised close to an Al atom.
+The resulting electron charge has partially the shape of
+a deformed 2p orbital pointing to the next neighbor O
+atom.
+In this case, the electron charge is entirely lo-
+calised around the same Al site, displaying the atomic
+character of the core exciton.
+IV.
+CONCLUSIONS
+In summary, we have presented a norm-conserving
+pseudopotential approach that permits one to evaluate
+optical and XANES spectra on the same footing, using
+the same basis set for valence and shallow-core electrons.
+We have validated the approach by comparison with full
+potential all-electron calculations, at three different lev-
+els of theory, independent-particle approximation, RPA
+and full excitonic calculation, within the BSE formalism.
+We have applied this approach to study the optical and
+semi-core excitations of corundum α-Al2O3, a promising
+material for its optical and structural properties. Both
+regimes, optical and XANES, present strong many-body
+effects that require the highest level of theory for an accu-
+rate and quantitative description. The BSE calculations
+show good agreement with experiments, when available,
+
+12
+FIG. 11: Exciton correlation function Ψλ(rh, re) for the lowest bright exciton in the optical spectrum and at the
+prepeak at the L1 edge. The position of the hole rh is fixed at r0
+h (see black ball). The color plots show the
+corresponding electron density distribution |Ψλ(r0
+h, re)|2 in the xy plane perpendicular to the z axis contaning the
+hole. In order to avoid nodes of the orbitals, the hole position has been slightly displaced from an oxygen atom for
+the optical exciton, and from an aluminum atom for the L1 edge. (This explains why the density distributions are
+not symmetric around r0
+h). The intensity follows a blue-cyan-green-yellow-orange-red gradient, and goes from 0
+(blue) to the maximum value of the square of the excitonic wavefunctions (red).
+but more importantly permit one to explain the physical
+origin of the various excitations, thanks to a thorough
+analysis of the excitons. The small anisotropy in the op-
+tical regime, for instance, reveals a different order of ex-
+citons in the z and perpendicular xy directions: the first
+exciton in bright along z, followed by dark excitons, while
+it is the contrary in the perpendicular xy direction. This
+splitting appears also for the L2,3 edge. The dark/bright
+character of the excitons in the optical, L1 and L2,3 edges
+is analysed both by projecting the excitonic eigenvectors
+on the LDA band structure, as well as by looking at the
+cumulative function, Eq. (3). The first analysis tool is
+particularly useful to understand the origin of each ex-
+citon, in terms of the single-particle transitions and of
+the atomic characters of the single bands; the cumula-
+tive function can reveal purely many-body effects, like
+the distructive interference that takes place at the L1
+edge, making the first exciton dark. In addition, the ex-
+citonic wavefunction, by showing the localization of the
+different excitons, can reveal counter-intuitive behaviour,
+like the electron localization on the oxygen atom, for the
+bright exciton in the optical spectrum, in contrast to a
+naive charge-transfer O→Al character.
+This work opens the way to the treatment of other
+shallow-core spectroscopies, like electron energy loss
+near-edge structures (ELNES). Moreover, the unified
+footing to tackle shallow core, valence, and conduction
+states, will be particularly useful to describe Resonant In-
+elastic X-ray Scattering (RIXS) and X-ray Raman Scat-
+tering (XRS).
+ACKNOWLEDGMENTS
+We thank the French Agence Nationale de la Recherche
+(ANR) for financial support (Grant Agreements No.
+ANR-19-CE30-0011). Computational time was granted
+by GENCI (Project No. 544).
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+Springer Nature 2021 LATEX template
+Grey area in Embedded WMLES on a
+transonic nacelle-aircraft configuration
+Marius Herr1*, Axel Probst2 and Rolf Radespiel1
+1*Institute of Fluid Mechanics, TU Braunschweig,
+Hermann-Blenk-Str. 37, Braunschweig, 38108, Lower Saxony,
+Germany.
+2Institute for Aerodynamics and Flow Technology, DLR,
+Bunsenstr. 10, G¨ottingen, 37073, Lower Saxony, Germany.
+*Corresponding author(s). E-mail(s): m.herr@tu-bs.de;
+Contributing authors: axel.probst@dlr.de; r.radespiel@tu-bs.de;
+Abstract
+A scale resolving hybrid RANS-LES technique is applied to an air-
+craft - nacelle configuration under transonic flow conditions using the
+unstructured, compressible TAU solver. Therefore a wall modelled LES
+methodology is locally applied to the nacelle lower surface in order to
+examine shock induced separation. In this context a synthetic turbu-
+lence generator (STG) is used to shorten the adaption region at the
+RANS – LES interface. Prior to the actual examinations, fundamen-
+tal features of the simulation technique are validated by simulations of
+decaying isotropic turbulence as well as a flat plate flow. For the aircraft
+- nacelle configuration at a Reynolds number of 3.3 million a sophisti-
+cated mesh with 420 million points was designed which refines 32 % of
+the outer casing surface of the nacelle. The results show a development
+of a well resolved turbulent boundary layer with a broad spectrum of
+turbulent scales which demonstrates the applicability of the mesh and
+method for aircraft configurations. Furthermore, the necessity of a low
+dissipation low dispersion scheme is demonstrated. However, the dis-
+tinct adaption region downstream of the STG limits the employment
+of the method in case of shock buffet for the given flow conditions.
+Keywords: hybrid RANS-LES, wall-modelled LES, synthetic turbulence,
+aircraft configuration, transonic flow, shock induced separation
+1
+arXiv:2301.05299v1  [physics.flu-dyn]  12 Jan 2023
+
+Springer Nature 2021 LATEX template
+2
+Grey area in Embedded WMLES on a nacelle-aircraft configuration
+1 Introduction
+Transonic flows about aircraft configurations exhibit complex, instationary
+flow phenomena such as oscillating shock fronts with boundary layer sepa-
+ration. This so-called buffet phenomenon causes unsteady aerodynamic loads
+which might endanger the flight safety. Therefore a fundamental understand-
+ing of the related flow physics is of particular interest to be able to find
+specific technical solutions which control this phenomenon. The present study
+examines a XRF-1 aircraft model which represents a wide-body long-range con-
+figuration and was designed by Airbus. An Ultra High Bypass Ratio (UHBR)
+nacelle is coupled to the model which represents a modern and efficient jet
+engine that is modelled as flow-through nacelle for wind tunnel testing. Due
+to the large circumference of the nacelle, a close coupling by means of a pylon
+to the wing lower side is necessary. This channel-like arrangement of nacelle,
+pylon, wing and fuselage causes the development of an accelerated flow which
+triggers the formation of transonic shocks within this area. Depending on the
+exact flow conditions these shocks evolve into buffet with significant loads.
+Initial investigations in the framework of the DFG (Deutsche Forschungsge-
+meinschaft) funded research group have shown a complex system of shock
+fronts [1]. As a first step toward representing this complex system with a sophis-
+ticated numerical method this study focuses on a single shock front located at
+the lower side of the nacelle.
+Numerous numerical investigations have investigated the problem of buf-
+fet onset with well established unsteady Reynolds-averaged Navier-Stokes
+(URANS) methods. However, it is well known that even highly developed
+Reynolds stress based URANS models show deficiencies in describing the
+dynamics of separated boundary layer as well as the aerodynamic effects of
+large flow separations [2]. Also, due to high, flight relevant Reynolds numbers
+a broad scale of turbulent structures arise for the given flow phenomenon.
+Therefore a simulation technique that provide both high spatial and temporal
+resolution is required. Direct Numerical Simulation (DNS) resolves all turbu-
+lent scales but is so far restricted to simple geometries at low Reynolds numbers
+due to its unfeasible computational effort for flight relevant flows. Therefore a
+Large Eddy Simulation (LES) technique is required which only resolves large
+turbulent scales whereas small, isotropic scales are modelled. Since an appli-
+cation of LES to the entire aircraft configuration is still computationally too
+expensive a hybrid RANS - LES technique is employed. In the present study the
+wall modelled LES (WMLES) method within the Improved Delayed Detached
+Eddy Simulation (IDDES) methodology is used [3]. Depending on the spatial
+discretisation, up to 5 % of the wall adjacent boundary layer is modelled by
+the RANS equations. Additionally, the area of WMLES is embedded around
+the transonic shock such that all relevant flow areas are enclosed. This cor-
+responds to 32 % of the outer casing surface of the nacelle. The remaining
+flow field of wing, body, pylon and nacelle is modelled with a URANS model.
+The embedded WMLES (EWMLES) requires an injection of synthetic turbu-
+lence at the RANS-LES interface which is located at the leading edge of the
+
+Springer Nature 2021 LATEX template
+Grey area in Embedded WMLES on a nacelle-aircraft configuration
+3
+nacelle for the present configuration. Otherwise, a so-called grey area would
+arise which describes a region of underresolved turbulence directly downstream
+of the RANS-LES boundary. To this end the synthetic turbulence generator
+(STG) devised by [4] is employed. Nevertheless, using this method, a transi-
+tional region from modelled to fully resolved turbulence is still present and is
+referred to as adaption region in this study. The analysis of this adaption region
+with regard to its length and behaviour of relevant flow quantities in this area
+are of major interest. Thus, especially the transient establishment of resolved
+turbulence within the WMLES area and the fundamental applicability of the
+method to the aircraft configuration are the focus of this study.
+The study is structured as follows. The employed WMLES model in
+conjunction with the STG is described in detail in subsection 2.1 and 2.2,
+respectively. Subsequently a thorough description of the employed low dissi-
+pation low dispersion (LD2) numerical scheme is given in 2.3. The following
+section 3 provides a basic validation of the Embedded WMLES based on the
+SST-RANS model by means of flows of decaying isotropic turbulence and a
+flow about a flat plate. The results of the application to the XRF-1 configu-
+ration are presented in section 4. An extensive description of the mesh design
+with regard to the extension of the WMLES area, the used refinement criteria
+and its application to the actual mesh environment are presented (Sec. 4.2).
+Results of the transient WMLES establishment are then shown and assessed
+in section 4.3. The analysis of temporally and spatially averaged flow quanti-
+ties in the area related to the STG is carried out (Sec. 4.4). Finally, sensitivity
+studies with regard to the position of the RANS-LES boundary (Sec. 4.5.1)
+and the effect of using a standard numerical scheme instead of the low dissipa-
+tion scheme (Sec. 4.5.2) is presented. This paper is closed by a final summary
+of all research findings (Sec. 5).
+2 Numerical Methods
+The flow simulations in this paper use the unstructured compressible DLR-
+TAU code [5] which numerically solves the flow and model equations on
+mixed-element grids (e.g. hexahedra, tetrahedra, prims) via the finite-volume
+approach. It applies 2nd-order discretization schemes for both space and time,
+together with low-Mach-number preconditioning for flows that are close to
+the incompressible limit. Implicit dual-time stepping allows adapting the time
+step in unsteady simulation to the physical requirements (i.e. related to the
+convective CFL-criterion), avoiding numerical stability restrictions.
+The relevant methods for embedded wall-modelled LES, i.e. the overall
+(hybrid) turbulence model, the method to generate and inject synthetic turbu-
+lence and the required local adaptation of the numerical scheme, are outlined
+in the following.
+
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+4
+Grey area in Embedded WMLES on a nacelle-aircraft configuration
+2.1 Hybrid RANS-LES Model
+The present embedded wall-modelled LES approach relies on the Improved
+Delayed Detached-Eddy Simulation (IDDES) [3] which combines local RANS,
+DES (i.e. RANS-LES) and wall-modelled LES (WMLES) functionalities in a
+seamless, automatic manner. This is achieved by a single hybrid length scale
+replacing the integral turbulent scale lRANS in the underlying RANS model,
+which is the two-equation SST model [6] in the present work. The hybrid length
+scale reads:
+lhyb = ˜fd (1 + fe) lRANS +
+�
+1 − ˜fd
+�
+lLES
+.
+(1)
+Here, the function ˜fd = max {(1 − fdt) , fB} is the main blending switch
+between the different modelling modes, where fdt and fB depend on local grid
+and flow properties (cf. [3]).
+In WMLES mode (fdt ≡ 1 and, thus, ˜fd ≡ fB), if resolved turbulent
+content enters an attached boundary layer, a RANS layer is kept near the
+wall and sized according to the local grid resolution, thus circumventing the
+extreme grid requirements of wall-resolved LES at high Reynolds numbers.
+However, since no wall-functions are applied in the present work, the equations
+need to be solved down to the wall with a (normalized) near-wall grid spacing
+of y+(1) ≤ 1. The additional elevating function fe is designed to reduce the
+well-known log-layer mismatch in WMLES.
+In the largest (outer) parts of the boundary layer, lhyb ≡ lLES = CDES∆,
+which approximates the behaviour of a Smagorinsky-type sub-grid model for
+LES. The model constant CDES is usually calibrated for canonical turbulent
+flow, such as decaying isotropic turbulence (DIT), see Sec. 3.1. However, since
+wall-bounded flows typically require a different calibration than free turbu-
+lence, another modification compared to standard DES/LES is introduced in
+the filter width ∆:
+∆ = ∆IDDES = min {max [Cw · dw, Cw · hmax, hwn] , ∆DES}
+,
+(2)
+where Cw = 0.15. In essence, this near-wall limitation of the filter width
+compensates for this flow-type dependency and allows using a unique CDES
+value for both wall-bounded and off-wall turbulent flow. More details on this
+modification are found in [3].
+For embedded WMLES, the IDDES in TAU can be locally forced to
+WMLES mode according to external user input, e.g. inside boxes or other suit-
+able geometric sub-areas of the flow domain. This is achieved by setting the
+function fdt to 1 downstream of the desired RANS-WMLES interface, thus
+safely reducing the eddy viscosity from RANS to WMLES level [7].
+2.2 Synthetic Turbulence Generation
+In this work, synthetic turbulent fluctuations at the streamwise RANS-LES
+interface are provided by the Synthetic Turbulence Generator (STG) of
+
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+Grey area in Embedded WMLES on a nacelle-aircraft configuration
+5
+Adamian and Travin [8] with extensions for volumetric forcing by Francois [9].
+This STG generates local velocity fluctuations from a superimposed set of N
+Fourier modes as:
+⃗u′
+ST = ⃗A ·
+√
+6
+N
+�
+n=1
+√qn
+�
+⃗σn cos
+�
+kn ⃗dn · ⃗r′ + φn + sn t′
+τ
+��
+,
+(3)
+where the direction vectors ⃗dn and ⃗σn ⊥ ⃗dn, the mode phase φn, and the mode
+frequency sn are randomly distributed. A realistic spectral energy distribution
+of the mode amplitudes qn is achieved by constructing a von K´arm´an model
+spectrum from RANS input data and a local grid cut-off. The RANS data,
+which is automatically extracted from just upstream the RANS/LES inter-
+face, is also used to scale the fluctuations via the Cholesky-decomposed RANS
+Reynolds-stress tensor ⃗A.
+For realistic temporal correlations in a volumetric forcing domain, the posi-
+tion vector ⃗r′ and the time t′ are modified in accordance with Taylor’s frozen
+velocity hypothesis, see [9] for details.
+Synthetic-Turbulence Injection
+To inject the synthetic fluctuations from Eq. (3), a forcing volume with a
+streamwise extent of about half the local boundary-layer thickness is marked
+just downstream of the RANS/LES interface. Inside this volume, a momentum
+source term is added [10] which approximates the partial time derivative of
+the synthetic fluctuations as:
+⃗Q = ∂ (ρ⃗u′
+ST )
+∂t
+≈ 3 (ρ⃗u′
+ST − ρ⃗u′n) −
+�
+ρ⃗u′n − ρ⃗u′n−1�
+2∆t
+.
+(4)
+This discretization corresponds to the 2nd-order backward difference scheme
+used for unsteady simulations with TAU. By computing the fluctuation values
+of the previous time steps from the actual flow field, i.e. as ⃗u′n = ⃗un − ⟨⃗u⟩ and
+⃗u′n−1 = ⃗un−1 −⟨⃗u⟩, the synthetic target field (Eq. 3) can be reproduced rather
+accurately in the simulation, even though running time averages are required.
+An additional Gauss-like blending function with a maximum value of 1 around
+the streamwise center of the forcing volume is multiplied to the source term
+in order to prevent abrupt variation of the forcing.
+2.3 Hybrid Low-Dissipation Low-Dispersion Scheme
+Since scale-resolving simulation methods like IDDES involve explicit modelling
+of the sub-grid stresses, the overall accuracy relies on low spatial discretization
+errors in the LES regions of a given grid. Concerning resolved turbulence, there
+are two types of error that mainly stem from the discretized convection of
+momentum: while numerical dissipation damps the turbulent fluctuations and
+
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+6
+Grey area in Embedded WMLES on a nacelle-aircraft configuration
+would lead to under-predicted Reynolds stress, numerical dispersion distorts
+the shape of resolved turbulent structures.
+For that reason, the present simulations apply a hybrid low-dissipation
+low-dispersion scheme (HLD2) [11], which combines different techniques to
+optimize the convection scheme for local scale-resolving simulations using
+unstructured finite-volume solvers.
+To provide low numerical dissipation, the spatial fluxes are calculated
+from Kok’s [12] skew-symmetric central convection operator, which allows for
+kinetic-energy conservation (i.e., it is non-dissipative) on curvilinear grids in
+the incompressible limit. For compressible flow on general unstructured grids,
+a classic blend of 2nd- / 4th-order artificial matrix-dissipation is added to
+ensure stability around shocks and in smooth flow regions. Compared to RANS
+computations, however, the 4th-order dissipation has been strongly reduced
+by manually optimizing its parameters in LES computations of the channel
+flow, yielding e.g. a global scaling factor of κ(4) = 1/1024 and a reduced
+Mach-number cut-off in the low-Mach-number preconditioning matrix.
+Moreover, to minimize the dispersion error of the second-order scheme, the
+skew-symmetric central fluxes are based on linearly-reconstructed face values
+φL,ij, φR,ij using the local Green-Gauss gradients ∇0φ. Exemplarily, a generic
+central flux term reads:
+φij,α = 1
+2 (φL,ij + φR,ij) = 1
+2 (φi + φj) + 1
+2α (∇0φi − ∇0φj) · dij
+,
+(5)
+where dij is the distance between the points i and j. With an extrapolation
+parameter of α = 0.36 the scheme was found to minimize the required points
+per wavelength for achieving a given error level in a 1-D wave problem, see
+[13] for details.
+Blended Scheme for Hybrid RANS-LES
+While the low-error properties of the LD2 scheme are essential for accurate
+LES and WMLES predictions with TAU [11], the pure RANS and outer flow
+regions in hybrid RANS-LES are less dependent on such numerical accuracy.
+Moreoever, although the LD2 scheme has been globally applied in hybrid
+RANS-LES, complex geometries like the present XRF-1 configuration and cor-
+responding unstructured grids may induce local numerical instabilities that
+are not damped by low-dissipative schemes. For this reason, we apply the LD2
+scheme in a hybrid form [11] where all parameters of the spatial scheme, Ψi,
+are locally computed from a blending formula:
+Ψi = (1 − σ) · Ψi,LD2 + σ · Ψi,Ref
+.
+(6)
+Here, Ψi,LD2 are the parameter values of the LD2 scheme (e.g. κ(4) = 1/1024,
+α = 0.36), whereas Ψi,Ref corresponds to standard central-scheme parameters
+typically used in RANS computations (e.g. κ(4) = 1/64, α = 0). The blending
+
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+Grey area in Embedded WMLES on a nacelle-aircraft configuration
+7
+function σ is adopted from [4] and discerns between the well-resolved vortex-
+dominated flow regions (LD2) and coarse-grid irrotational regions (Ref ).
+By now, the hybrid LD2 scheme (HLD2) has been successfully applied
+in a number of hybrid RANS-LES computations ranging from canonical
+flows on structured grids [11] to complex high-lift aircraft on mixed-element
+unstructured meshes [14].
+3 Basic Validation of Embedded WMLES
+Before analyzing the embedded WMLES approach from Sec. 2 for a complex
+transonic aircraft configuration with UHBR nacelle in Sec. 4, we investigate
+and demonstrate its basic scale-resolving functionalities in fundamental test
+cases, i.e. decaying isotropic turbulence for pure LES and a developing flat-
+plate boundary layer for WMLES.
+3.1 Decaying Isotropic Turbulence
+Although SST-based IDDES is a well-known hybrid model present in many
+CFD codes, a proper verification for a given flow solver and the applied
+numerical scheme requires fundamental tests of the different modelling modes.
+This includes the pure LES functionality, where the hybrid model acts
+as Smagorinsky-type sub-grid model and mostly relies on the ”outer-flow”
+calibration constant of SST-based IDDES, i.e. CDES = 0.61.1
+For this reason, we present for the first time TAU simulations of decaying
+isotropic turbulence (DIT) using SST-IDDES with the LD2 scheme and com-
+pare the results with classic experimental data from [15]. In particular, the
+turbulent-kinetic-energy (TKE) spectra at two different time levels after the
+start of decay, i.e. t = 0.87 s and t = 2.0 s, are considered. Additionally, to
+emphasize the effect of the LD2 scheme, further SST-IDDES simulations are
+performed using a reference central-scheme with higher artificial dissipation
+(cf. Eq. 6 in Sec. 2.3).
+As for the computational setup, a cubic domain with normalized edge
+length of 2π is discretized by Cartesian meshes with 323, 643 and 1283 cells,
+respectively. Periodic boundary conditions are applied in all three directions.
+The initial velocity field has been generated by a Kraichnan-type synthetic
+turbulence approach [16] and retains the TKE spectrum of the experiment at
+t = 0 s. Due to the compressible formulation of the DLR-TAU code, appropri-
+ate initial density and pressure fields are derived from the isentropic relations of
+compressible fluids, describing the change of state from stagnation (Ma∞ = 0)
+to the local Mach number, i.e. ρ/ρ∞ = f (Ma) and p/p∞ = f (Ma). Moreover,
+the initial fields of modeled TKE and specific dissipation rate ω are computed
+in a preliminary steady-state SST-IDDES computation, where all equations
+except for the hybrid turbulence model are frozen. The temporal resolutions
+1Note that the calibration constant in SST-based DES-variants takes a different value close to
+walls, but this region is usually treated in RANS mode anyway..
+
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+8
+Grey area in Embedded WMLES on a nacelle-aircraft configuration
+Fig. 1 TKE spectra of decaying isotropic turbulence (DIT) for two different times along
+with experimental data [15]. Results for the LD2 scheme (left) and a reference central-scheme
+(right) are shown.
+of ∆t/s ∈ { 5·10−3, 5·10−3, 2·10−3} for the coarse, middle and fine grid were
+determined in time-step convergence studies.
+Fig. 1 (left) shows the results for the SST-IDDES with LD2 scheme which
+demonstrate a good agreement with the experimental results for all spatial
+resolutions and both time levels. For the reference central-scheme however,
+the picture is different. Although there are agreements with the experimental
+results for small wave numbers scales k+ ≤ 8 for all resolutions and time levels,
+deviations arise for larger wave numbers. These deviations are growing with
+increasing wave number and finally result in a significant underestimation of
+the TKE for all setups.
+As a result we successfully demonstrated the LES functionality of SST-
+IDDES in conjunction with the LD2 scheme. The low dissipation feature of
+the numerical scheme was confirmed and additionally emphasized by reference
+simulations with higher artificial dissipation.
+3.2 Developing Flat Plate Boundary Layer
+For a basic assessment of the full embedded WMLES functionality, we consider
+the test case of a developing flat-plate boundary layer, which transitions from
+RANS to WMLES at a fixed streamwise position. It starts with zero thickness
+at the inflow and is computed in SST-RANS mode up to the position, where
+the momentum-thickness Reynolds number reaches Reθ = 3040. Here, a zonal
+switch to WMLES within IDDES is placed, along with a synthetic-turbulence
+forcing region of about half a boundary layer thickness in streamwise direction,
+see Sec. 2.2.
+A hybrid grid with 5.8 million points and hexahedral cells in the WMLES
+area is used, which ensures ∆x+ ≈ 100 − 200, ∆y+ ≈ 1, ∆z+ ≈ 50 like the
+structured grid used in [17]. More relevant for WMLES, the streamwise spacing
+fulfills ∆x ≤ δ/10 throughout the flow domain, where δ is the approximate
+local boundary layer thickness. The normalized timestep (in wall units) is
+
+E+
+10-2
+10
+Experiment t = 0.84
+Experiment t = 2
+SST -- IDDES, 323, LD2
+SST -- IDDES, 643, LD2
+SST -- IDDES, 1283, LD2
+k*
+20
+40
+60E+
+10
+10
+Experiment t = 0.84
+Experiment t = 2
+SST -- IDDES, 323, ref. Scheme
+SST -- IDDES, 643, ref. scheme
+SST -- IDDES. 128, ref. scheme
+k*
+20
+40 
+60Springer Nature 2021 LATEX template
+Grey area in Embedded WMLES on a nacelle-aircraft configuration
+9
+Fig. 2 Evolution of averaged skin friction along streamwise position x of the flat plate test
+case.
+∆t+ ≈ 0.4 and safely fulfills the convective CFL criterion (CFLconv < 1) in
+the whole LES region.
+The statistical input data for the STG methods is given by external input
+from a precursor RANS profile at Reθ = 3040 which has been augmented with
+an anisotropic normal-stress approximation according to [18].
+The spanwise and temporal averaged results of the skin friction distribu-
+tion mean-cf are depicted in Fig. 2 along with the Coles-Fernholz correlation
+[19]. After an initial overshoot of mean-cf at the position of the STG, mean-cf
+shows good agreement with the Coles-Fernholz correlation and remains within
+an acceptable error margin of 5 %. Note that the adaption region downstream
+of the STG is hardly visible but still present. This region is defined as underpre-
+diction of mean-cf compared to the previous mean-cf level directly upstream
+of the STG. The adaption-length which respresents the distance between the
+position of the STG and the first peak in mean-cf downstream of the overshoot
+amounts 7 δST G where δST G is the boundary layer thickness at the position of
+the STG. Within this adaption region the sum of modelled and resolved tur-
+bulent stresses are lower than the previous level of modelled turbulence of the
+RANS region which results in an underprediction of mean-cf [20].
+Finally, this examination confirms the embedded WMLES functionality of
+SST-IDDES with STG for a flat plate flow. Thus this methodic is basically
+verified for comparable geometry sections at the XRF-1-UHBR configuration.
+4 Grey-Area Investigation on Nacelle-Aircraft
+Configuration
+4.1 Geometry, Flow Conditions and RANS Mesh
+The actual target configuration consists of a half model of a modern trans-
+port aircraft configuration in conjunction with a through flow nacelle (cf. Fig.
+3). The employed XRF-1 aircraft model represents a wide-body long-range
+
+0.004
+0.003
+mean-cf
+0.002
+0.001
+Coles-Fernholz
+SST - IDDES + STG (LD2)
+0
+20
+40
+60
+X/Springer Nature 2021 LATEX template
+10
+Grey area in Embedded WMLES on a nacelle-aircraft configuration
+research configuration and is designed by Airbus. A Ultra High Bypass Ratio
+(UHBR) nacelle is integrated with the aid of a pylon and positioned close to
+the wing lower side. The UHBR design consists of an outer casing and a core
+body with plug. The casing is shaped circularly with a cross section similar to
+an airfoil. Both, nacelle and a specifically designed pylon were developed by
+DLR [1].
+In order to find a suitable flow condition with shock induced separation in
+the surrounding of the nacelle surface a comprehensive numerical study was
+performed where various high speed off-design conditions were assessed. As
+key parameter for the occurrence of transonic shocks at a Reynolds number
+of Re = 3.3 million a low angle of attack (α) was identified. For a farfield
+Mach number of 0.84 and α = −4◦ shock induced separation is present at
+the wing lower side, the pylon and the nacelle. A single, locally separated
+transonic shock could be found at the outer surface of the nacelle lower side
+(cf. Fig. 4). Thus, a flow condition which allows to examine an isolated shock
+with subsequent boundary layer separation in the context of a nacelle-aircraft
+configuration was found.
+In a prelinimary work a high quality RANS mesh for the XRF-1 - UHBR
+half model was designed and constructed by projects partners of the research
+unit at the University of Stuttgart and DLR. The surface RANS mesh mainly
+consists of structured areas which are extruded to hexahedral blocks. These
+are designed to contain the entire RANS boundary layer with a safety factor
+of 2. The wall adjacent cell spacing fulfills y+(1) ≤ 0.4 and a growth rate of
+1.12 is applied in wall normal direction. A h-type mesh topology is employed
+at the intersections of the aircraft components to be able to accurately resolve
+flow features in these areas. The farfield region is discreticed by tetrahedra
+and extends to 50 wingspans in all coordinate directions. The total grid size
+before refinement amounts 112 million points.
+4.2 Grid Design for Embedded WMLES
+In the following the mesh design for the WMLES refinement region is intro-
+duced. A sophisticated meshing strategy, that aims to reduce the grid size
+as far as possible but follows basic refinement and extension constraints for
+WMLES, is developed. This is necessary in order to limit mesh size and result-
+ing computing time to a reasonable level. Special care was taken to the mesh
+resolution of all coordinate directions (∆x, ∆y and ∆z) which depend on the
+local boundary layer thickness δ. Additionally, a potential shock movement is
+considered with regard to the refinement extension as well as mesh resolution.
+The refinement region is embedded within the previously described RANS
+mesh with the aid of unstructured bands in the surface mesh (cf. Fig. 4 and
+Fig. 5). This strategy allows to drastically increase the resolution within the
+structured boundary layer such that the surrounding RANS region remains
+unchanged. An unstructured nearfield block, which is also present in the pure
+RANS mesh, serves as an interface between the hexahedral blocks and the
+
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+Grey area in Embedded WMLES on a nacelle-aircraft configuration
+11
+Fig. 3 Bottom view of XRF-1 - aircraft configuration with UHBR nacelle. The nacelle
+lower side includes the mesh refinement region for embedded WMLES.
+farfield, exhibits a mesh decay rate of 0.85. The total mesh size of the combina-
+tion of RANS mesh and refinement region for WMLES comprises 420 million
+points.
+4.2.1 Extension of the refinement region
+To describe locations on the nacelle surface more precisely a cylindrical coor-
+dinate system r, ϕ and x/c is introduced, where c represents the nacelle chord
+length. Its reference point r = 0, x/c = 0 is located in the nacelle center within
+a cross section that includes the entire nacelle leading edge. ϕ is set to 0◦ at
+the intersection between nacelle and pylon and increases in clockwise direction
+that 90◦ points towards the fuselage.
+According to [21] the first step in designing hybrid RANS LES mesh for
+DES based algorithms is the definition of the RANS and LES regions for the
+given configuration. Since the aim of this research topic is the application of a
+WMLES methodology to a flow region with shock induced separation, all flow
+regions directly related to this phenomenon are of interest and should be highly
+resolved. The primary region is the area of recirculation (AOR) downstream
+of the shock position (cf. Fig. 4 left). Flow regions related to this are the
+attached boundary layer upstream of the AOR and separated boundary layer
+downstream of the AOR until the trailing edge of the nacelle. To this end the
+average shock front position and extension of the AOR are calculated by a
+preceding SST-RANS calculation. Fig. 4 (left) shows a surface plot of the skin
+friction coefficient (cf) where the cf is only plotted for cf < 0 which serves as
+an indicator of the AOR. The refinement region in spanwise direction (ϕ) is
+chosen such that the entire area of recirculation is included with some margins
+in ϕ-direction and extends 105◦ starting from 120◦ until 225◦ (cf. Fig. 4).
+
+X
+ZSpringer Nature 2021 LATEX template
+12
+Grey area in Embedded WMLES on a nacelle-aircraft configuration
+Fig. 4
+Bottom view of the UHBR-nacelle. Left: Area of recirculation of SST-RANS solu-
+tion for Ma∞ = 0.84 and α = −4◦. The shown RANS surface mesh already includes the
+boundaries for the refinement region in form of unstructured streaks. Right: Extension of
+refinement area with stepwise increase in streamwise direction. The colorbar visualizes the
+cell surface area where yellow and purple represent large and low areas, respectively.
+Since the boundary layers thickness is not only a function of x but also
+of ϕ we introduce the new variables δϕ,max(x) and δϕ,min(x) which refer to
+the maximum and minimum boundary layer thickness for a given streamwise
+position x. In x/c direction the refinement is applied between xa/c = 0.06
+and xb/c = 1. The choice of xa/c = 0.06 as the most upstream position
+is the result of the dependence of mesh resolution on the boundary layer
+thickness δϕ,min(x). The smaller the boundary layer thickness δϕ,min(x) at
+location xa the smaller the required cell lengths ∆ζ(xa) for ζ ∈ {r, ϕ, x}
+since ∆ζ(x) ≤ δϕ,min(x)/10. The refinement in wall normal direction r is
+applied for wall distances that hold dw(x) ≤ 1.2 · δϕ,max(x) in the interval
+0.06 ≤ x/c ≤ 0.16 and dw ≤ 1.5 · δϕ,max(x) within 0.16 ≤ x/c ≤ 1. Thus dw/c
+ranges from 0.2% at x/c = 0.06 to 15% at the trailing edge (cf. Fig. 4 right).
+Although these distances are smaller than dw ≤ 2 · δ(x) suggested by [22] we
+show in Sec. 4.3 that the whole resolved boundary layer remains within the
+refined area with distance drefined(x) over the entire simulated time period.
+Additionally, the extension of the refinement area in r-direction also consid-
+ers a potential oscillation of the boundary layer separation point around its
+average position at xs/c = 0.13 (SST-RANS solution). We assumed an oscil-
+lation amplitude of ±0.03 c which also allows to employ this mesh in case of
+shock buffet. As a consequence, at position x/c = 0.16 a refinement distance
+of drefined(0.16c) = 1.2 · δϕ,max(0.19c) is used.
+4.2.2 Resolution of the refinement region
+The resolution in x-direction depends on the local boundary layer thick-
+ness and is set to a limit of ∆x(x) ≤ δϕ,min(x)/10 which leads to a total
+number of 1350 points in x-direction from the leading edge to the trail-
+ing edge. Again an oscillation of separation due to shock buffet point is
+considered. Thus it is assumed to have a attached boundary layer until
+
+C,<0
+XZSpringer Nature 2021 LATEX template
+Grey area in Embedded WMLES on a nacelle-aircraft configuration
+13
+Fig. 5 Surface mesh of refinement region on lower side of UHBR nacelle. Left:
+Discrete
+coarsening of ∆ϕ is apparent which subdivides the refinement area into five subregions.
+Right:
+Vertical unstructured (triangular based) streak enables to refine locally and keep
+surrounding RANS resolution untouched. Horizontal unstructured stripe allows to coarsen
+the refinement region in ϕ-direction.
+xs/c = 0.13 + 0.03 leading to reduced boundary layer thickness compared to
+the preliminary SST-RANS solution. Therefore the boundary layer thickness
+at x/c = 0.16 is estimated to δϕ,min(x/c = 0.08) · 24/5 according to turbu-
+lent boundary layer theory. As before the resolution in ϕ-direction is limited
+to r∆ϕ(x) ≤ δϕ,min(x)/10. In contrast to the resolution in x-direction the
+adaption of ∆ϕ(x) to δϕ,min(x) is realised in a discrete manner. Therefore the
+refinement region is separated into five subregions with its boundaries located
+at x/c ∈ {0.06; 0.16; 0.25; 0.4; 0.82; 1} (cf. Fig. 5). ∆ϕ(x) remains con-
+stant within each subregion Ωi and is set to r∆ϕ(x ∈ Ωi) = δϕ,min(xi)/10 with
+xi defined as the most upstream position of Ωi. With this protocol the res-
+olution in ϕ-direction is always smaller than δϕ,min(x)/10 which results into
+{4350; 1660; 870; 603; 250} points in ϕ-direction within the correspond-
+ing subregion. Without this stepwise increase of ∆ϕ the total grid number
+would increase by a factor of 3 to 1.2 · 109 points. Again a potential move-
+ment of the boundary layer separation point is considered and therefore
+r∆ϕ(x = 0.16c) =
+1
+10δϕ,min(x = 0.08c) · 24/5. In r-direction the wall normal
+spacing of the wall adjacent cells is limited to r+(1) = 0.4. The cells of the
+entire refinement area are extruded geometrically with a growth factor of 1.12
+until ∆r = ∆x(x = 0.06c) is reached and ∆r is initially kept constant to obtain
+locally isotropic cells. Since the distance of the refinement region drefined(x)
+increases in x-direction in a cascading manner (cf. Fig. 4 (right) and 6) the
+geometric growth is continued for refinement areas with larger wall distances.
+Exemplarily, ∆r is further increased to ∆r = ∆x(x = 0.16c) for wall distances
+in the interval drefined(x = 0.16c) ≤ r ≤ drefined(x = 0.25c) and applied
+where 0.16 ≤ x/c ≤ 1. Subsequently ∆r is again increased until ∆r = ∆x(x =
+
+X
+Y
+ZSpringer Nature 2021 LATEX template
+14
+Grey area in Embedded WMLES on a nacelle-aircraft configuration
+Fig. 6 Cross section of nacelle lower side at ϕ = 180◦. Subregion Ω1 (0.06 ≤ x/c ≤ 0.16)
+of the refinement region includes 200 Mio. cells which corresponds to 48% of the entire grid
+size.
+0.25c) for wall distances in the intervall drefined(x = 0.25c) ≤ r ≤ drefined(x =
+0.4c) and applied where 0.25 �� x/c ≤ 1. This protocol is repeated until ∆r
+amounts ∆r = ∆x(x = 0.82c) for drefined(x = 0.82c) ≤ r ≤ drefined(x = 1c)
+and 0.82 ≤ x/c ≤ 1. Finally, the total number of grid points in wall normal
+direction comprises {113; 168; 183; 230; 258} points within the corresponding
+subregion.
+4.3 Results of Transient WMLES Establishment
+As initial solution for the SST-IDDES a converged SST-RANS solution
+was employed. The physical time step size amounts ∆t = 5.5 · 10−8 s =
+1/16750 CTU where 1 CTU = u∞ · c represents a single convective time unit
+(CTU). ∆t is chosen that CFL < 1 is fulfilled for all grid cells.
+Fig. 7 represents the temporal evolution of the Mach number in a cross
+section at ϕ = 180◦ and four different times. With regard to the turbulent
+boundary layer thickness δ it should be noted that δ is entirely located within
+the refinement volume with sufficient distance to its boundary (indicated by
+black lines). After the depicted maximal extension at 0.5 CTU the boundary
+layer thickness significantly decreases at later times. This decrease appears
+to be related with the shock movement in downstream direction since this
+correlation is also observed for various transonic flows of wing profiles [23].
+As mentioned before the root of the shock front xs is moving from its initial
+SST-RANS position xs(t0) = 0.13c downstream to xs(t1 CTU) = 0.17c and
+remains at the same position until xs(t1.5 CTU). Although xs is located further
+downstream as we assumed for the mesh design (0.1 ≤ xs/c ≤ 0.16) one has
+to note that such shock displacements are common in transient simulations
+(e.g. t ≤ 7.5 CTU). The shock position will most likely move upstream again
+for more advanced simulation times.
+Another perspective on the temporal evolution is given in Fig. 8. Here the
+cf-distribution is shown at four different times. This figure confirms that the
+resolved turbulence develops over the entire refinement area. The transonic
+shock front is visible in form of a sudden decrease in cf. As in Fig. 7 it can
+be seen that the whole front is moving downstream until it remains in an area
+of 0.16 ≤ xs/c ≤ 0.2. A minor numerical effect appears at the lateral edge
+
+-0.7
+-0.75
+-0.8
+-0.85
+-0.9
+0.4
+0.8
+0
+0.2
+0.6
+X/cSpringer Nature 2021 LATEX template
+Grey area in Embedded WMLES on a nacelle-aircraft configuration
+15
+Fig. 7
+Ma-number fields within a cross section of the refinement volume at ϕ = 180◦ for
+four different times.
+of the refined mesh in ϕ-direction where underresolved turbulence is present.
+This is due to the fact that the STG does not directly connect to the lateral
+RANS zones at the edges of the refinement region. Therefore two small gaps
+appear where little resolved and significantly reduced modelled turbulence
+exists which result in low values of cf. This artefact can easily be circumvented
+in future simulations by narrowing the LES zone in spanwise direction and
+thus generate modelled turbulence in the respective regions. Nevertheless, the
+
+Ma 0.1
+0.5
+0.9
+1.3
+1.7
+-0.76
+-0.78
+-0.8
+N
+-0.82
+-0.84
+0
+0.05
+0.1
+0.15
+0.2
+0.25
+0.3
+0.35
+0.02 CTU
+x/c-0.76
+-0.78
+C
+-0.8
+N
+-0.82
+-0.84
+0
+0.05
+0.1
+0.15
+0.2
+0.25
+0.3
+0.35
+0.5 CTU
+x/c-0.76
+-0.78
+-0.8
+N
+-0.82
+-0.84
+0
+0.05
+0.1
+0.15
+0.2
+0.25
+0.3
+0.35
+1 CTU
+x/c-0.76
+-0.78
+-0.8
+N
+-0.82
+-0.84
+0
+0.05
+0.1
+0.15
+0.2
+0.25
+0.3
+0.35
+1.5 CTU
+x/cSpringer Nature 2021 LATEX template
+16
+Grey area in Embedded WMLES on a nacelle-aircraft configuration
+described phenomenon is limited to the boundaries and does not affect the
+actual focus region.
+To give an impression of the vortex structure of the resolved turbulence an
+isosurface of the Q-criterion (Q = 1010) at t = 1.5 CTU is depicted in Fig.
+9. As already observed in Fig. 8 an extensive formation of turbulent struc-
+tures within the refinement region is present. These structures are growing
+with increasing streamwise position and partially evolve into horseshoe vortices
+which corresponds to expected flow behaviour.
+4.4 Investigation of grey area
+In the following a quantitave analysis of the grey area / adaption region is
+performed. Therefore the flow field was averaged with regard to time and
+spanwise direction ϕ. The temporal average was applied for 0.42 ≤ t/CTU ≤
+1.5. The start time t = 0.42 is chosen such that the resolved turbulence is
+completely established within the focus region (0.06 ≤ x/c ≤ 0.25) and no
+remains of the initial RANS-solution are present in this area (cf. Fig. 8 at
+t = 0.5 CTU). The spanwise average was applied over the refinement section
+such that the areas of underresolved turbulence at its margins were omitted
+(ϕ ∈ [125◦; 220◦]).
+Fig. 10 (top) shows the result of the EWMLES mean pressure distribution
+(mean-cp) along with the initial RANS solution. Good agreement between
+these curves are present for x/c ≤ 0.13 where x/c = 0.13 is the average location
+of the shock front of the SST-RANS solution which results into a sudden rise in
+mean-cp. It is apparent that this agreement also persists for positions upstream
+of the STG (x/c ≤ 0.06) which indicates that no upstream effect of the STG
+exists. With regard to the EWMLES shock position the already described
+shift in downstream direction is also present in this depiction and located at
+x/c = 0.15. Due to the comparatively early start in the averaging of mean-cp
+it is not reasonable to compare the curves for x/c ≥ 0.3 since transient effects
+from the switch from RANS to EWMLES still exist in this area.
+A further quantitive flow comparison between SST-RANS and EWMLES is
+given in Fig. 10 (bottom) which shows mean skin friction distributions (mean-
+cf). In the flow region upstream of the STG (x/c ≤ 0.06) good agreement
+are visible again which confirms the previously mentioned absence of potential
+STG upstream effects. However, for 0.06 ≤ x/c ≤ 0.16 remarkable deviations
+appear. One observes a significant drop in mean-cf directly downstream of
+the STG and its increase with a peak value at x/c = 0.13 and a mean-cf-
+level which is comparable to the mean-cf value at the STG position. Although
+a similar behaviour is present for the flat plate flow as described in Sec. 3.2
+the flat plate variations in mean-cf are of significantly smaller. The adaption
+length which measures the distance between STG position and subsequent
+peak in mean-cf amounts 46 δST G where δST G represents the boundary layer
+thickness at the STG position. In case of the flat plate flow this adaption
+length only amounts 6 δST G (cf. Fig. 2). A further analysis of these deviations
+with reference to the flat plate flow are given in Sec. 4.6. Considering now the
+
+Springer Nature 2021 LATEX template
+Grey area in Embedded WMLES on a nacelle-aircraft configuration
+17
+Fig. 8 Temporal evolution of cf-distribution within the refinement area on projected nacelle
+surface.
+
+0.000
+0.002
+0.003
+0.005
+0.007
+0.008
+0.010
+0.65
+y/c
+0.6
+0.55
+0.5
+0.45
+0.4
+0.35
+0.3
+0.25
+0.2
+0.15
+0.1
+0.05
+0
+0.1
+0.2
+0.3
+0.4
+0.5
+0.6
+0.7
+0.8
+0.9
+1
+0.02 CTU
+X/C0.65
+C
+0.6
+0.55
+0.5
+0.45
+0.4
+0.35
+0.3
+0.25
+0.2
+0.15
+0.1
+0.05
+0
+0.1
+0.2
+0.3
+0.4
+0.5
+0.6
+0.7
+0.8
+0.9
+0.5 CTU
+X/C0.65
+0.6
+0.55
+0.5
+0.45
+0.4
+0.35
+0.3
+0.25
+0.2
+0.15
+0.1
+0.05
+0
+0.1
+0.2
+0.3
+0.4
+0.5
+0.6
+0.7
+0.8
+0.9
+1 CTU
+X/C0.65
+y/c
+0.6
+0.55
+0.5
+0.45
+0.4
+0.35
+0.3
+0.25
+0.2
+0.15
+0.1
+0.05
+0
+0.1
+0.2
+0.3
+0.4
+0.5
+0.6
+0.7
+0.8
+0.9
+1
+1.5 CTU
+X/CSpringer Nature 2021 LATEX template
+18
+Grey area in Embedded WMLES on a nacelle-aircraft configuration
+Fig. 9 Isosurface of Q-Criterion (Q = 1010) at nacelle lower surface for LD2 scheme at
+t = 1.5 CTU.
+region where 0.16 ≤ x/c ≤ 0.25 we observe that the region of recirculation has
+disappeared, at least for this transient period of time averaging since mean-cf
+is always positive. Furthermore additional distortions in the EWMLES mean-
+cf distribution appear at x/c = 0.25 and x/c = 0.40 which corresponds to
+locations of the ∆ϕ coarsening steps of the mesh (cf. Sec. 4.2.2). This indicates
+that the local mesh resolutions of r∆ϕ = δϕ,min/10 might be locally at the
+lower limit at these positions.
+4.5 Sensitivity studies
+4.5.1 Positioning of the RANS-LES interface
+Preliminary grid number estimations for different locations of the RANS-LES
+interface in x-direction (xST G) demonstrated a strong dependence of xST G
+and the total grid number. A shift of this boundary in downstream direction
+allows to reduce the total grid number significantly. Exemplarily, moving xST G
+by 0.02c enables to reduce the total grid size about 100 Mio points without
+violating the applied extension and resolution constraints for the refinement
+area. This dependence is a consequence of the shortening of the refinement area
+in x-direction by which the subregion with the highest cell density is narrowed.
+Also, due to the dependence of ∆ϕΩ1 on δϕ,min(xST G) in subregion Ω1 it is
+possible to increase ∆ϕΩ1 in the entire interval x/c ∈ [xST G; 0.16] (cf. 4.2.2).
+This dependency on the STG position suggests to place the RANS-LES
+boundary as close as possible to the shock front and examine its effect on the
+flow solution. Based on the original assumption that the adaption length of the
+STG amounts less than 10 δST G we estimated xST G/c = 0.08 as latest possible
+position in order to avoid direct interactions with the shock front. Additionally,
+for this estimation a potential shock movement in upstream direction until
+xs,min = 0.1 was taken into account. For the following examinations we used
+
+Ma 0.2
+0.6
+1.4
+1.8Springer Nature 2021 LATEX template
+Grey area in Embedded WMLES on a nacelle-aircraft configuration
+19
+Fig. 10 Quantitave comparison of time and spanwise averaged pressure - (top) and skin
+friction distributions (bottom) between the initial RANS and EWMLES solutions.
+the same mesh as before to verify a basic applicability of a late RANS-LES
+interface.
+Fig. 11 shows mean-cp and mean-cf distributions of the EWMLES results
+for xST G/c = 0.08 (green curves) where the same averaging procedure as in
+Sec. 4.4 is employed. It is striking that the mean-cp distribution is almost
+identical to the previous xST G/c = 0.06 result (red) with maximum deviations
+of two line thicknesses for x/c ≥ 0.16. However, with respect to mean-cf and
+its adaption area downstream of the STG distinct differences compared to the
+xST G/c = 0.06 result exist. Firstly, the initial decay is significantly weaker than
+before. Furthermore, its adaption length is reduced and only amounts 19 δST G
+so that its peak is located at almost the same position as for the xST G/c = 0.06
+result. The peak value though, is significantly reduced and corresponding to
+the initial RANS solution directly upstream of the shock position. A further
+discussion of these features of the adaption regions is given in Sec. 4.6. It is
+remarkable that for x/c ≥ 0.16 the subsequent mean-cf evolution is almost
+identical to the xST G/c = 0.06 result which demonstrates an independence of
+the flow solution with regard to the location of the RANS-LES interface.
+4.5.2 Impact of Numerical Scheme
+A further objective of our research was to compare the effect of different numer-
+ical schemes for the central discretisation of viscous fluxes which is applied in
+the refinement region (LES). In addition to the already employed LD2 scheme
+(Sec. 2.3) a reference central-scheme (Eq. 6 in Sec. 2.3) is applied on the same
+
+RANS
+EWMLES, STG 0.06, LD2
+-1.2
+-1
+mean-cp
+-0.8
+-0.6
+-0.4
+-0.2
+0
+0.1
+0.2
+0.3
+0.4
+0.5
+0.6
+x/c
+-0.75
+N
+-0.8
+-0.85
+0
+0.1
+0.2
+0.3
+0.4
+0.5
+0.6
+x/c0.006
+RANS
+EWMLES. STG 0.06, LD2
+0.005
+0.004
+mean-cf
+0.003
+0.002
+0.001
+0
+0
+0.1
+0.2
+0.3
+0.4
+0.5
+0.6
+x/c
+-0.75
+N
+-0.8
+-0.85
+0
+0.1
+0.2
+0.3
+0.4
+0.5
+0.6
+x/cSpringer Nature 2021 LATEX template
+20
+Grey area in Embedded WMLES on a nacelle-aircraft configuration
+Fig. 11 Effect of positioning of the RANS-LES interface on averaged surface pressure and
+skin friction distributions.
+numerical setup as in Sec. 4.4. Although the necessity of the high quality LD2
+scheme against the reference scheme has been demonstrated with the aid of
+the DIT-testcase in 3.1 it is not obvious how the reference scheme performs for
+transonic flows on a 3D configuration. To give a qualitative impression of the
+flowfield the Q-Criterion at Q = 1010 for a snapshot at t = 1.5 CTU is shown
+in Fig. 12 which can directly compared to Fig. 9. The comparison shows that
+the previous formation of turbulent structures is now partially interrupted.
+Especially the region directly downstream of the STG lacks turbulent struc-
+tures. It is striking that coarser structures such as the clearly visible horseshoe
+vortexes are preserved whereas tiny structures are vanished. This is in direct
+agreement with the results from the DIT testcase which demonstrates that
+small turbulent scales are strongly damped by the reference scheme (cf. Fig.1).
+These observations are also present in the analysis of the average skin fric-
+tion distribution (blue curve in Fig. 13). Whereas the mean surface pressure
+is hardly affected by the numerical scheme, mean-cf shows large deviations.
+Especially the decay downstream of the STG indicates a lack of resolved
+turbulence. Additionally, compared to the LD2 results the mean-cf level is
+underestimated in the area downstream of the shock - boundary layer interac-
+tion (0.35 ≤ x/c ≤ 0.6). This confirms the previous observation of Fig. 12 of
+underresolved turbulence throughout the entire refinement region.
+
+RANS
+EWMLES, STG 0.06, LD2
+-1.2
+EWMLES, STG 0.08, LD2
+-1
+mean-cp
+-0.8
+-0.6
+-0.4
+-0.2
+0
+0.1
+0.2
+0.3
+0.4
+0.5
+0.6
+x/c
+-0.75
+N
+-0.8
+-0.85
+0
+0.1
+0.2
+0.3
+0.4
+0.5
+0.6
+x/c0.006
+RANS
+EWMLES, STG 0.06. LD2
+0.005
+EWMLES, STG 0.08, LD2
+0.004
+mean-cf
+0.003
+0.002
+0.001
+0
+0
+0.1
+0.2
+0.3
+0.4
+0.5
+0.6
+x/c
+-0.75
+N
+-0.8
+-0.85
+0
+0.1
+0.2
+0.3
+0.4
+0.5
+0.6
+x/cSpringer Nature 2021 LATEX template
+Grey area in Embedded WMLES on a nacelle-aircraft configuration
+21
+Fig. 12 Isosurface of Q-Criterion (Q = 1010) for reference central-scheme at nacelle lower
+at t = 1.5 CTU.
+Fig. 13 Effect of different numerical schemes on averaged surface pressure and skin friction
+distributions.
+4.6 Reynolds number and mesh resolution effect on STG
+adaption region
+In the following we address the so far unsound behaviour of the adaption
+region downstream of the STG arising for all shown configurations. As already
+described before the adaption region displays the largest deviations with regard
+to adaption length as well as maximal and minimal mean-cf-deviations for the
+
+Ma 0.2
+0.6
+1.4
+1.8RANS
+-1.2
+EWMLES. STG 0.06. LD2
+EWMLES. STG 0.08. LD2
+EWMLES, STG 0.06, Reference
+-1
+mean-cp
+-0.8
+-0.6
+-0.4
+-0.2
+0
+0.1
+0.2
+0.3
+0.4
+0.5
+0.6
+x/c
+-0.75
+Ni
+-0.8
+-0.85
+0
+0.1
+0.2
+0.3
+0.4
+0.5
+0.6
+x/c0.006
+RANS
+EWMLES. STG 0.06. LD2
+0.005
+EWMLES. STG 0.08. LD2
+EWMLES, STG 0.06, Reference
+0.004
+mean-cf
+0.003
+0.002
+0.001
+0
+0
+0.1
+0.2
+0.3
+0.4
+0.5
+0.6
+x/c
+-0.75
+Ni
+-0.8
+-0.85
+0
+0.1
+0.2
+0.3
+0.4
+0.5
+0.6
+x/cSpringer Nature 2021 LATEX template
+22
+Grey area in Embedded WMLES on a nacelle-aircraft configuration
+nacelle at xST G = 0.06c. These features reduce for xST G = 0.08c and almost
+vanish but are still present for the flat plate test case (cf. Fig. 2 and 11). A
+closer look into the flow properties and mesh resolution at the location of the
+STG suggests a dependency on Reδ,ST G (Tab. 1). Here, Reδ,ST G is defined as a
+Reynolds number referring to the local boundary layer thickness δST G as well
+as velocity and kinematic viscosity at the outer edge of δST G. This Reynolds
+number, which directly impacts the input statistics of the STG, has its lowest
+number for the nacelle case at xST G = 0.06c (4989) and increases for xST G =
+0.06c (6975) and the flat plate flow (24200). The ratio of turbulent- and laminar
+viscosity (max (µt/µl)) which serves as measure of modelled turbulence shows
+a comparable trend. Since low Reynolds numbers enhance the stability of the
+boundary layer and hence suppress turbulent fluctuations, this might lead to a
+damping of the injected turbulent structures. As a consequence the boundary
+layer evolves into a flow with significantly reduced turbulence which is visible
+in a strongly reduced level of mean-cf. Thus, it appears that the distinct
+adaption region can be traced back to a low-Reynolds number effect.
+Another reason might be due to the mesh resolution ∆y which amounts
+δ/20 for the flat plate flow and coarsens to δ/16 and δ/12 for xST G = 0.08c
+and xST G = 0.06c, respectively (cf. Tab. 1). Since a resolution of ∆y = δ/20 is
+actually defined as coarsest resolution in this flow direction the here observed
+somewhat coarser resolutions might perturb a proper development of the
+turbulent boundary layer [3].
+Therefore further examinations of the transonic nacelle flow for higher Re∞
+(resulting in larger Reδ) as well as finer resolutions ∆y will be performed in
+future work in order to provide a verification of the here detected limits of
+synthetic turbulence generation at locally low Reynolds numbers.
+Re∞
+δST G/m
+Reδ,ST G
+∆x
+∆y
+max (µt/µl)
+Flat Plate
+4.7 Mio
+0.006
+24200
+δ/10
+δ/20
+87
+Nacelle
+3.3 Mio
+0.00024
+4989
+δ/11.2
+δ/11.76
+9
+xST G = 0.06c
+Nacelle
+3.3 Mio
+0.00033
+6975
+δ/13.75
+δ/16.17
+10
+xST G = 0.08c
+Table 1 Comparison of several local flow quantities at the location of the synthetic
+turbulence generator for all presented configurations.
+5 Conclusions
+A scale-resolving WMLES methodology in conjunction with the SST tur-
+bulence model was applied to the XRF-1 aircraft configuration with UHBR
+nacelle at transonic flow conditions. The method was applied locally at the
+
+Springer Nature 2021 LATEX template
+Grey area in Embedded WMLES on a nacelle-aircraft configuration
+23
+nacelle surface in order to examine shock induced separation. A Synthetic
+Turbulence Generator (STG) was employed to enhance the transition from
+modelled to resolved turbulence at the RANS-LES interface.
+Prior to the actual examination on the aircraft configurations basic func-
+tionalities of the methodology were successfully verified for flows of decaying
+isotropic turbulence and a flow over a flat plate for Reθ = 3030.
+With regard to the target configuration a sophisticated mesh which refines
+32 % of the nacelle outer surfaces and comprises 420 million grid points was
+constructed. The main features of the mesh design are the dependence of mesh
+resolution (∆x, ∆y and ∆z) on the local boundary layer thickness and the
+consideration of a potential shock movement due to buffet.
+Analysis of the transient process of the simulation showed a well resolved
+formation of turbulent structures over almost the entire refinement region with
+a broad spectrum of turbulent scales. It has been demonstrated that these
+features are also the result of the employed LD2 scheme. For a reference central-
+scheme with higher artificial dissipation, small turbulent scales are damped
+leading to globally underresolved turbulence.
+Another outcome of this study is the observation that the STG - adaption
+region correlates to the local Reynolds number as well as mesh resolution in
+spanwise direction. For decreasing Reynolds numbers and coarser mesh resolu-
+tions an increasing adaption length and more distinct decay in the skin friction
+distribution were observed. We note that the methodology is only applicable
+if the STG adaption region does not interfere with the transonic shock front
+and therefore sufficient distance to the shock is required. This distance might
+not be given in case of an upstream moving shock which would arise for strong
+shock buffet at the given Reynolds number. Therefore further research on the
+transonic nacelle flow for higher Reynolds numbers as well as finer resolutions
+will be performed in future work to verify a potential reduction of the adaption
+length.
+Acknowledgments.
+The authors gratefully acknowledge the Deutsche
+Forschungsgemeinschaft DFG (German Research Foundation) for funding this
+work in the framework of the research unit FOR 2895. The authors thank the
+Helmholtz Gemeinschaft HGF (Helmholtz Association), Deutsches Zentrum
+f¨ur Luft- und Raumfahrt DLR (German AerospaceCenter) and Airbus for pro-
+viding the wind tunnel model and financing the wind tunnel measurements
+Additionally, the authors gratefully acknowledge the computing time granted
+by the Resource Allocation Board and provided on the supercomputer Lise and
+Emmy at NHR@ZIB and NHR@G¨ottingen as part of the NHR infrastructure.
+The calculations for this research were conducted with computing resources
+under the project nii00164.
+Declarations
+• Funding: This study was funded by DFG (German Research Foundation).
+
+Springer Nature 2021 LATEX template
+24
+Grey area in Embedded WMLES on a nacelle-aircraft configuration
+• Competing interests: The authors have no competing interests to declare
+that are relevant to the content of this article.
+• Ethics approval: Not applicable
+• Consent to participate: Not applicable
+• Consent for publication: Not applicable
+• Availability of data and materials: Not applicable
+• Code availability: Not applicable
+• Authors’ contributions: Not applicable
+References
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+

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+arXiv:2301.02633v1  [math.DG]  6 Jan 2023
+COUNTEREXAMPLES TO A DIVERGENCE LOWER BOUND FOR THE
+COVARIANT DERIVATIVE OF SKEW-SYMMETRIC 2-TENSOR FIELDS
+STEFANO BORGHINI AND LORENZO MAZZIERI
+Abstract. In [3] an estimate for suitable skew-symmetric 2-tensors was claimed.
+Soon after,
+this estimate has been exploited to claim powerful classification results: most notably, it has been
+employed to propose a proof of a Black Hole Uniqueness Theorem for vacuum static spacetimes
+with positive scalar curvature [6] and in connection with the Besse Conjecture [8]. In the present
+note we point out an issue in the argument proposed in [3] and we provide a counterexample to
+the estimate.
+1. Introduction
+The Black Hole Uniqueness Theorem for three-dimensional static solutions with positive scalar
+curvature and the Besse Conjecture for solutions to the Critical Point Equation are two very
+famous and related open problems in contemporary geometric analysis. Very recently, some very
+remarkable advances have been claimed on both of these problems in a series of papers [1, 2, 3, 6,
+7, 8]. In this short note, we point out an issue in the approach proposed in the above mentioned
+papers, providing counterexamples.
+To introduce the problems of interest together with some notation, let us recall that a three-
+dimensional static solution is a triple (M, g, f) satisfying
+fRic = ∇2f + R
+2 f g ,
+∆f = −R
+2 f ,
+(1.1)
+where (M, g) is a Riemannian manifold, f is a smooth function and Ric and R denote the Ricci
+tensor and the scalar curvature of g, respectively. When R is positive, it is natural to suppose that
+(M, g) is a compact manifold with boundary and that f is vanishing on the boundary. A strictly
+related problem is the so called Critical Point Equation, which consists in the following system
+(1 + f)
+�
+Ric − R
+n g
+�
+= ∇2f +
+R
+n(n − 1) g ,
+∆f = −
+R
+n − 1f
+(1.2)
+where the unknowns are given by the triple (M, g, f), with (M, g) a closed Riemannian manifold
+and f a smooth function.
+In [3], the authors aim at classifying solutions to the Critical Point Equation subject to the
+condition of having Positive Isotropic Curvature. To this end, they consider the differential 2-form
+ω = df ∧ ι∇fz ,
+where z indicates the traceless Ricci tensor, and they claim that it must vanish. Notice that,
+using (1.2), the differential 2-form ω can be rewritten as
+ω =
+1
+2(1 + f)df ∧ d|∇f|2 ,
+where | · | is the norm computed with respect to the metric g. If ω ≡ 0, then, using again the
+equation (1.2), one can prove that the Cotton tensor of g must also vanish, by a direct computation.
+It follows that either n = 3 and g is Locally Conformally Flat, or else n ≥ 4 and g has harmonic
+Weyl tensor. In both cases, the classification follows easily. The same strategy is adopted in [6]1,
+1Notice that this reference has been withdrawn by the authors during the preparation of the present note.
+1
+
+2
+S. BORGHINI AND L. MAZZIERI
+where this time the differential 2-form ω is defined as
+ω =
+1
+2f df ∧ d|∇f|2 ,
+with g and f satisfying (1.1). In both cases, the vanishing of ω is deduced through an integration
+by parts argument – which we describe in Subsection 2.2 below, in the case of static metrics –
+making a substantial use of the key estimate
+|∇ω|2 ≥ |δω|2 ,
+(1.3)
+which the authors claim to hold at all points of M where ω is not vanishing (see Lemma 5.5 in [3]).
+The proposed proof of (1.3) does not make use of the full strength of either (1.1) or (1.2). In fact,
+it is based on a local computation, in which the global structure of M is not playing any role. As
+such, if correct, it should work for every differential 2-form having the structure
+ω = λ(f) df ∧ d|∇f|2 .
+(1.4)
+for some smooth function λ = λ(f), independently of the validity of (1.1) or (1.2). Aim of the
+present note is to disprove the claim that every ω as in (1.4), defined on an open subset of a
+Riemannian manifold (M, g), satisfies estimate (1.3).
+In Section 3 we point out the issue in the original proof of (1.3), given in [3, Lemma 5.5], whereas
+in Section 4 we provide effective counterexamples to the claim. Namely, we show that
+For every smooth real function λ ̸≡ 0, there exist a smooth Riemannian metric g and a smooth
+function f such that |∇ω|2 < |δω|2, with ω = λ(f) df ∧ d|∇f|2.
+For the sake of completeness, we discuss in Section 2 how the validity of an estimate like (1.3)
+can be exploited to deduce that ω must vanish everywhere.
+2. Analysis of a skew-symmetric 2-tensor field
+To make our computations more transparent, we prefer to work with the tensor-fields formalism.
+However one can also work with the formalism of differential forms as done in [3]. Instead of ω
+defined as in (2.1), we consider the skew-symmetric 2-tensor field P, given by
+P = λ(f)
+�
+df ⊗ d|∇f|2 − d|∇f|2 ⊗ df
+�
+,
+(2.1)
+with λ, f and g as above. In this formalism, we have that estimate (1.3) is equivalent to
+|∇P|2 ≥ 2 |divP|2 ,
+(2.2)
+as 2 |∇ω|2 = |∇P|2 (the factor two comes from the slight difference in the definition of norms on dif-
+ferential forms and tensor, namely |∇ω|2 = �
+j<k
+�
+i(∇iωjk)2, whereas |∇P|2 = �
+j,k
+�
+i(∇iPjk)2)
+and δω = −divP. Notice that, replacing the constant 2 with the smaller constant 1/n, one gets
+the always valid lower bound |∇P|2 ≥ (1/n) |divP|2. Furthermore, exploiting the special struc-
+ture (2.1) of P, one can significantly improve on this bound, obtaining (n − 1)|∇P|2 ≥ 2 |divP|2
+(see the appendix). On the other hand, estimate (2.2) is too strong and cannot hold in general, as
+we will discuss below.
+2.1. Two differential identities. Here we discuss some basic though fundamental properties of
+a skew-symmetric 2-tensor P having the form (2.1).
+Proposition 2.1. Let (M, g) be a n-dimensional Riemannian manifold and let f ∈ C ∞(M).
+Then, the skew-symmetric 2-tensor field P defined as in (2.1), for some smooth real function λ,
+satisfies the identity
+∇P(X, Y, Z) + ∇P(Y, Z, X) + ∇P(Z, X, Y ) = 0 .
+Remark 1. In terms of the differential 2-form ω defined as in (1.4), the above identity is telling
+us that ω is closed, as observed in [3, Lemma 5.4]. Observe that, if ω is as in (1.4), then it is
+straightforward to realize that dω = (dλ/df) df ∧ df ∧ d|∇f|2 = 0.
+
+COUNTEREXAMPLES TO A DIVERGENCE LOWER BOUND FOR THE COVARIANT DERIVATIVE OF SKEW-SYMMETRIC 2-TENSOR FIELDS3
+Proof. For simplicity we work with normal coordinates {x1, . . . , xn}. A simple computation gives
+∇iPjk =
+˙λ
+λPjk∇if + λ
+�
+∇2
+ijf∇k|∇f|2 − ∇j|∇f|2∇2
+ikf
+�
++ λ
+�
+∇jf∇2
+ik|∇f|2 − ∇2
+ij|∇f|2∇kf
+�
+.
+It is now a matter of computation to check that the sums over rotating indexes of the three pieces
+on the right hand side give zero. We compute
+Pjk∇if + Pki∇jf + Pij∇kf = λ
+�
+∇if∇jf∇k|∇f|2 − ∇if∇j|∇f|2∇kf
++ ∇jf∇kf∇i|∇f|2 − ∇jf∇k|∇f|2∇if + ∇kf∇if∇j|∇f|2 − ∇kf∇i|∇f|2∇jf
+�
+= 0 .
+Similarly, one has
+∇2
+ijf∇k|∇f|2 − ∇j|∇f|2∇2
+ikf + ∇2
+jkf∇i|∇f|2 − ∇k|∇f|2∇2
+jif + ∇2
+kif∇j|∇f|2 − ∇i|∇f|2∇2
+kjf = 0 ,
+∇jf∇2
+ik|∇f|2 − ∇2
+ij|∇f|2∇kf + ∇kf∇2
+ji|∇f|2 − ∇2
+jk|∇f|2∇if + ∇if∇2
+kj|∇f|2 − ∇2
+ki|∇f|2∇jf = 0 .
+It follows then that
+∇iPjk + ∇jPki + ∇kPij = 0 ,
+as claimed.
+□
+Another interesting property of P is that it satisfies a Bochner-type formula, as it is established
+in the following proposition.
+Proposition 2.2. Let (M, g) be a n-dimensional Riemannian manifold and let f ∈ C ∞(M).
+Then, the skew-symmetric 2-tensor field P defined as in (2.1), for some smooth real function λ,
+satisfies the identity
+1
+2∆|P|2 = |∇P|2 + 2⟨P | ∇(div P)⟩ +
+2R
+(n − 1)(n − 2)|P|2 + 2n − 4
+n − 2RjsPskPjk + 2WijksPisPjk .
+Proof. We perform our computations with respect to normal coordinates.
+Exploiting Proposi-
+tion 2.1 and the skew-symmetry of P, we compute
+∆|P|2 = 2∇i(Pjk∇iPjk)
+= 2|∇P|2 + 2Pjk∆Pjk
+= 2|∇P|2 − 2Pjk∇2
+ijPki − 2Pjk∇2
+ikPij
+= 2|∇P|2 + 4Pjk∇2
+ijPik
+= 2|∇P|2 + 4Pjk
+�
+∇2
+jiPik + RijisPsk + RijksPis
+�
+= 2|∇P|2 + 4Pjk (∇j(div P)k + RjsPsk + RijksPis) .
+To obtain the claimed identity, it is now enough to substitute the general formula for the Riemann
+tensor
+Rijks = −
+R
+(n − 1)(n − 2)(gikgjs − gisgjk) +
+1
+n − 2 (Rikgjs − Risgjk + gikRjs − gisRjk) + Wijks
+in the computation above.
+□
+The differential identity obtained in the previous proposition simplifies significantly when n = 3,
+since in this case the Weyl tensor vanishes and we get
+1
+2∆|P|2 = |∇P|2 + 2⟨P | ∇(div P)⟩ + R|P|2 − 2RjsPskPjk .
+(2.3)
+
+4
+S. BORGHINI AND L. MAZZIERI
+2.2. Application to 3-dimensional static solutions. In [6] a classification result for 3-dimensional
+static metrics with positive scalar curvature was proposed, building on the above Bochner-type
+formula and on the validity of estimate (3.4). For completeness, here we retrace their proof.
+Using formula (1.1), we can substitute the Ricci tensor in (2.3), getting
+1
+2∆|P|2 = |∇P|2 + 2⟨P | ∇(div P)⟩ + R
+2 |P|2 + 2
+f P(∇f, div P) − 1
+2f ⟨∇f | ∇|P|2⟩ ,
+(2.4)
+which can be rewritten as
+1
+2div(f|P|2) = f|∇P|2 + 2f⟨P | ∇(div P)⟩ + R
+2 f |P|2 + 2P(∇f, div P) .
+Since M is compact and f = 0 on ∂M, integrating by parts we obtain then
+0 =
+ˆ
+M
+�
+f|∇P|2 − 2f|div P|2 + R
+2 f |P|2
+�
+dµ .
+Here one can appreciate the strength of estimate (2.2). Indeed, if (2.2) is in force and R > 0, then
+|P|2 must vanish identically and we obtain the following
+Proposition 2.3. Let (M, g, f) be a compact three-dimensional static solution with positive scalar
+curvature and nonempty boundary. Assume that f = 0 on ∂M and positive in the interior. If
+estimate (2.2) holds for some P as in (2.1), then P must vanish identically and one has
+df ⊗ d|∇f|2 = d|∇f|2 ⊗ df .
+This is a crucial step in the strategy outlined in [6]. As anticipated, they exploit the identity
+P = 0 in combination with the static equation to deduce that the Cotton tensor must vanish. The
+classification follows, invoking a well known result by Kobayashi [4] and Lafontaine [5].
+As we are going to see in the next sections, it is not clear how to establish the validity of (2.2)
+in general, however we will prove in the appendix that the weaker lower bound |∇P|2 ≥ |divP|2
+holds true. This leads to
+ˆ
+M
+f|div P|2dµ ≥
+ˆ
+M
+R
+2 f |P|2dµ .
+Building on this integral inequality, one might classify three-dimensional static metrics with posi-
+tive scalar curvature admitting a divergence-free P-tensor.
+3. The issue in the proof of the estimate
+Here we retrace the proof of estimate (1.3) originally proposed in [3, Lemma 5.5], pointing out
+the main issue in the argument.
+As a first step, the authors find a local orthonormal frame with respect to which the tensor P
+has a nice structure. This part of the proof appears to be correct and it is an interesting fact
+on its own that will also be helpful in the appendix, so we include it here as a lemma. In the
+following statement it is helpful to consider the vector valued 1-form A : TM → TM defined by
+P(X, Y ) = g(AX, Y ). In coordinates: Aj
+i = gjmPim.
+Lemma 3.1. Let (M, g) be a n-dimensional Riemannian manifold. Let f ∈ C ∞(M) and let P be
+the tensor defined by (2.1). Let x ∈ M be a point with |P|(x) ̸= 0. Then in a small neighborhood
+U of x it holds |P| ̸= 0, |∇f| ̸= 0, |A∇f| ̸= 0 and there exists a smooth orthonormal frame
+{E1, . . . , En} with E1 = ∇f/|∇f| and E2 = AE1/|AE1|. With respect to this frame, the tensor P
+rewrites as
+P = u
+�
+θ1 ⊗ θ2 − θ2 ⊗ θ1�
+,
+(3.1)
+where u is a smooth function and {θ1, . . . , θn} is the dual coframe of {E1, . . . , En} (namely,
+θi(Ej) = δi
+j at any point in U).
+
+COUNTEREXAMPLES TO A DIVERGENCE LOWER BOUND FOR THE COVARIANT DERIVATIVE OF SKEW-SYMMETRIC 2-TENSOR FIELDS5
+Proof. A proof of this fact is given in [3], however we write here a shorter self contained version.
+We first construct the orthonormal frame in the lemma.
+Consider a neighborhood U of x
+in which |P| ̸= 0.
+From the definition (2.1) of P, it is clear that |∇f| ̸= 0 in U as well.
+In
+particular the vector E1 = ∇f/|∇f| is well defined in U. We complete E1 to an orthonormal
+frame {E1, �E2, . . . , �En} in U. Since g(E1, �Ei) = 0 for i ≥ 2, we have ∇ �
+Eif = 0 for any i ≥ 2, hence
+P( �Ei, �Ej) = λ(f)
+�
+∇ �
+Eif ∇ �
+Ej|∇f|2 − ∇ �
+Ei|∇f|2 ∇ �
+Ejf
+�
+= 0 ,
+(3.2)
+for any i, j ≥ 2. Since |P| ̸= 0 in U, then at any point in U it holds g(AE1, �Ej) = P(E1, �Ej) ̸= 0
+for some j. In particular AE1 ̸= 0 in U. Since g(AE1, E1) = P(E1, E1) = 0, it follows that AE1 is
+orthogonal to E1. In particular, the vector E2 = AE1/|AE1| is well defined and orthonormal to E1
+on the whole U. We can then complete E1, E2 to an orthonormal frame {E1, . . . , En} in U. This
+is precisely the orthonormal frame described in the statement of the lemma. Notice in particular
+that
+P(E1, Ej) = g(AE1, Ej) = |AE1| g(E2, Ej) = |AE1| δ2j .
+In view of (3.2), we deduce that the only nonzero entries of P are P(E1, E2) = −P(E2, E1).
+Formula (3.1) follows.
+□
+Next, the authors compute |∇P|2 and |div P|2 with respect to this frame. The computations
+regarding |∇P|2 appear to be correct. On the other hand, it seems to us that the expression of
+the divergence term worked out by the authors contains a mistake. A simple calculation (see the
+appendix for more details) gives
+(div P)(E1) = −E2(u) +
+n
+�
+i=3
+⟨∇EiEi | E2⟩ u = −E2(u) +
+n
+�
+i=3
+⟨Ei | [E2, Ei]⟩ u ,
+(div P)(E2) = E1(u) −
+n
+�
+i=3
+⟨∇EiEi | E1⟩ u = E1(u) +
+n
+�
+i=3
+⟨Ei | [E1, Ei]⟩ u ,
+(div P)(Ek) = ⟨Ek | [E1, E2]⟩ u ,
+k ≥ 3 .
+(3.3)
+It is worth pointing out that the frame {E1, . . . , En} was constructed with a pointwise argument.
+The frame is easily seen to be smooth, but it is important to observe that it is not necessarily
+induced from a local coordinate system. In particular, the Lie brackets [Ei, Ej] are not necessarily
+vanishing. This seems to be the core of the issue: in fact, the authors claim that
+div P = −E2(u)θ1 + E1(u)θ2 .
+(3.4)
+In view of (3.3), this formula appears to be incorrect whenever the Lie brackets do not vanish.
+Remark 2. In [3], and more precisely in the final page of the proof of [3, Lemma 5.5] this formula is
+written as δω = E2(u)θ1 − E1(u)θ2. As already observed, ω corresponds to our P in the formalism
+of the differential forms, and the codifferential δ is clearly related to the divergence through the
+formula δω = −divP.
+4. Counterexamples to estimate (1.3)
+We work in dimension 3 for simplicity, but similar counterexamples might be constructed in
+higher dimension as well. Consider local coordinates {r, x1, x2} defined on an open set, a positive
+smooth function φ = φ(r) and the warped product metric
+g = dr �� dr + φ2(dx1 ⊗ dx1 + dx2 ⊗ dx2) .
+Let then f ∈ C ∞(M) be a smooth function of the form f = ψ ◦ x1, for some smooth nonconstant
+real function ψ. Let us consider then a skew-symmetric 2-tensor field P as in (2.1), for some choice
+of λ. In local coordinates, we have that the components of P are given by
+Pαβ = λ
+�
+∇αf∇2
+βηf − ∇βf∇2
+αηf
+�
+gησ∇σf = λ ψ′
+φ2
+�
+∇αf∇2
+1βf − ∇βf∇2
+1αf
+�
+,
+
+6
+S. BORGHINI AND L. MAZZIERI
+where the greek indexes are running in {r, 1, 2}. Here and in what follows we will denote with ′
+the derivatives with respect to x1 and with a dot the derivatives with respect to r. The Christoffel
+symbols of the metric g are as follows
+Γr
+rr = Γr
+ri = Γi
+rr = Γk
+ij = 0 ,
+Γr
+ij = −φ ˙φδij ,
+Γj
+ri =
+˙φ
+φδj
+i ,
+where the latin indexes are running in {1, 2}. It then follows easily that the only nonzero compo-
+nents of the Hessian are
+∇2
+11f = ψ′′ ,
+∇2
+1rf = −
+˙φ
+φ ψ′ ,
+and that
+P = λ
+˙φ
+φ3 (ψ′)3 �
+dr ⊗ dx1 − dx1 ⊗ dr
+�
+.
+Notice that we are in a setting similar to the one of Section 3, except that our frame
+{∂/∂r, ∂/∂x1, ∂/∂x2}
+is not orthonormal. Hence, to check that our P has the structure prescribed in (3.1), one should
+write its local expression, with respect to an orthonormal frame.
+This latter can be obtained
+setting E1 = (1/φ)∂/∂x1, E2 = ∂/∂r, E3 = (1/φ)∂/∂x2. Its dual orthonormal co-frame is then
+given by θ1 = φdx1, θ2 = dr, θ3 = φdx2. It is easy to check that this frame satisfies the properties
+described in Lemma 3.1 and that
+P = −λ
+˙φ
+φ4 (ψ′)3 �
+θ1 ⊗ θ2 − θ2 ⊗ θ1�
+.
+However, we prefer to perform our computations with respect to the frame fields induced by the
+local coordinates (r, x1, x2). In this framework, it is easy to show that the only nonzero components
+of ∇P are
+∇rP1r = −
+� ¨φ
+φ3 − 4
+˙φ2
+φ4
+�
+λ (ψ′)3 ,
+∇1P1r = −
+˙φ
+φ3 (λ (ψ′)3)′ ,
+∇2P12 = −
+˙φ2
+φ2 λ (ψ′)3 .
+It easily follows that
+divP = −
+˙φ
+φ5 (λ (ψ′)3)′ dr + λ (ψ′)3
+� ¨φ
+φ3 − 3
+˙φ2
+φ4
+�
+dx1 .
+Here it is possible to notice the discrepancy between our computations and formula (3.4), as
+computing the right hand side of that formula would give
+−
+˙φ
+φ5 (λ (ψ′)3)′ dr + λ (ψ′)3
+� ¨φ
+φ3 − 4
+˙φ2
+φ4
+�
+dx1 ,
+which looks very similar, but does not correspond to the correct value of divP. Computing the
+squared norms of ∇P and divP, one finally arrives at
+|∇P|2 − 2|divP|2 = 4λ2 ˙φ2(ψ′)6
+φ8
+�
+4
+˙φ2
+φ2 −
+¨φ
+φ
+�
+.
+To make this difference negative, it is then sufficient to specify a choice of the functions λ, ψ and
+φ such that the right hand side is negative. In particular, it is sufficient to choose φ in such a way
+that the quantity in round brackets is negative. This can be achieved, for example, setting
+φ = (r + c)−1/k,
+for some k > 3 and some c > 0 .
+
+COUNTEREXAMPLES TO A DIVERGENCE LOWER BOUND FOR THE COVARIANT DERIVATIVE OF SKEW-SYMMETRIC 2-TENSOR FIELDS7
+It follows that, with this choice of φ, for any λ and any f = ψ ◦ x1, the estimate (2.2) does not
+hold. Hence, the lower bound (1.3) is false as well.
+Appendix
+For completeness, let us point out the correct relation always holding between |∇P| and |divP|.
+Let (M, g) be a n-dimensional Riemannian manifold, n ≥ 3. As in Section 3, we take a point x with
+|P|(x) ̸= 0 and we consider the local orthonormal frame {E1, . . . , En} provided by Lemma 3.1. We
+recall that, with respect to this frame, the tensor P takes the following form
+P = u
+�
+θ1 ⊗ θ2 − θ2 ⊗ θ1�
+.
+(4.1)
+Exploiting the compatibility of ∇ with the metric g, for any i, j, k we have
+0 = Ei (g(Ej, Ek)) = g(∇EiEj, Ek) + g(Ej, ∇EiEk) ,
+and in particular
+g(∇EiEk, Ek) = 0 ,
+g(∇EiEi, Ek) = −g(Ei, ∇EiEk) = −g(Ei, [Ei, Ek]) .
+We are now ready to compute the components of ∇P. Since P(Ei, Ej) = 0 whenever {i, j} ̸= {1, 2},
+we have
+∇EiP(E1, E2) = Ei (P(E1, E2)) − P(∇EiE1, E2) − P(E1, ∇EiE2)
+= Ei(u) − g(∇EiE1, E1)P(E1, E2) − g(∇EiE2, E2)P(E1, E2)
+= Ei(u) .
+Similarly, for any k ≥ 3, we have
+∇EiP(E1, Ek) = Ei(P(E1, Ek)) − P(∇EiE1, Ek) − P(E1, ∇EiEk)
+= − g(∇EiEk, E2)P(E1, E2)
+= − g(∇EiEk, E2) u ,
+and
+∇EiP(E2, Ek) = Ei(P(E2, Ek)) − P(∇EiE2, Ek) − P(E2, ∇EiEk)
+= − g(∇EiEk, E1)P(E2, E1)
+= g(∇EiEk, E1) u .
+Similarly, one computes ∇EiP(E1, E1) = ∇EiP(E2, E2) = 0 and ∇EiP(Ej, Ek) = 0 whenever j, k
+are ≥ 3. It is now easy to compute the divergence of P:
+(div P)(E1) = −E2(u) +
+n
+�
+i=3
+⟨∇EiEi | E2⟩ u = −E2(u) +
+n
+�
+i=3
+⟨Ei | [E2, Ei]⟩ u ,
+(div P)(E2) = E1(u) −
+n
+�
+i=3
+⟨∇EiEi | E1⟩ u = E1(u) −
+n
+�
+i=3
+⟨Ei | [E1, Ei]⟩ u ,
+(div P)(Ei) = −g(∇E1Ei, E2) u + g(∇E2Ei, E1) u ,
+i ≥ 3 .
+Using the inequality (�k
+i=1 xi)2 ≤ k �k
+i=1 x2
+i , a simple calculation then gives
+|divP|2
+n − 1
+≤
+2
+�
+k=1
+�
+Ek(u)2 +
+n
+�
+i=3
+⟨Ei | [Ei, Ek]⟩2u2
+�
++
+2
+n − 1
+n
+�
+i=3
+�
+⟨∇E1Ei | E2⟩2 + ⟨∇E2Ei | E1⟩2�
+u2
+≤
+2
+�
+k=1
+�
+Ek(u)2 +
+n
+�
+i=3
+⟨Ei | [Ei, Ek]⟩2u2
+�
++
+n
+�
+i=3
+�
+⟨∇E1Ei | E2⟩2 + ⟨∇E2Ei | E1⟩2�
+u2 .
+
+8
+S. BORGHINI AND L. MAZZIERI
+On the other hand
+1
+2|∇P|2 ≥
+2
+�
+k=1
+�
+(∇EkP(E1, E2))2 +
+n
+�
+i=3
+(∇EiP(Ek, Ei))2 +
+n
+�
+i=3
+(∇EkP(Ek, Ei))2
+�
+=
+2
+�
+k=1
+�
+Ek(u)2 +
+n
+�
+i=3
+⟨Ei | [Ei, Ek]⟩2u2
+�
++
+n
+�
+i=3
+�
+⟨∇E1Ei | E2⟩2 + ⟨∇E2Ei | E1⟩2�
+u2 .
+In conclusion, we have shown the following.
+Proposition 4.1. Let (M, g) be a n-dimensional Riemannian manifold, n ≥ 3. Let f ∈ C ∞(M)
+and let P be the tensor defined by (2.1). Then, at any point of M it holds
+|∇P|2 ≥
+2
+n − 1 |divP|2 .
+(4.2)
+Proof. Estimate (4.2) follows immediately from the computations above at any point where P has
+the form (2.1), that is, at any point where |P| ̸= 0. Let then x be a point where |P| = 0. If |P|
+vanishes identically in a neighborhood of x, then |∇P| = |div P| = 0 in that neighborhood, and
+inequality (4.2) is trivially satisfied. Otherwise there exists a sequence of points xi converging to
+x with |P|(xi) ̸= 0. Since estimate (4.2) holds at the points xi, then it must hold at x as well by
+continuity.
+□
+Acknowledgements. The authors would like to thank R. Beig, P. T. Chru´sciel and W. Simon
+for stimulating discussions about the classification of static vacuum spacetimes. The authors are
+members of the Gruppo Nazionale per l’Analisi Matematica, la Probabilit`a e le loro Applicazioni
+(GNAMPA) of the Istituto Nazionale di Alta Matematica (INdAM).
+References
+[1] S. Hwang, M. Santos, and G. Yun. Closed generalized Einstein manifolds with positive isotropic curvature. arXiv
+preprint arXiv:2108.10675, 2021.
+[2] S. Hwang and G. Yun. Vacuum static spaces with positive isotropic curvature. arXiv preprint arXiv:2103.15818,
+2021.
+[3] S. Hwang and G. Yun. Besse conjecture with positive isotropic curvature. Annals of Global Analysis and Geometry,
+pages 1–26, 2022.
+[4] O. Kobayashi. A differential equation arising from scalar curvature function. J. Math. Soc. Japan, 34(4):665–675,
+1982.
+[5] J. Lafontaine. Sur la g´eom´etrie d’une g´en´eralisation de l’´equation diff´erentielle d’Obata. J. Math. Pures Appl.
+(9), 62(1):63–72, 1983.
+[6] X. Xu and J. Ye. Closed three-dimensional vacuum static spaces. Inventiones mathematicae, pages 1–17, 2022.
+[7] G. Yun and S. Hwang. V-static spaces with positive isotropic curvature. arXiv preprint arXiv:2103.16039, 2021.
+[8] G. Yun and S. Hwang. Critical point equation on three-dimensional manifolds and the Besse conjecture. arXiv
+preprint arXiv:2208.10887, 2022.
+S. Borghini, Universit`a degli Studi di Trento, via Sommarive 14, 38123 Povo (TN), Italy
+Email address: stefano.borghini@unitn.it
+L. Mazzieri, Universit`a degli Studi di Trento, via Sommarive 14, 38123 Povo (TN), Italy
+Email address: lorenzo.mazzieri@unitn.it
+

8tE0T4oBgHgl3EQfwgFY/content/tmp_files/load_file.txt

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+filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE0T4oBgHgl3EQfwgFY/content/2301.02633v1.pdf,len=339
+page_content='arXiv:2301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE0T4oBgHgl3EQfwgFY/content/2301.02633v1.pdf'}
+page_content='02633v1  [math.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE0T4oBgHgl3EQfwgFY/content/2301.02633v1.pdf'}
+page_content='DG]  6 Jan 2023  COUNTEREXAMPLES TO A DIVERGENCE LOWER BOUND FOR THE  COVARIANT DERIVATIVE OF SKEW-SYMMETRIC 2-TENSOR FIELDS  STEFANO BORGHINI AND LORENZO MAZZIERI  Abstract.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE0T4oBgHgl3EQfwgFY/content/2301.02633v1.pdf'}
+page_content=' In [3] an estimate for suitable skew-symmetric 2-tensors was claimed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE0T4oBgHgl3EQfwgFY/content/2301.02633v1.pdf'}
+page_content='  Soon after,  this estimate has been exploited to claim powerful classification results: most notably, it has been  employed to propose a proof of a Black Hole Uniqueness Theorem for vacuum static spacetimes  with positive scalar curvature [6] and in connection with the Besse Conjecture [8].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE0T4oBgHgl3EQfwgFY/content/2301.02633v1.pdf'}
+page_content=' In the present  note we point out an issue in the argument proposed in [3] and we provide a counterexample to  the estimate.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE0T4oBgHgl3EQfwgFY/content/2301.02633v1.pdf'}
+page_content='  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE0T4oBgHgl3EQfwgFY/content/2301.02633v1.pdf'}
+page_content=' Introduction  The Black Hole Uniqueness Theorem for three-dimensional static solutions with positive scalar  curvature and the Besse Conjecture for solutions to the Critical Point Equation are two very  famous and related open problems in contemporary geometric analysis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE0T4oBgHgl3EQfwgFY/content/2301.02633v1.pdf'}
+page_content=' Very recently, some very  remarkable advances have been claimed on both of these problems in a series of papers [1, 2, 3, 6,  7, 8].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE0T4oBgHgl3EQfwgFY/content/2301.02633v1.pdf'}
+page_content=' In this short note, we point out an issue in the approach proposed in the above mentioned  papers, providing counterexamples.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE0T4oBgHgl3EQfwgFY/content/2301.02633v1.pdf'}
+page_content='  To introduce the problems of interest together with some notation, let us recall that a three-  dimensional static solution is a triple (M, g, f) satisfying  fRic = ∇2f + R  2 f g ,  ∆f = −R  2 f ,  (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE0T4oBgHgl3EQfwgFY/content/2301.02633v1.pdf'}
+page_content='1)  where (M, g) is a Riemannian manifold, f is a smooth function and Ric and R denote the Ricci  tensor and the scalar curvature of g, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE0T4oBgHgl3EQfwgFY/content/2301.02633v1.pdf'}
+page_content=' When R is positive, it is natural to suppose that  (M, g) is a compact manifold with boundary and that f is vanishing on the boundary.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE0T4oBgHgl3EQfwgFY/content/2301.02633v1.pdf'}
+page_content=' A strictly  related problem is the so called Critical Point Equation, which consists in the following system  (1 + f)  �  Ric − R  n g  �  = ∇2f +  R  n(n − 1) g ,  ∆f = −  R  n − 1f  (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE0T4oBgHgl3EQfwgFY/content/2301.02633v1.pdf'}
+page_content='2)  where the unknowns are given by the triple (M, g, f), with (M, g) a closed Riemannian manifold  and f a smooth function.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE0T4oBgHgl3EQfwgFY/content/2301.02633v1.pdf'}
+page_content='  In [3], the authors aim at classifying solutions to the Critical Point Equation subject to the  condition of having Positive Isotropic Curvature.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE0T4oBgHgl3EQfwgFY/content/2301.02633v1.pdf'}
+page_content=' To this end, they consider the differential 2-form  ω = df ∧ ι∇fz ,  where z indicates the traceless Ricci tensor, and they claim that it must vanish.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE0T4oBgHgl3EQfwgFY/content/2301.02633v1.pdf'}
+page_content=' Notice that,  using (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE0T4oBgHgl3EQfwgFY/content/2301.02633v1.pdf'}
+page_content='2), the differential 2-form ω can be rewritten as  ω =  1  2(1 + f)df ∧ d|∇f|2 ,  where | · | is the norm computed with respect to the metric g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE0T4oBgHgl3EQfwgFY/content/2301.02633v1.pdf'}
+page_content=' If ω ≡ 0, then, using again the  equation (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE0T4oBgHgl3EQfwgFY/content/2301.02633v1.pdf'}
+page_content='2), one can prove that the Cotton tensor of g must also vanish, by a direct computation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE0T4oBgHgl3EQfwgFY/content/2301.02633v1.pdf'}
+page_content='  It follows that either n = 3 and g is Locally Conformally Flat, or else n ≥ 4 and g has harmonic  Weyl tensor.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE0T4oBgHgl3EQfwgFY/content/2301.02633v1.pdf'}
+page_content=' In both cases, the classification follows easily.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE0T4oBgHgl3EQfwgFY/content/2301.02633v1.pdf'}
+page_content=' The same strategy is adopted in [6]1,  1Notice that this reference has been withdrawn by the authors during the preparation of the present note.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE0T4oBgHgl3EQfwgFY/content/2301.02633v1.pdf'}
+page_content='  1  2  S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE0T4oBgHgl3EQfwgFY/content/2301.02633v1.pdf'}
+page_content=' BORGHINI AND L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE0T4oBgHgl3EQfwgFY/content/2301.02633v1.pdf'}
+page_content=' MAZZIERI  where this time the differential 2-form ω is defined as  ω =  1  2f df ∧ d|∇f|2 ,  with g and f satisfying (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE0T4oBgHgl3EQfwgFY/content/2301.02633v1.pdf'}
+page_content='1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE0T4oBgHgl3EQfwgFY/content/2301.02633v1.pdf'}
+page_content=' In both cases, the vanishing of ω is deduced through an integration  by parts argument – which we describe in Subsection 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE0T4oBgHgl3EQfwgFY/content/2301.02633v1.pdf'}
+page_content='2 below, in the case of static metrics –  making a substantial use of the key estimate  |∇ω|2 ≥ |δω|2 ,  (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE0T4oBgHgl3EQfwgFY/content/2301.02633v1.pdf'}
+page_content='3)  which the authors claim to hold at all points of M where ω is not vanishing (see Lemma 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE0T4oBgHgl3EQfwgFY/content/2301.02633v1.pdf'}
+page_content='5 in [3]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE0T4oBgHgl3EQfwgFY/content/2301.02633v1.pdf'}
+page_content='  The proposed proof of (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE0T4oBgHgl3EQfwgFY/content/2301.02633v1.pdf'}
+page_content='3) does not make use of the full strength of either (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE0T4oBgHgl3EQfwgFY/content/2301.02633v1.pdf'}
+page_content='1) or (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE0T4oBgHgl3EQfwgFY/content/2301.02633v1.pdf'}
+page_content='2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE0T4oBgHgl3EQfwgFY/content/2301.02633v1.pdf'}
+page_content=' In fact,  it is based on a local computation, in which the global structure of M is not playing any role.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE0T4oBgHgl3EQfwgFY/content/2301.02633v1.pdf'}
+page_content=' As  such, if correct, it should work for every differential 2-form having the structure  ω = λ(f) df ∧ d|∇f|2 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE0T4oBgHgl3EQfwgFY/content/2301.02633v1.pdf'}
+page_content='  (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE0T4oBgHgl3EQfwgFY/content/2301.02633v1.pdf'}
+page_content='4)  for some smooth function λ = λ(f), independently of the validity of (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE0T4oBgHgl3EQfwgFY/content/2301.02633v1.pdf'}
+page_content='1) or (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE0T4oBgHgl3EQfwgFY/content/2301.02633v1.pdf'}
+page_content='2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE0T4oBgHgl3EQfwgFY/content/2301.02633v1.pdf'}
+page_content=' Aim of the  present note is to disprove the claim that every ω as in (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE0T4oBgHgl3EQfwgFY/content/2301.02633v1.pdf'}
+page_content='4), defined on an open subset of a  Riemannian manifold (M, g), satisfies estimate (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE0T4oBgHgl3EQfwgFY/content/2301.02633v1.pdf'}
+page_content='3).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE0T4oBgHgl3EQfwgFY/content/2301.02633v1.pdf'}
+page_content='  In Section 3 we point out the issue in the original proof of (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE0T4oBgHgl3EQfwgFY/content/2301.02633v1.pdf'}
+page_content='3), given in [3, Lemma 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE0T4oBgHgl3EQfwgFY/content/2301.02633v1.pdf'}
+page_content='5], whereas  in Section 4 we provide effective counterexamples to the claim.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE0T4oBgHgl3EQfwgFY/content/2301.02633v1.pdf'}
+page_content=' Namely, we show that  For every smooth real function λ ̸≡ 0, there exist a smooth Riemannian metric g and a smooth  function f such that |∇ω|2 < |δω|2, with ω = λ(f) df ∧ d|∇f|2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE0T4oBgHgl3EQfwgFY/content/2301.02633v1.pdf'}
+page_content='  For the sake of completeness, we discuss in Section 2 how the validity of an estimate like (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE0T4oBgHgl3EQfwgFY/content/2301.02633v1.pdf'}
+page_content='3)  can be exploited to deduce that ω must vanish everywhere.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE0T4oBgHgl3EQfwgFY/content/2301.02633v1.pdf'}
+page_content='  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE0T4oBgHgl3EQfwgFY/content/2301.02633v1.pdf'}
+page_content=' Analysis of a skew-symmetric 2-tensor field  To make our computations more transparent, we prefer to work with the tensor-fields formalism.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE0T4oBgHgl3EQfwgFY/content/2301.02633v1.pdf'}
+page_content='  However one can also work with the formalism of differential forms as done in [3].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE0T4oBgHgl3EQfwgFY/content/2301.02633v1.pdf'}
+page_content=' Instead of ω  defined as in (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE0T4oBgHgl3EQfwgFY/content/2301.02633v1.pdf'}
+page_content='1), we consider the skew-symmetric 2-tensor field P, given by  P = λ(f)  �  df ⊗ d|∇f|2 − d|∇f|2 ⊗ df  �  ,  (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE0T4oBgHgl3EQfwgFY/content/2301.02633v1.pdf'}
+page_content='1)  with λ, f and g as above.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE0T4oBgHgl3EQfwgFY/content/2301.02633v1.pdf'}
+page_content=' In this formalism, we have that estimate (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE0T4oBgHgl3EQfwgFY/content/2301.02633v1.pdf'}
+page_content='3) is equivalent to  |∇P|2 ≥ 2 |divP|2 ,  (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE0T4oBgHgl3EQfwgFY/content/2301.02633v1.pdf'}
+page_content='2)  as 2 |∇ω|2 = |∇P|2 (the factor two comes from the slight difference in the definition of norms on dif-  ferential forms and tensor, namely |∇ω|2 = �  j<k  �  i(∇iωjk)2, whereas |∇P|2 = �  j,k  �  i(∇iPjk)2)  and δω = −divP.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE0T4oBgHgl3EQfwgFY/content/2301.02633v1.pdf'}
+page_content=' Notice that, replacing the constant 2 with the smaller constant 1/n, one gets  the always valid lower bound |∇P|2 ≥ (1/n) |divP|2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE0T4oBgHgl3EQfwgFY/content/2301.02633v1.pdf'}
+page_content=' Furthermore, exploiting the special struc-  ture (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE0T4oBgHgl3EQfwgFY/content/2301.02633v1.pdf'}
+page_content='1) of P, one can significantly improve on this bound, obtaining (n − 1)|∇P|2 ≥ 2 |divP|2  (see the appendix).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE0T4oBgHgl3EQfwgFY/content/2301.02633v1.pdf'}
+page_content=' On the other hand, estimate (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE0T4oBgHgl3EQfwgFY/content/2301.02633v1.pdf'}
+page_content='2) is too strong and cannot hold in general, as  we will discuss below.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE0T4oBgHgl3EQfwgFY/content/2301.02633v1.pdf'}
+page_content='  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE0T4oBgHgl3EQfwgFY/content/2301.02633v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE0T4oBgHgl3EQfwgFY/content/2301.02633v1.pdf'}
+page_content=' Two differential identities.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE0T4oBgHgl3EQfwgFY/content/2301.02633v1.pdf'}
+page_content=' Here we discuss some basic though fundamental properties of  a skew-symmetric 2-tensor P having the form (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE0T4oBgHgl3EQfwgFY/content/2301.02633v1.pdf'}
+page_content='1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE0T4oBgHgl3EQfwgFY/content/2301.02633v1.pdf'}
+page_content='  Proposition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE0T4oBgHgl3EQfwgFY/content/2301.02633v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE0T4oBgHgl3EQfwgFY/content/2301.02633v1.pdf'}
+page_content=' Let (M, g) be a n-dimensional Riemannian manifold and let f ∈ C ∞(M).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE0T4oBgHgl3EQfwgFY/content/2301.02633v1.pdf'}
+page_content='  Then, the skew-symmetric 2-tensor field P defined as in (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE0T4oBgHgl3EQfwgFY/content/2301.02633v1.pdf'}
+page_content='1), for some smooth real function λ,  satisfies the identity  ∇P(X, Y, Z) + ∇P(Y, Z, X) + ∇P(Z, X, Y ) = 0 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE0T4oBgHgl3EQfwgFY/content/2301.02633v1.pdf'}
+page_content='  Remark 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE0T4oBgHgl3EQfwgFY/content/2301.02633v1.pdf'}
+page_content=' In terms of the differential 2-form ω defined as in (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE0T4oBgHgl3EQfwgFY/content/2301.02633v1.pdf'}
+page_content='4), the above identity is telling  us that ω is closed, as observed in [3, Lemma 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE0T4oBgHgl3EQfwgFY/content/2301.02633v1.pdf'}
+page_content='4].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE0T4oBgHgl3EQfwgFY/content/2301.02633v1.pdf'}
+page_content=' Observe that, if ω is as in (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE0T4oBgHgl3EQfwgFY/content/2301.02633v1.pdf'}
+page_content='4), then it is  straightforward to realize that dω = (dλ/df) df ∧ df ∧ d|∇f|2 = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE0T4oBgHgl3EQfwgFY/content/2301.02633v1.pdf'}
+page_content='  COUNTEREXAMPLES TO A DIVERGENCE LOWER BOUND FOR THE COVARIANT DERIVATIVE OF SKEW-SYMMETRIC 2-TENSOR FIELDS3  Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE0T4oBgHgl3EQfwgFY/content/2301.02633v1.pdf'}
+page_content=' For simplicity we work with normal coordinates {x1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE0T4oBgHgl3EQfwgFY/content/2301.02633v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE0T4oBgHgl3EQfwgFY/content/2301.02633v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE0T4oBgHgl3EQfwgFY/content/2301.02633v1.pdf'}
+page_content=' , xn}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE0T4oBgHgl3EQfwgFY/content/2301.02633v1.pdf'}
+page_content=' A simple computation gives  ∇iPjk =  ˙λ  λPjk∇if + λ  �  ∇2  ijf∇k|∇f|2 − ∇j|∇f|2∇2  ikf  �  + λ  �  ∇jf∇2  ik|∇f|2 − ∇2  ij|∇f|2∇kf  �  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE0T4oBgHgl3EQfwgFY/content/2301.02633v1.pdf'}
+page_content='  It is now a matter of computation to check that the sums over rotating indexes of the three pieces  on the right hand side give zero.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE0T4oBgHgl3EQfwgFY/content/2301.02633v1.pdf'}
+page_content=' We compute  Pjk∇if + Pki∇jf + Pij∇kf = λ  �  ∇if∇jf∇k|∇f|2 − ∇if∇j|∇f|2∇kf  + ∇jf∇kf∇i|∇f|2 − ∇jf∇k|∇f|2∇if + ∇kf∇if∇j|∇f|2 − ∇kf∇i|∇f|2∇jf  �  = 0 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE0T4oBgHgl3EQfwgFY/content/2301.02633v1.pdf'}
+page_content='  Similarly, one has  ∇2  ijf∇k|∇f|2 − ∇j|∇f|2∇2  ikf + ∇2  jkf∇i|∇f|2 − ∇k|∇f|2∇2  jif + ∇2  kif∇j|∇f|2 − ∇i|∇f|2∇2  kjf = 0 ,  ∇jf∇2  ik|∇f|2 − ∇2  ij|∇f|2∇kf + ∇kf∇2  ji|∇f|2 − ∇2  jk|∇f|2∇if + ∇if∇2  kj|∇f|2 − ∇2  ki|∇f|2∇jf = 0 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE0T4oBgHgl3EQfwgFY/content/2301.02633v1.pdf'}
+page_content='  It follows then that  ∇iPjk + ∇jPki + ∇kPij = 0 ,  as claimed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE0T4oBgHgl3EQfwgFY/content/2301.02633v1.pdf'}
+page_content='  □  Another interesting property of P is that it satisfies a Bochner-type formula, as it is established  in the following proposition.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE0T4oBgHgl3EQfwgFY/content/2301.02633v1.pdf'}
+page_content='  Proposition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE0T4oBgHgl3EQfwgFY/content/2301.02633v1.pdf'}
+page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE0T4oBgHgl3EQfwgFY/content/2301.02633v1.pdf'}
+page_content=' Let (M, g) be a n-dimensional Riemannian manifold and let f ∈ C ∞(M).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE0T4oBgHgl3EQfwgFY/content/2301.02633v1.pdf'}
+page_content='  Then, the skew-symmetric 2-tensor field P defined as in (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE0T4oBgHgl3EQfwgFY/content/2301.02633v1.pdf'}
+page_content='1), for some smooth real function λ,  satisfies the identity  1  2∆|P|2 = |∇P|2 + 2⟨P | ∇(div P)⟩ +  2R  (n − 1)(n − 2)|P|2 + 2n − 4  n − 2RjsPskPjk + 2WijksPisPjk .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE0T4oBgHgl3EQfwgFY/content/2301.02633v1.pdf'}
+page_content='  Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE0T4oBgHgl3EQfwgFY/content/2301.02633v1.pdf'}
+page_content=' We perform our computations with respect to normal coordinates.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE0T4oBgHgl3EQfwgFY/content/2301.02633v1.pdf'}
+page_content='  Exploiting Proposi-  tion 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE0T4oBgHgl3EQfwgFY/content/2301.02633v1.pdf'}
+page_content='1 and the skew-symmetry of P, we compute  ∆|P|2 = 2∇i(Pjk∇iPjk)  = 2|∇P|2 + 2Pjk∆Pjk  = 2|∇P|2 − 2Pjk∇2  ijPki − 2Pjk∇2  ikPij  = 2|∇P|2 + 4Pjk∇2  ijPik  = 2|∇P|2 + 4Pjk  �  ∇2  jiPik + RijisPsk + RijksPis  �  = 2|∇P|2 + 4Pjk (∇j(div P)k + RjsPsk + RijksPis) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE0T4oBgHgl3EQfwgFY/content/2301.02633v1.pdf'}
+page_content='  To obtain the claimed identity, it is now enough to substitute the general formula for the Riemann  tensor  Rijks = −  R  (n − 1)(n − 2)(gikgjs − gisgjk) +  1  n − 2 (Rikgjs − Risgjk + gikRjs − gisRjk) + Wijks  in the computation above.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE0T4oBgHgl3EQfwgFY/content/2301.02633v1.pdf'}
+page_content='  □  The differential identity obtained in the previous proposition simplifies significantly when n = 3,  since in this case the Weyl tensor vanishes and we get  1  2∆|P|2 = |∇P|2 + 2⟨P | ∇(div P)⟩ + R|P|2 − 2RjsPskPjk .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE0T4oBgHgl3EQfwgFY/content/2301.02633v1.pdf'}
+page_content='  (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE0T4oBgHgl3EQfwgFY/content/2301.02633v1.pdf'}
+page_content='3)  4  S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE0T4oBgHgl3EQfwgFY/content/2301.02633v1.pdf'}
+page_content=' BORGHINI AND L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE0T4oBgHgl3EQfwgFY/content/2301.02633v1.pdf'}
+page_content=' MAZZIERI  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE0T4oBgHgl3EQfwgFY/content/2301.02633v1.pdf'}
+page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE0T4oBgHgl3EQfwgFY/content/2301.02633v1.pdf'}
+page_content=' Application to 3-dimensional static solutions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE0T4oBgHgl3EQfwgFY/content/2301.02633v1.pdf'}
+page_content=' In [6] a classification result for 3-dimensional  static metrics with positive scalar curvature was proposed, building on the above Bochner-type  formula and on the validity of estimate (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE0T4oBgHgl3EQfwgFY/content/2301.02633v1.pdf'}
+page_content='4).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE0T4oBgHgl3EQfwgFY/content/2301.02633v1.pdf'}
+page_content=' For completeness, here we retrace their proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE0T4oBgHgl3EQfwgFY/content/2301.02633v1.pdf'}
+page_content='  Using formula (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE0T4oBgHgl3EQfwgFY/content/2301.02633v1.pdf'}
+page_content='1), we can substitute the Ricci tensor in (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE0T4oBgHgl3EQfwgFY/content/2301.02633v1.pdf'}
+page_content='3), getting  1  2∆|P|2 = |∇P|2 + 2⟨P | ∇(div P)⟩ + R  2 |P|2 + 2  f P(∇f, div P) − 1  2f ⟨∇f | ∇|P|2⟩ ,  (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE0T4oBgHgl3EQfwgFY/content/2301.02633v1.pdf'}
+page_content='4)  which can be rewritten as  1  2div(f|P|2) = f|∇P|2 + 2f⟨P | ∇(div P)⟩ + R  2 f |P|2 + 2P(∇f, div P) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE0T4oBgHgl3EQfwgFY/content/2301.02633v1.pdf'}
+page_content='  Since M is compact and f = 0 on ∂M, integrating by parts we obtain then  0 =  ˆ  M  �  f|∇P|2 − 2f|div P|2 + R  2 f |P|2  �  dµ .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE0T4oBgHgl3EQfwgFY/content/2301.02633v1.pdf'}
+page_content='  Here one can appreciate the strength of estimate (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE0T4oBgHgl3EQfwgFY/content/2301.02633v1.pdf'}
+page_content='2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE0T4oBgHgl3EQfwgFY/content/2301.02633v1.pdf'}
+page_content=' Indeed, if (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE0T4oBgHgl3EQfwgFY/content/2301.02633v1.pdf'}
+page_content='2) is in force and R > 0, then  |P|2 must vanish identically and we obtain the following  Proposition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE0T4oBgHgl3EQfwgFY/content/2301.02633v1.pdf'}
+page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE0T4oBgHgl3EQfwgFY/content/2301.02633v1.pdf'}
+page_content=' Let (M, g, f) be a compact three-dimensional static solution with positive scalar  curvature and nonempty boundary.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE0T4oBgHgl3EQfwgFY/content/2301.02633v1.pdf'}
+page_content=' Assume that f = 0 on ∂M and positive in the interior.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE0T4oBgHgl3EQfwgFY/content/2301.02633v1.pdf'}
+page_content=' If  estimate (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE0T4oBgHgl3EQfwgFY/content/2301.02633v1.pdf'}
+page_content='2) holds for some P as in (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE0T4oBgHgl3EQfwgFY/content/2301.02633v1.pdf'}
+page_content='1), then P must vanish identically and one has  df ⊗ d|∇f|2 = d|∇f|2 ⊗ df .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE0T4oBgHgl3EQfwgFY/content/2301.02633v1.pdf'}
+page_content='  This is a crucial step in the strategy outlined in [6].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE0T4oBgHgl3EQfwgFY/content/2301.02633v1.pdf'}
+page_content=' As anticipated, they exploit the identity  P = 0 in combination with the static equation to deduce that the Cotton tensor must vanish.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE0T4oBgHgl3EQfwgFY/content/2301.02633v1.pdf'}
+page_content=' The  classification follows, invoking a well known result by Kobayashi [4] and Lafontaine [5].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE0T4oBgHgl3EQfwgFY/content/2301.02633v1.pdf'}
+page_content='  As we are going to see in the next sections, it is not clear how to establish the validity of (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE0T4oBgHgl3EQfwgFY/content/2301.02633v1.pdf'}
+page_content='2)  in general, however we will prove in the appendix that the weaker lower bound |∇P|2 ≥ |divP|2  holds true.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE0T4oBgHgl3EQfwgFY/content/2301.02633v1.pdf'}
+page_content=' This leads to  ˆ  M  f|div P|2dµ ≥  ˆ  M  R  2 f |P|2dµ .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE0T4oBgHgl3EQfwgFY/content/2301.02633v1.pdf'}
+page_content='  Building on this integral inequality, one might classify three-dimensional static metrics with posi-  tive scalar curvature admitting a divergence-free P-tensor.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE0T4oBgHgl3EQfwgFY/content/2301.02633v1.pdf'}
+page_content='  3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE0T4oBgHgl3EQfwgFY/content/2301.02633v1.pdf'}
+page_content=' The issue in the proof of the estimate  Here we retrace the proof of estimate (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE0T4oBgHgl3EQfwgFY/content/2301.02633v1.pdf'}
+page_content='3) originally proposed in [3, Lemma 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE0T4oBgHgl3EQfwgFY/content/2301.02633v1.pdf'}
+page_content='5], pointing out  the main issue in the argument.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE0T4oBgHgl3EQfwgFY/content/2301.02633v1.pdf'}
+page_content='  As a first step, the authors find a local orthonormal frame with respect to which the tensor P  has a nice structure.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE0T4oBgHgl3EQfwgFY/content/2301.02633v1.pdf'}
+page_content=' This part of the proof appears to be correct and it is an interesting fact  on its own that will also be helpful in the appendix, so we include it here as a lemma.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE0T4oBgHgl3EQfwgFY/content/2301.02633v1.pdf'}
+page_content=' In the  following statement it is helpful to consider the vector valued 1-form A : TM → TM defined by  P(X, Y ) = g(AX, Y ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE0T4oBgHgl3EQfwgFY/content/2301.02633v1.pdf'}
+page_content=' In coordinates: Aj  i = gjmPim.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE0T4oBgHgl3EQfwgFY/content/2301.02633v1.pdf'}
+page_content='  Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE0T4oBgHgl3EQfwgFY/content/2301.02633v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE0T4oBgHgl3EQfwgFY/content/2301.02633v1.pdf'}
+page_content=' Let (M, g) be a n-dimensional Riemannian manifold.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE0T4oBgHgl3EQfwgFY/content/2301.02633v1.pdf'}
+page_content=' Let f ∈ C ∞(M) and let P be  the tensor defined by (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE0T4oBgHgl3EQfwgFY/content/2301.02633v1.pdf'}
+page_content='1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE0T4oBgHgl3EQfwgFY/content/2301.02633v1.pdf'}
+page_content=' Let x ∈ M be a point with |P|(x) ̸= 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE0T4oBgHgl3EQfwgFY/content/2301.02633v1.pdf'}
+page_content=' Then in a small neighborhood  U of x it holds |P| ̸= 0, |∇f| ̸= 0, |A∇f| ̸= 0 and there exists a smooth orthonormal frame  {E1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE0T4oBgHgl3EQfwgFY/content/2301.02633v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE0T4oBgHgl3EQfwgFY/content/2301.02633v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE0T4oBgHgl3EQfwgFY/content/2301.02633v1.pdf'}
+page_content=' , En} with E1 = ∇f/|∇f| and E2 = AE1/|AE1|.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE0T4oBgHgl3EQfwgFY/content/2301.02633v1.pdf'}
+page_content=' With respect to this frame, the tensor P  rewrites as  P = u  �  θ1 ⊗ θ2 − θ2 ⊗ θ1�  ,  (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE0T4oBgHgl3EQfwgFY/content/2301.02633v1.pdf'}
+page_content='1)  where u is a smooth function and {θ1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE0T4oBgHgl3EQfwgFY/content/2301.02633v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE0T4oBgHgl3EQfwgFY/content/2301.02633v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE0T4oBgHgl3EQfwgFY/content/2301.02633v1.pdf'}
+page_content=' , θn} is the dual coframe of {E1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE0T4oBgHgl3EQfwgFY/content/2301.02633v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE0T4oBgHgl3EQfwgFY/content/2301.02633v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE0T4oBgHgl3EQfwgFY/content/2301.02633v1.pdf'}
+page_content=' , En} (namely,  θi(Ej) = δi  j at any point in U).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE0T4oBgHgl3EQfwgFY/content/2301.02633v1.pdf'}
+page_content='  COUNTEREXAMPLES TO A DIVERGENCE LOWER BOUND FOR THE COVARIANT DERIVATIVE OF SKEW-SYMMETRIC 2-TENSOR FIELDS5  Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE0T4oBgHgl3EQfwgFY/content/2301.02633v1.pdf'}
+page_content=' A proof of this fact is given in [3], however we write here a shorter self contained version.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE0T4oBgHgl3EQfwgFY/content/2301.02633v1.pdf'}
+page_content='  We first construct the orthonormal frame in the lemma.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE0T4oBgHgl3EQfwgFY/content/2301.02633v1.pdf'}
+page_content='  Consider a neighborhood U of x  in which |P| ̸= 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE0T4oBgHgl3EQfwgFY/content/2301.02633v1.pdf'}
+page_content='  From the definition (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE0T4oBgHgl3EQfwgFY/content/2301.02633v1.pdf'}
+page_content='1) of P, it is clear that |∇f| ̸= 0 in U as well.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE0T4oBgHgl3EQfwgFY/content/2301.02633v1.pdf'}
+page_content='  In  particular the vector E1 = ∇f/|∇f| is well defined in U.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE0T4oBgHgl3EQfwgFY/content/2301.02633v1.pdf'}
+page_content=' We complete E1 to an orthonormal  frame {E1, �E2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE0T4oBgHgl3EQfwgFY/content/2301.02633v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE0T4oBgHgl3EQfwgFY/content/2301.02633v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE0T4oBgHgl3EQfwgFY/content/2301.02633v1.pdf'}
+page_content=' , �En} in U.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE0T4oBgHgl3EQfwgFY/content/2301.02633v1.pdf'}
+page_content=' Since g(E1, �Ei) = 0 for i ≥ 2, we have ∇ �  Eif = 0 for any i ≥ 2, hence  P( �Ei, �Ej) = λ(f)  �  ∇ �  Eif ∇ �  Ej|∇f|2 − ∇ �  Ei|∇f|2 ∇ �  Ejf  �  = 0 ,  (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE0T4oBgHgl3EQfwgFY/content/2301.02633v1.pdf'}
+page_content='2)  for any i, j ≥ 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE0T4oBgHgl3EQfwgFY/content/2301.02633v1.pdf'}
+page_content=' Since |P| ̸= 0 in U, then at any point in U it holds g(AE1, �Ej) = P(E1, �Ej) ̸= 0  for some j.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE0T4oBgHgl3EQfwgFY/content/2301.02633v1.pdf'}
+page_content=' In particular AE1 ̸= 0 in U.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE0T4oBgHgl3EQfwgFY/content/2301.02633v1.pdf'}
+page_content=' Since g(AE1, E1) = P(E1, E1) = 0, it follows that AE1 is  orthogonal to E1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE0T4oBgHgl3EQfwgFY/content/2301.02633v1.pdf'}
+page_content=' In particular, the vector E2 = AE1/|AE1| is well defined and orthonormal to E1  on the whole U.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE0T4oBgHgl3EQfwgFY/content/2301.02633v1.pdf'}
+page_content=' We can then complete E1, E2 to an orthonormal frame {E1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE0T4oBgHgl3EQfwgFY/content/2301.02633v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE0T4oBgHgl3EQfwgFY/content/2301.02633v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE0T4oBgHgl3EQfwgFY/content/2301.02633v1.pdf'}
+page_content=' , En} in U.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE0T4oBgHgl3EQfwgFY/content/2301.02633v1.pdf'}
+page_content=' This  is precisely the orthonormal frame described in the statement of the lemma.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE0T4oBgHgl3EQfwgFY/content/2301.02633v1.pdf'}
+page_content=' Notice in particular  that  P(E1, Ej) = g(AE1, Ej) = |AE1| g(E2, Ej) = |AE1| δ2j .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE0T4oBgHgl3EQfwgFY/content/2301.02633v1.pdf'}
+page_content='  In view of (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE0T4oBgHgl3EQfwgFY/content/2301.02633v1.pdf'}
+page_content='2), we deduce that the only nonzero entries of P are P(E1, E2) = −P(E2, E1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE0T4oBgHgl3EQfwgFY/content/2301.02633v1.pdf'}
+page_content='  Formula (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE0T4oBgHgl3EQfwgFY/content/2301.02633v1.pdf'}
+page_content='1) follows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE0T4oBgHgl3EQfwgFY/content/2301.02633v1.pdf'}
+page_content='  □  Next, the authors compute |∇P|2 and |div P|2 with respect to this frame.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE0T4oBgHgl3EQfwgFY/content/2301.02633v1.pdf'}
+page_content=' The computations  regarding |∇P|2 appear to be correct.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE0T4oBgHgl3EQfwgFY/content/2301.02633v1.pdf'}
+page_content=' On the other hand, it seems to us that the expression of  the divergence term worked out by the authors contains a mistake.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE0T4oBgHgl3EQfwgFY/content/2301.02633v1.pdf'}
+page_content=' A simple calculation (see the  appendix for more details) gives  (div P)(E1) = −E2(u) +  n  �  i=3  ⟨∇EiEi | E2⟩ u = −E2(u) +  n  �  i=3  ⟨Ei | [E2, Ei]⟩ u ,  (div P)(E2) = E1(u) −  n  �  i=3  ⟨∇EiEi | E1⟩ u = E1(u) +  n  �  i=3  ⟨Ei | [E1, Ei]⟩ u ,  (div P)(Ek) = ⟨Ek | [E1, E2]⟩ u ,  k ≥ 3 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE0T4oBgHgl3EQfwgFY/content/2301.02633v1.pdf'}
+page_content='  (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE0T4oBgHgl3EQfwgFY/content/2301.02633v1.pdf'}
+page_content='3)  It is worth pointing out that the frame {E1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE0T4oBgHgl3EQfwgFY/content/2301.02633v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE0T4oBgHgl3EQfwgFY/content/2301.02633v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE0T4oBgHgl3EQfwgFY/content/2301.02633v1.pdf'}
+page_content=' , En} was constructed with a pointwise argument.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE0T4oBgHgl3EQfwgFY/content/2301.02633v1.pdf'}
+page_content='  The frame is easily seen to be smooth, but it is important to observe that it is not necessarily  induced from a local coordinate system.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE0T4oBgHgl3EQfwgFY/content/2301.02633v1.pdf'}
+page_content=' In particular, the Lie brackets [Ei, Ej] are not necessarily  vanishing.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE0T4oBgHgl3EQfwgFY/content/2301.02633v1.pdf'}
+page_content=' This seems to be the core of the issue: in fact, the authors claim that  div P = −E2(u)θ1 + E1(u)θ2 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE0T4oBgHgl3EQfwgFY/content/2301.02633v1.pdf'}
+page_content='  (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE0T4oBgHgl3EQfwgFY/content/2301.02633v1.pdf'}
+page_content='4)  In view of (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE0T4oBgHgl3EQfwgFY/content/2301.02633v1.pdf'}
+page_content='3), this formula appears to be incorrect whenever the Lie brackets do not vanish.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE0T4oBgHgl3EQfwgFY/content/2301.02633v1.pdf'}
+page_content='  Remark 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE0T4oBgHgl3EQfwgFY/content/2301.02633v1.pdf'}
+page_content=' In [3], and more precisely in the final page of the proof of [3, Lemma 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE0T4oBgHgl3EQfwgFY/content/2301.02633v1.pdf'}
+page_content='5] this formula is  written as δω = E2(u)θ1 − E1(u)θ2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE0T4oBgHgl3EQfwgFY/content/2301.02633v1.pdf'}
+page_content=' As already observed, ω corresponds to our P in the formalism  of the differential forms, and the codifferential δ is clearly related to the divergence through the  formula δω = −divP.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE0T4oBgHgl3EQfwgFY/content/2301.02633v1.pdf'}
+page_content='  4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE0T4oBgHgl3EQfwgFY/content/2301.02633v1.pdf'}
+page_content=' Counterexamples to estimate (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE0T4oBgHgl3EQfwgFY/content/2301.02633v1.pdf'}
+page_content='3)  We work in dimension 3 for simplicity, but similar counterexamples might be constructed in  higher dimension as well.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE0T4oBgHgl3EQfwgFY/content/2301.02633v1.pdf'}
+page_content=' Consider local coordinates {r, x1, x2} defined on an open set, a positive  smooth function φ = φ(r) and the warped product metric  g = dr ⊗ dr + φ2(dx1 ⊗ dx1 + dx2 ⊗ dx2) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE0T4oBgHgl3EQfwgFY/content/2301.02633v1.pdf'}
+page_content='  Let then f ∈ C ∞(M) be a smooth function of the form f = ψ ◦ x1, for some smooth nonconstant  real function ψ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE0T4oBgHgl3EQfwgFY/content/2301.02633v1.pdf'}
+page_content=' Let us consider then a skew-symmetric 2-tensor field P as in (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE0T4oBgHgl3EQfwgFY/content/2301.02633v1.pdf'}
+page_content='1), for some choice  of λ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE0T4oBgHgl3EQfwgFY/content/2301.02633v1.pdf'}
+page_content=' In local coordinates, we have that the components of P are given by  Pαβ = λ  �  ∇αf∇2  βηf − ∇βf∇2  αηf  �  gησ∇σf = λ ψ′  φ2  �  ∇αf∇2  1βf − ∇βf∇2  1αf  �  ,  6  S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE0T4oBgHgl3EQfwgFY/content/2301.02633v1.pdf'}
+page_content=' BORGHINI AND L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE0T4oBgHgl3EQfwgFY/content/2301.02633v1.pdf'}
+page_content=' MAZZIERI  where the greek indexes are running in {r, 1, 2}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE0T4oBgHgl3EQfwgFY/content/2301.02633v1.pdf'}
+page_content=' Here and in what follows we will denote with ′  the derivatives with respect to x1 and with a dot the derivatives with respect to r.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE0T4oBgHgl3EQfwgFY/content/2301.02633v1.pdf'}
+page_content=' The Christoffel  symbols of the metric g are as follows  Γr  rr = Γr  ri = Γi  rr = Γk  ij = 0 ,  Γr  ij = −φ ˙φδij ,  Γj  ri =  ˙φ  φδj  i ,  where the latin indexes are running in {1, 2}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE0T4oBgHgl3EQfwgFY/content/2301.02633v1.pdf'}
+page_content=' It then follows easily that the only nonzero compo-  nents of the Hessian are  ∇2  11f = ψ′′ ,  ∇2  1rf = −  ˙φ  φ ψ′ ,  and that  P = λ  ˙φ  φ3 (ψ′)3 �  dr ⊗ dx1 − dx1 ⊗ dr  �  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE0T4oBgHgl3EQfwgFY/content/2301.02633v1.pdf'}
+page_content='  Notice that we are in a setting similar to the one of Section 3, except that our frame  {∂/∂r, ∂/∂x1, ∂/∂x2}  is not orthonormal.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE0T4oBgHgl3EQfwgFY/content/2301.02633v1.pdf'}
+page_content=' Hence, to check that our P has the structure prescribed in (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE0T4oBgHgl3EQfwgFY/content/2301.02633v1.pdf'}
+page_content='1), one should  write its local expression, with respect to an orthonormal frame.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE0T4oBgHgl3EQfwgFY/content/2301.02633v1.pdf'}
+page_content='  This latter can be obtained  setting E1 = (1/φ)∂/∂x1, E2 = ∂/∂r, E3 = (1/φ)∂/∂x2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE0T4oBgHgl3EQfwgFY/content/2301.02633v1.pdf'}
+page_content=' Its dual orthonormal co-frame is then  given by θ1 = φdx1, θ2 = dr, θ3 = φdx2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE0T4oBgHgl3EQfwgFY/content/2301.02633v1.pdf'}
+page_content=' It is easy to check that this frame satisfies the properties  described in Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE0T4oBgHgl3EQfwgFY/content/2301.02633v1.pdf'}
+page_content='1 and that  P = −λ  ˙φ  φ4 (ψ′)3 �  θ1 ⊗ θ2 − θ2 ⊗ θ1�  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE0T4oBgHgl3EQfwgFY/content/2301.02633v1.pdf'}
+page_content='  However, we prefer to perform our computations with respect to the frame fields induced by the  local coordinates (r, x1, x2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE0T4oBgHgl3EQfwgFY/content/2301.02633v1.pdf'}
+page_content=' In this framework, it is easy to show that the only nonzero components  of ∇P are  ∇rP1r = −  � ¨φ  φ3 − 4  ˙φ2  φ4  �  λ (ψ′)3 ,  ∇1P1r = −  ˙φ  φ3 (λ (ψ′)3)′ ,  ∇2P12 = −  ˙φ2  φ2 λ (ψ′)3 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE0T4oBgHgl3EQfwgFY/content/2301.02633v1.pdf'}
+page_content='  It easily follows that  divP = −  ˙φ  φ5 (λ (ψ′)3)′ dr + λ (ψ′)3  � ¨φ  φ3 − 3  ˙φ2  φ4  �  dx1 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE0T4oBgHgl3EQfwgFY/content/2301.02633v1.pdf'}
+page_content='  Here it is possible to notice the discrepancy between our computations and formula (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE0T4oBgHgl3EQfwgFY/content/2301.02633v1.pdf'}
+page_content='4), as  computing the right hand side of that formula would give  −  ˙φ  φ5 (λ (ψ′)3)′ dr + λ (ψ′)3  � ¨φ  φ3 − 4  ˙φ2  φ4  �  dx1 ,  which looks very similar, but does not correspond to the correct value of divP.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE0T4oBgHgl3EQfwgFY/content/2301.02633v1.pdf'}
+page_content=' Computing the  squared norms of ∇P and divP, one finally arrives at  |∇P|2 − 2|divP|2 = 4λ2 ˙φ2(ψ′)6  φ8  �  4  ˙φ2  φ2 −  ¨φ  φ  �  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE0T4oBgHgl3EQfwgFY/content/2301.02633v1.pdf'}
+page_content='  To make this difference negative, it is then sufficient to specify a choice of the functions λ, ψ and  φ such that the right hand side is negative.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE0T4oBgHgl3EQfwgFY/content/2301.02633v1.pdf'}
+page_content=' In particular, it is sufficient to choose φ in such a way  that the quantity in round brackets is negative.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE0T4oBgHgl3EQfwgFY/content/2301.02633v1.pdf'}
+page_content=' This can be achieved, for example, setting  φ = (r + c)−1/k,  for some k > 3 and some c > 0 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE0T4oBgHgl3EQfwgFY/content/2301.02633v1.pdf'}
+page_content='  COUNTEREXAMPLES TO A DIVERGENCE LOWER BOUND FOR THE COVARIANT DERIVATIVE OF SKEW-SYMMETRIC 2-TENSOR FIELDS7  It follows that, with this choice of φ, for any λ and any f = ψ ◦ x1, the estimate (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE0T4oBgHgl3EQfwgFY/content/2301.02633v1.pdf'}
+page_content='2) does not  hold.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE0T4oBgHgl3EQfwgFY/content/2301.02633v1.pdf'}
+page_content=' Hence, the lower bound (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE0T4oBgHgl3EQfwgFY/content/2301.02633v1.pdf'}
+page_content='3) is false as well.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE0T4oBgHgl3EQfwgFY/content/2301.02633v1.pdf'}
+page_content='  Appendix  For completeness, let us point out the correct relation always holding between |∇P| and |divP|.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE0T4oBgHgl3EQfwgFY/content/2301.02633v1.pdf'}
+page_content='  Let (M, g) be a n-dimensional Riemannian manifold, n ≥ 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE0T4oBgHgl3EQfwgFY/content/2301.02633v1.pdf'}
+page_content=' As in Section 3, we take a point x with  |P|(x) ̸= 0 and we consider the local orthonormal frame {E1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE0T4oBgHgl3EQfwgFY/content/2301.02633v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE0T4oBgHgl3EQfwgFY/content/2301.02633v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE0T4oBgHgl3EQfwgFY/content/2301.02633v1.pdf'}
+page_content=' , En} provided by Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE0T4oBgHgl3EQfwgFY/content/2301.02633v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE0T4oBgHgl3EQfwgFY/content/2301.02633v1.pdf'}
+page_content=' We  recall that, with respect to this frame, the tensor P takes the following form  P = u  �  θ1 ⊗ θ2 − θ2 ⊗ θ1�  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE0T4oBgHgl3EQfwgFY/content/2301.02633v1.pdf'}
+page_content='  (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE0T4oBgHgl3EQfwgFY/content/2301.02633v1.pdf'}
+page_content='1)  Exploiting the compatibility of ∇ with the metric g, for any i, j, k we have  0 = Ei (g(Ej, Ek)) = g(∇EiEj, Ek) + g(Ej, ∇EiEk) ,  and in particular  g(∇EiEk, Ek) = 0 ,  g(∇EiEi, Ek) = −g(Ei, ∇EiEk) = −g(Ei, [Ei, Ek]) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE0T4oBgHgl3EQfwgFY/content/2301.02633v1.pdf'}
+page_content='  We are now ready to compute the components of ∇P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE0T4oBgHgl3EQfwgFY/content/2301.02633v1.pdf'}
+page_content=' Since P(Ei, Ej) = 0 whenever {i, j} ̸= {1, 2},  we have  ∇EiP(E1, E2) = Ei (P(E1, E2)) − P(∇EiE1, E2) − P(E1, ∇EiE2)  = Ei(u) − g(∇EiE1, E1)P(E1, E2) − g(∇EiE2, E2)P(E1, E2)  = Ei(u) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE0T4oBgHgl3EQfwgFY/content/2301.02633v1.pdf'}
+page_content='  Similarly, for any k ≥ 3, we have  ∇EiP(E1, Ek) = Ei(P(E1, Ek)) − P(∇EiE1, Ek) − P(E1, ∇EiEk)  = − g(∇EiEk, E2)P(E1, E2)  = − g(∇EiEk, E2) u ,  and  ∇EiP(E2, Ek) = Ei(P(E2, Ek)) − P(∇EiE2, Ek) − P(E2, ∇EiEk)  = − g(∇EiEk, E1)P(E2, E1)  = g(∇EiEk, E1) u .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE0T4oBgHgl3EQfwgFY/content/2301.02633v1.pdf'}
+page_content='  Similarly, one computes ∇EiP(E1, E1) = ∇EiP(E2, E2) = 0 and ∇EiP(Ej, Ek) = 0 whenever j, k  are ≥ 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE0T4oBgHgl3EQfwgFY/content/2301.02633v1.pdf'}
+page_content=' It is now easy to compute the divergence of P:  (div P)(E1) = −E2(u) +  n  �  i=3  ⟨∇EiEi | E2⟩ u = −E2(u) +  n  �  i=3  ⟨Ei | [E2, Ei]⟩ u ,  (div P)(E2) = E1(u) −  n  �  i=3  ⟨∇EiEi | E1⟩ u = E1(u) −  n  �  i=3  ⟨Ei | [E1, Ei]⟩ u ,  (div P)(Ei) = −g(∇E1Ei, E2) u + g(∇E2Ei, E1) u ,  i ≥ 3 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE0T4oBgHgl3EQfwgFY/content/2301.02633v1.pdf'}
+page_content='  Using the inequality (�k  i=1 xi)2 ≤ k �k  i=1 x2  i , a simple calculation then gives  |divP|2  n − 1  ≤  2  �  k=1  �  Ek(u)2 +  n  �  i=3  ⟨Ei | [Ei, Ek]⟩2u2  �  +  2  n − 1  n  �  i=3  �  ⟨∇E1Ei | E2⟩2 + ⟨∇E2Ei | E1⟩2�  u2  ≤  2  �  k=1  �  Ek(u)2 +  n  �  i=3  ⟨Ei | [Ei, Ek]⟩2u2  �  +  n  �  i=3  �  ⟨∇E1Ei | E2⟩2 + ⟨∇E2Ei | E1⟩2�  u2 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE0T4oBgHgl3EQfwgFY/content/2301.02633v1.pdf'}
+page_content='  8  S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE0T4oBgHgl3EQfwgFY/content/2301.02633v1.pdf'}
+page_content=' BORGHINI AND L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE0T4oBgHgl3EQfwgFY/content/2301.02633v1.pdf'}
+page_content=' MAZZIERI  On the other hand  1  2|∇P|2 ≥  2  �  k=1  �  (∇EkP(E1, E2))2 +  n  �  i=3  (∇EiP(Ek, Ei))2 +  n  �  i=3  (∇EkP(Ek, Ei))2  �  =  2  �  k=1  �  Ek(u)2 +  n  �  i=3  ⟨Ei | [Ei, Ek]⟩2u2  �  +  n  �  i=3  �  ⟨∇E1Ei | E2⟩2 + ⟨∇E2Ei | E1⟩2�  u2 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE0T4oBgHgl3EQfwgFY/content/2301.02633v1.pdf'}
+page_content='  In conclusion, we have shown the following.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE0T4oBgHgl3EQfwgFY/content/2301.02633v1.pdf'}
+page_content='  Proposition 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE0T4oBgHgl3EQfwgFY/content/2301.02633v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE0T4oBgHgl3EQfwgFY/content/2301.02633v1.pdf'}
+page_content=' Let (M, g) be a n-dimensional Riemannian manifold, n ≥ 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE0T4oBgHgl3EQfwgFY/content/2301.02633v1.pdf'}
+page_content=' Let f ∈ C ∞(M)  and let P be the tensor defined by (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE0T4oBgHgl3EQfwgFY/content/2301.02633v1.pdf'}
+page_content='1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE0T4oBgHgl3EQfwgFY/content/2301.02633v1.pdf'}
+page_content=' Then, at any point of M it holds  |∇P|2 ≥  2  n − 1 |divP|2 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE0T4oBgHgl3EQfwgFY/content/2301.02633v1.pdf'}
+page_content='  (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE0T4oBgHgl3EQfwgFY/content/2301.02633v1.pdf'}
+page_content='2)  Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE0T4oBgHgl3EQfwgFY/content/2301.02633v1.pdf'}
+page_content=' Estimate (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE0T4oBgHgl3EQfwgFY/content/2301.02633v1.pdf'}
+page_content='2) follows immediately from the computations above at any point where P has  the form (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE0T4oBgHgl3EQfwgFY/content/2301.02633v1.pdf'}
+page_content='1), that is, at any point where |P| ̸= 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE0T4oBgHgl3EQfwgFY/content/2301.02633v1.pdf'}
+page_content=' Let then x be a point where |P| = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE0T4oBgHgl3EQfwgFY/content/2301.02633v1.pdf'}
+page_content=' If |P|  vanishes identically in a neighborhood of x, then |∇P| = |div P| = 0 in that neighborhood, and  inequality (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE0T4oBgHgl3EQfwgFY/content/2301.02633v1.pdf'}
+page_content='2) is trivially satisfied.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE0T4oBgHgl3EQfwgFY/content/2301.02633v1.pdf'}
+page_content=' Otherwise there exists a sequence of points xi converging to  x with |P|(xi) ̸= 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE0T4oBgHgl3EQfwgFY/content/2301.02633v1.pdf'}
+page_content=' Since estimate (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE0T4oBgHgl3EQfwgFY/content/2301.02633v1.pdf'}
+page_content='2) holds at the points xi, then it must hold at x as well by  continuity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE0T4oBgHgl3EQfwgFY/content/2301.02633v1.pdf'}
+page_content='  □  Acknowledgements.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE0T4oBgHgl3EQfwgFY/content/2301.02633v1.pdf'}
+page_content=' The authors would like to thank R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE0T4oBgHgl3EQfwgFY/content/2301.02633v1.pdf'}
+page_content=' Beig, P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE0T4oBgHgl3EQfwgFY/content/2301.02633v1.pdf'}
+page_content=' T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE0T4oBgHgl3EQfwgFY/content/2301.02633v1.pdf'}
+page_content=' Chru´sciel and W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE0T4oBgHgl3EQfwgFY/content/2301.02633v1.pdf'}
+page_content=' Simon  for stimulating discussions about the classification of static vacuum spacetimes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE0T4oBgHgl3EQfwgFY/content/2301.02633v1.pdf'}
+page_content=' The authors are  members of the Gruppo Nazionale per l’Analisi Matematica, la Probabilit`a e le loro Applicazioni  (GNAMPA) of the Istituto Nazionale di Alta Matematica (INdAM).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE0T4oBgHgl3EQfwgFY/content/2301.02633v1.pdf'}
+page_content='  References  [1] S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE0T4oBgHgl3EQfwgFY/content/2301.02633v1.pdf'}
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+page_content=' arXiv preprint arXiv:2103.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE0T4oBgHgl3EQfwgFY/content/2301.02633v1.pdf'}
+page_content='15818,  2021.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE0T4oBgHgl3EQfwgFY/content/2301.02633v1.pdf'}
+page_content='  [3] S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE0T4oBgHgl3EQfwgFY/content/2301.02633v1.pdf'}
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+page_content=' Yun.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE0T4oBgHgl3EQfwgFY/content/2301.02633v1.pdf'}
+page_content=' Besse conjecture with positive isotropic curvature.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE0T4oBgHgl3EQfwgFY/content/2301.02633v1.pdf'}
+page_content=' Annals of Global Analysis and Geometry,  pages 1–26, 2022.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE0T4oBgHgl3EQfwgFY/content/2301.02633v1.pdf'}
+page_content='  [4] O.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE0T4oBgHgl3EQfwgFY/content/2301.02633v1.pdf'}
+page_content=' Kobayashi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE0T4oBgHgl3EQfwgFY/content/2301.02633v1.pdf'}
+page_content=' A differential equation arising from scalar curvature function.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE0T4oBgHgl3EQfwgFY/content/2301.02633v1.pdf'}
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+page_content='borghini@unitn.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE0T4oBgHgl3EQfwgFY/content/2301.02633v1.pdf'}
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+page_content=' Mazzieri, Universit`a degli Studi di Trento, via Sommarive 14, 38123 Povo (TN), Italy  Email address: lorenzo.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE0T4oBgHgl3EQfwgFY/content/2301.02633v1.pdf'}
+page_content='mazzieri@unitn.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE0T4oBgHgl3EQfwgFY/content/2301.02633v1.pdf'}
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+Indistinguishable telecom band photons from a single erbium ion in the solid state
+Salim Ourari,1, ∗ Łukasz Dusanowski,1, ∗ Sebastian P. Horvath,1, ∗ Mehmet T. Uysal,1, ∗
+Christopher M. Phenicie,1 Paul Stevenson,1, † Mouktik Raha,1 Songtao Chen,1, ‡
+Robert J. Cava,2 Nathalie P. de Leon,1 and Jeff D. Thompson1, §
+1Department of Electrical and Computer Engineering, Princeton University, Princeton, NJ 08544, USA
+2Department of Chemistry, Princeton University, Princeton, NJ 08544, USA
+Atomic defects in the solid state are a key component of quantum repeater networks for long-
+distance quantum communication [1]. Recently, there has been significant interest in rare earth ions
+[2–4], in particular Er3+ for its telecom-band optical transition [5–7], but their application has been
+hampered by optical spectral diffusion precluding indistinguishable single photon generation. In
+this work we implant Er3+ into CaWO4, a material that combines a non-polar site symmetry, low
+decoherence from nuclear spins [8], and is free of background rare earth ions, to realize significantly
+reduced optical spectral diffusion. For shallow implanted ions coupled to nanophotonic cavities with
+large Purcell factor, we observe single-scan optical linewidths of 150 kHz and long-term spectral
+diffusion of 63 kHz, both close to the Purcell-enhanced radiative linewidth of 21 kHz. This enables
+the observation of Hong-Ou-Mandel interference [9] between successively emitted photons with high
+visibility, measured after a 36 km delay line. We also observe spin relaxation times T1 = 3.7 s and
+T2 > 200 µs, with the latter limited by paramagnetic impurities in the crystal instead of nuclear
+spins. This represents a significant step towards the construction of telecom-band quantum repeater
+networks with single Er3+ ions.
+Long-distance quantum networks are an enabling technology for quantum communication, distributed quantum
+computing and entanglement-enhanced sensing and metrology [10]. The rate of direct entanglement transmission with
+photons decreases exponentially with distance, but this can be overcome using quantum repeaters with memories
+[11].
+In particular, single atom-like defects in the solid state [1] have been used to demonstrate key milestones
+including spin-photon entanglement [12, 13] and single-photon transistors [14], remote entanglement of spins [15],
+entanglement purification [16] and memory-enhanced quantum communication [17]. A challenge to deploying these
+techniques in long-distance networks is that atomic systems typically operate at transition frequencies outside of the
+low-loss window of optical fibers, requiring wavelength conversion for long-distance propagation [18, 19].
+The rare earth ion Er3+ has a telecom-band optical transition at a wavelength of 1.5 µm that is widely exploited
+for solid-state optical amplifiers, and in dilute ensembles, as a quantum memory for light [20, 21]. Er3+ ions can
+have long spin [8, 22] and optical [23] coherence in a variety of host crystals, a property shared with other rare earth
+ions [24–26]. In recent years, micro- and nano-scale optical resonators have enabled the observation of enhanced
+single photon emission from Er3+ and other rare earth ions [3–5, 7, 27], which has subsequently enabled single-shot
+spin readout [4, 28] and coupling to nearby nuclear spins that could serve as ancilla qubits [29–31]. However, a
+central challenge to the development of quantum repeaters with single rare earth ions is spectral diffusion, which
+is particularly pronounced in nanophotonic devices used to achieve fast optical emission from single rare earth ions
+[4, 5, 7]. To date, indistinguishable single photon emission from a single rare earth ion has not been observed.
+Rare earth ions also provide a unique opportunity for materials engineering, as they can be incorporated into a
+wide range of host crystals while preserving their basic properties, including the optical transition wavelength and
+spin configuration [32–35]. An ideal host material would incorporate Er3+ on a non-polar site to suppress linear
+electric field shifts of the optical transition, and have a low concentration of nuclear spins, other magnetic impurities
+and particularly trace rare earth ions to allow long spin coherence and low fluorescence background [36].
+In this work, we demonstrate indistinguishable single photon emission from a single Er3+ ion coupled to a
+nanophotonic optical cavity. This is enabled by shallow ion implantation of Er3+ into CaWO4, a host material
+satisfying the above criteria and for which long electron spin coherence has recently been demonstrated in Er3+
+ensembles at millikelvin temperatures [8]. By coupling the ions to silicon nanophotonic circuits, we observe individual
+ions with single-scan optical linewidths of 150 kHz, and emission rate enhancement by a factor of P = 850 via the
+Purcell effect. Using a 36 km delay line, we observe Hong-Ou-Mandel (HOM) interference between successively
+emitted photons with a visibility of V = 80(4)%. We also demonstrate spin initialization and single-shot readout
+with a fidelity F = 0.972, as well as the preservation of electron spin coherence for more than 200 µs, limited by
+paramagnetic impurities in the sample. This demonstration is a key step for the development of quantum repeaters
+based on single rare earth ions, and Er3+ in particular.
+Our samples are produced by introducing erbium into commercially available high purity CaWO4 using ion
+implantation with an energy of 35 keV, targeting a depth of 10 nm. In a test sample implanted with a high Er3+
+fluence of 1×1012 ions/cm2, we observe an ensemble optical spectrum at T = 4 K consistent with substitutional Er3+
+arXiv:2301.03564v1  [quant-ph]  9 Jan 2023
+
+2
+on the Ca2+ site with S4 symmetry (Fig. 1a) [37, 38]. After annealing at 300 ◦C in air, the inhomogeneous optical
+linewidth of the Z1-Y1 transition at 1532.63 nm is 730 MHz (Fig. 1b). This is comparable to previously reported
+linewidths in bulk-doped samples (approximately 0.5-1 GHz [35, 39]), suggesting that the implantation damage is
+effectively removed by annealing.
+To resolve individual ions, we implant a second sample at a lower fluence of 5 × 109 ions/cm2. Single ions are
+probed using a silicon photonic crystal cavity that is fabricated on a separate silicon-on-insulator wafer, and then
+bonded to the top surface of the CaWO4 substrate (Fig. 1c-d) [5]. The device and sample are cooled to T = 0.47 K
+in a 3He cryostat, with optical and microwave access provided with a scanning probe head [40]. We probe single
+ions in the device using photoluminescence excitation (PLE) spectroscopy, by sweeping the frequency of a pulsed
+laser and observing the time-delayed fluorescence through the cavity with a superconducting nanowire single photon
+detector (SNSPD). The spectrum contains clearly resolved lines from individual Er3+ ions (Fig. 1e). The number
+of lines is roughly consistent with the expected number of ions in the cavity area A = 1.3 µm2, suggesting a high
+conversion efficiency. The following experiments are performed on the ion indicated by the arrow.
+Coupling the Er3+ ion to the cavity allows for optical preparation and measurement of the electron spin. We apply
+a magnetic field of |B| = 600 G to lift the degeneracy of the S = 1/2 ground and excited states, resulting in two spin-
+conserving transitions (A,B) and two spin-flip transitions (C,D) as shown in Fig. 2a-b (the magnetic moments for the
+ground and excited state are described in the supplementary information [41]). Tuning the cavity to the A transition
+enhances the decay rate of the excited state, shortening the lifetime from 6.3 ms to τ = 7.4 µs, corresponding to a
+Purcell factor of P = 850 (Fig. 2c). To enable spin readout, we engineer a cycling transition by selectively enhancing
+the A transition relative to D by a combination of detuning from the cavity and preferential orientation of the
+transition dipole moment with respect to the cavity polarization [28], resulting in ΓA/ΓD ≈ 1030(10) [41]. Spin
+initialization is performed by optical pumping on the A or B transitions while simultaneously driving the excited
+state MWe transition [42]. In Fig. 2d, we demonstrate spin initialization and readout with an average fidelity of
+F = 0.972 (Fig. 2d). The combination of high collection efficiency and low background from other Er3+ ions allows
+for high-contrast optical Rabi oscillations (Fig. 2e). After a π pulse, a single photon is detected with a probability
+P1 = 0.035, on top of a background count rate of Pb = P1/117. Both P1 and the signal-to-background ratio are
+larger than what is obtained with frequency converted NV centers by more than an order of magnitude [19], enabled
+by the high quantum efficiency and collection efficiency of the Er-cavity system.
+The linewidth of the spin-conserving transitions is determined using PLE spectroscopy. To avoid optical pumping,
+the excitation laser has two tones separated by approximately 1 GHz to drive the A and B transitions simultaneously.
+The typical linewidth of a single scan (1 minute) is approximately 150 kHz, while the line center has an r.m.s.
+fluctuation of 63 kHz over 12 hours (Fig. 2f). This represents a 100-fold improvement over previously reported
+linewidths for individual Er3+ ions in nanophotonic cavities [5, 7, 42], and is to our knowledge the narrowest optical
+transition observed for a solid-state defect in a nanophotonic device. We note that similar linewidths have been
+observed for single Er3+ ions in 19 µm thick Y2SiO5 membranes [6]. The single-scan linewidth is 7 times larger
+than the Purcell-enhanced radiative linewidth of the A transition, Γr = 1/τ = 2π × 21.4 kHz, however, photon
+echo experiments suggest that this linewidth is dominated by slow dynamics [41], such that indistinguishable photon
+emission may be possible on short timescales or with active feedback.
+We perform HOM two-photon interference measurements [9] on time-delayed photons using an unbalanced Mach-
+Zehnder interferometer (MZI) with a ∆L = 36 km delay line in one arm (Fig. 3a). By tuning the repetition rate of
+the excitation pulses to match the delay time of the long arm (∆t = 175 µs), successive photons may arrive at the
+final beamsplitter simultaneously, and HOM interference will suppress the probability of detecting one photon at each
+output if the photons are indistinguishable. Experimentally, we observe strongly suppressed coincidences (Fig. 3b),
+indicating a high degree of indistinguishability. In a control experiment, we artificially broaden the photon in the
+short arm using a fiber stretcher driven by a noise source, restoring the coincidence rate expected for distinguishable
+photons (Fig. 3c). We measure an HOM coincidence rate of R = 2 min−1, defined as the rate of simultaneous photon
+detection in the distinguishable photon case, corresponding to a per-shot coincidence probability of Pc = 8.5 × 10−6.
+The indistinguishability is quantified by the visibility V [43], given by V = 1 − 2A0/A|i|≥2, where A0 is the
+integrated counts under the central peak and A|i|≥2 is the average integrated counts in each side peak (Fig. 3b). The
+visibility is maximized for a coincidence window approaching zero, however the number of photons within this window
+(the acceptance fraction) will also be small (Fig. 3e). For coincidences with photon detection times t1, t2 separated
+by |t2 − t1| < 2τ (corresponding to an acceptance fraction of 63%), the raw visibility is over 70%, rising to 90% when
+the accidental coincidences from dark counts and ambient background are subtracted. Integrating under the entire
+peak in Fig. 3d and subtracting accidental coincidences gives V = 80(4)%. The residual distinguishability has a
+significant contribution (4%) due to the MZI output beamsplitter ratio deviating from 50:50. Therefore, we conclude
+that the effective linewidth over hundreds of microseconds is only slightly larger than the radiative linewidth [41].
+Lastly, we study the properties of the Er3+ spin, which has the potential to serve as a quantum memory for spin-
+
+3
+photon entanglement. In bulk Er3+:CaWO4 , the magnetic moment is anisotropic with gc = 1.25 and ga = 8.38 [38].
+However, for the individual ions studied in this work, we observe significantly distorted magnetic moments, including
+a variation of g in the aa-plane. These deviations can be reproduced with the inclusion of a small axial crystal field
+term [41], which may arise from proximity to the surface or the presence of a nearby defect.
+The spin relaxation time is T1 = 3.7 s, in line with previous reports [8], and is limited by the direct process with
+a T1 ∝ 1/B5 dependence [41]. Ramsey and Hahn echo experiments give T ∗
+2 = 247 ns and T2 = 44 µs, respectively
+(Fig. 4c-d). An XY64 dynamical decoupling sequence allows coherence to be preserved for longer than 200 µs (Fig. 4e),
+while also showing collapses and revivals due to the 183W nuclear spin bath.
+The Hahn echo T2 is improved by one order of magnitude from Er3+:Y2SiO5 under similar conditions [42], but the
+coherence is still significantly shorter than predictions based on CCE simulations accounting for the 183W nuclear
+spin bath (I = 1/2, 14.3% abundance). This implicates paramagnetic impurities in the host crystal or on the surface
+as the primary source of decoherence, with an inferred density of approximately 3 × 1016 cm−3 [41]. Indeed, longer
+spin echo coherence times of T2 = 23 ms were observed for bulk Er3+ ensembles in CaWO4 in Ref [8] by operating
+at dilution refrigerator temperatures to freeze out paramagnetic impurities. Although the coherence is not limited
+by the nuclear spin bath, dips in coherence due to a single strongly coupled 183W spin are observed in Fig. 4d.
+The results demonstrated in this work will enable spin-photon entanglement and HOM interference between
+multiple Er3+ emitters with postselection using a narrow coincidence window or active tracking of the transition
+frequencies. In future work, the radiative linewidth can be further increased using cavities with higher Q [44] or
+smaller mode volume [45].
+Furthermore, more careful annealing or surface preparation may reduce the spectral
+diffusion.
+The flexibility to incorporate Er3+ via ion implantation, instead of during growth, will allow future
+exploration of CaWO4 samples produced and refined using diverse techniques. Reducing the impurity concentration
+may also improve the optical linewidth: scaling the ground state magnetic linewidth 1/T ∗
+2 to the optical transition
+implies a significant magnetic noise contribution of 2π × 46 kHz.
+In this work, we have demonstrated an engineered material, ion-implanted Er3+:CaWO4, that enables indistin-
+guishable single photon generation from a single rare earth ion in the telecom band. We attribute the improved
+performance to the higher Er3+ site symmetry (compared to previous observations of single Er3+ ions [5, 7, 27]).
+Spectral multiplexing of many ions per node [42], using quantum eraser techniques to overcome static frequency
+differences [46], will enable higher repetition rates over long fiber segments, while simultaneously reducing the co-
+herence time requirements [47]. Additional storage capacity and functionality may be obtained from ancilla nuclear
+spin registers, as recently demonstrated for several rare-earth ion systems [30, 31], and ion implantation may allow
+for the creation of spatially modulated density profiles with strong magnetic ion-ion interactions.
+Acknowledgements:
+We acknowledge helpful conversations with Charles Thiel, Philippe Goldner, and Miloš
+Rančić. This work was primarily supported by the U.S. Department of Energy, Office of Science, National Quantum
+Information Science Research Centers, Co-design Center for Quantum Advantage (C2QA) under contract number
+DE-SC0012704.
+We also acknowledge support from the DOE Early Career award (for modeling of decoherence
+mechanisms and spin interactions), as well as AFOSR (FA9550-18-1-0334 and YIP FA9550-18-1-0081), the Eric
+and Wendy Schmidt Transformative Technology Fund, and DARPA DRINQS (D18AC00015) for establishing the
+materials spectroscopy pipeline and developing integrated nanophotonic devices.
+We acknowledge the use of
+Princeton’s Imaging and Analysis Center, which is partially supported by the PCCM, an NSF MRSEC (DMR-
+1420541), as well as the Princeton Micro-Nano Fabrication Center.
+Note: While finalizing this manuscript, we became aware of recent reporting the detection of single Er3+ ions in
+CaWO4 using magnetic resonance techniques [48].
+
+4
+−500 −250
+0
+250
+500
+Laser frequency (MHz)
+0.000
+0.005
+0.010
+Counts per pulse
+~10 nm
+CaWO4
+implanted Er3+ 
+Z
+X
+silicon cavity
+a
+b
+d
+e
+c
+20 µm
+Y X
+5 µm
+2 µm
+Y
+X
+Y
+X
+(i)
+(i)
+(ii)
+(ii)
+e
+Ca
+W
+O
+Er
+a = 5.2 Å
+c = 11.4 Å
+a
+−6
+−3
+0
+3
+6
+Laser frequency (GHz) 
+- 1532.62 nm
+Fluorescence (a.u.)
+Z
+Y
+X
+FIG. 1. Er3+:CaWO4 device architecture. a CaWO4 crystal structure, with a substitutional Er3+ impurity in an S4 Ca2+
+site. b A dense implanted Er3+:CaWO4 ensemble has an inhomogeneous optical linewidth of 730 MHz on the Z1-Y1 transition.
+In addition to the central peak, we observe hyperfine structure from 167Er with nuclear spin I = 7/2. c Scanning electron
+microscope image of a representative silicon nanophotonic device, consisting of a photonic crystal grating coupler [inset (i)]
+that tapers adiabatically into a bus waveguide connected to a photonic crystal nanobeam cavity [inset (ii)]. d Erbium ions
+are implanted targeting a depth of 10 nm, and couple evanescently to the silicon photonic crystal on the surface. e PLE
+spectrum of Er3+ ions coupled to the cavity, with resolved single ion lines. The red arrow indicates the ion used for subsequent
+experiments.
+−500 −250
+0
+250
+500
+Detuning (kHz)
+4.50
+4.75
+5.00
+Time (hours)
+−500 −250
+0
+250 500
+Detuning (kHz)
+0
+2
+4
+6
+8
+10
+12
+Time (hours)
+A
+B
+D
+C
+MWg
+MWe
+|↑e⟩ 
+|↓e⟩
+|↑g⟩
+|↓g⟩
+1532 nm
+7 GHz
+6 GHz
+−10
+−5
+0
+5
+10
+Frequency (GHz)
+Reflection
+A
+B
+C
+D
+a
+b
+c
+d
+e
+f
+0
+300
+600
+Optical pulse width (ns)
+0.000
+0.018
+0.035
+Counts per pulse
+init. |↑g⟩
+init. |↓g⟩
+10
+0 10
+1 10
+2 10
+3 10
+4
+Time (µs)
+Fluorescence (a.u.)
+0
+5
+10
+15
+Number of photons
+10
+−4
+10
+−3
+10
+−2
+10
+−1
+10
+0
+Probability
+FIG. 2. Efficient photon collection from a cavity-coupled ion. a Er3+ level structure. In a magnetic field, Er3+ has
+four distinct optical transitions. The field strength is |B| = 600 G, oriented in the aa-plane, 22o from the X-axis. b Reflection
+spectrum of the cavity showing a full-width, half-maximum linewidth of κ = 1.0 GHz (Q = 1.9 × 105), which is tuned into
+resonance with the A transition. c The lifetime of the |↑e⟩ excited state is reduced to 7.4 µs (blue), which is 850 times shorter
+than the bulk lifetime of 6.3 ms (orange). d Histogram of photon counts obtained during spin readout after initializing in
+|↑g⟩ and |↓g⟩. The average readout fidelity is F = 0.972, using a threshold of one photon. The solid line is a fit to a Poisson
+distribution with average photon number ¯n = 6.4. e Optical Rabi oscillation on transition A. The peak single photon emission
+probability is P1 = 0.035. f Repeated PLE scans show an average single-scan linewidth of 150 kHz, and long-term diffusion
+of the line center of 63 kHz.
+
+8888
+888888888888888888888888885
+c
+75:25
+BS
+50:50
+BS
+36 km
+fiber
+SNSPD
+SNSPD
+fiber
+stretcher
+PC
+PC
+∆t
+VOA
+FG
+d
+a
+e
+−525
+−350
+−175
+0
+175
+350
+525
+0
+20
+40
+60
+80
+HOM coincidences
+−525
+−350
+−175
+0
+175
+350
+525
+Detection time difference t1 − t2 (μs)
+0
+20
+40
+60
+80
+HOM coincidences
+b
+0.5
+0.6
+0.7
+0.8
+0.9
+1.0
+Visibility
+0.0
+0.5
+1.0
+1.5
+2.0
+Coincidence window / (2T1)
+0.0
+0.2
+0.4
+0.6
+0.8
+1.0
+Acceptance fraction
+A0
+A1
+A-1
+A-2
+A-3
+A2
+A3
+−30
+−15
+0
+15
+30
+Detection time difference t1 − t2 (μs)
+0.0
+0.1
+0.2
+0.3
+0.4
+0.5
+Norm. HOM coincidences
+FIG. 3. Generation of indistinguishable photons. a Schematic of the HOM interferometer, indicating beamsplitters (BS),
+a variable optical attenuator (VOA), polarization controllers (PC) and a fiber stretcher driven by a noise source (FG) to tune the
+distinguishability. b Histogram of coincidences detected in a 4 hour measurement period. The Hong-Ou-Mandel effect results
+in a suppressed probability of coincidences at zero delay, indicating indistinguishable single-photon emission. c Histogram
+of coincidences in a control experiment with a noise source applied to the fiber stretcher, destroying the indistinguishability
+and Hong–Ou–Mandel interference. d Zoom-in around the zero time delay HOM interference pattern. The red line shows
+a model including background counts (black dashed line) and pure dephasing of the optical transition.
+The blue dashed
+line shows a simple model for the control experiment assuming perfect distinguishability, while the solid blue line shows a
+model incorporating the finite bandwidth of the noise source [41]. e Interference visibility (top) and relative coincidence rate
+(bottom) as a function of the coincidence window, before (green) and after (red) subtracting the accidental coincidences from
+the detector and ambient background dark counts.
+
+6
+c
+e
+a
+b
+0.0
+0.4
+0.8
+Free evolution time (µs) 
+0.50
+0.75
+1.00
+Population |↑g⟩
+0
+5
+10
+Wait time (s)
+0.50
+0.75
+1.00
+Population |↑g⟩
+0
+150
+300
+MWg pulse width (ns)
+0.0
+0.5
+1.0
+Population |↑g⟩
+d
+0
+200
+400
+600
+800
+1000
+1200
+Free evolution time (µs) 
+0.4
+0.6
+0.8
+1.0
+Population |↑g⟩
+0
+50
+100
+Free evolution time (µs) 
+0.50
+0.75
+1.00
+Population |↓g⟩
+FIG. 4. Spin dynamics. a Rabi oscillations after initializing into |↑g⟩ (blue) or |↓g⟩ (orange). The spin transition frequency
+fMWg = 7.0 GHz.
+b Spin relaxation after initialization into |↑g⟩.
+An exponential fit yields T1 = 3.7(3) s.
+c Ramsey
+measurement. Fitting to e−(t/T ∗
+2 )n reveals a T ∗
+2 = 247(9) ns with n = 2.2(3). d A Hahn echo measurement shows dips
+in coherence resulting from the 183W nuclear spin bath. The grey lines show CCE simulations for randomly selected 183W
+configurations, where each configuration includes a single strongly coupled 183W spin required to reproduce the dips. The blue
+lines include an additional, phenomenological stretched decay with T2 = 44 µs. e Applying an XY64 dynamical decoupling
+sequence extends the spin coherence to longer times. Here, the grey lines show CCE simulations for the same 183W bath
+configurations in panel (d), while the blue lines have an additional phenomenological decay of 460 µs.
+
+7
+∗ These authors contributed equally to this work.
+† Present address: Department of Physics, Northeastern University, Boston, Massachusetts 02115, USA
+‡ Present address: Department of Electrical and Computer Engineering, Rice University, Houston, Texas 77005, USA
+§ jdthompson@princeton.edu
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+
+Supplementary information for indistinguishable telecom band photons from a single
+erbium ion in the solid state
+Salim Ourari,1, ∗ Łukasz Dusanowski,1, ∗ Sebastian P. Horvath,1, ∗ Mehmet T. Uysal,1, ∗ Christopher M. Phenicie,1
+Paul Stevenson,1 Mouktik Raha,1 Songtao Chen,1 Robert J. Cava,2 Nathalie P. de Leon,1 and Jeff D. Thompson1, †
+1Department of Electrical and Computer Engineering, Princeton University, Princeton, NJ 08544, USA
+2Department of Chemistry, Princeton University, Princeton, NJ 08544, USA
+CONTENTS
+I. Photon collection efficiency
+1
+II. CaWO4 sample preparation
+2
+III. Site-selective excitation spectroscopy
+2
+IV. Spin state initialization and readout
+3
+V. Engineering an optical cycling transition
+3
+VI. Photon echo investigation of the optical coherence
+4
+VII. HOM fit functions
+4
+VIII. Distortion of the magnetic moment tensor
+7
+IX. Dependence of T1 on magnetic field
+8
+X. Spin coherence modeling
+8
+XI. Estimating concentration of paramagnetic impurities
+10
+References
+12
+I.
+PHOTON COLLECTION EFFICIENCY
+This section details the efficiency of components in our photonic circuit that impact the total photon collection
+efficiency. We group these losses into four terms: internal losses in the cavity, in the grating coupler and waveguide
+taper, transmission through passive optical components, and the quantum efficiency of the SNSPDs.
+The internal losses in the cavity are determined from the reflection spectrum. In particular, the contrast of the
+reflection on and off cavity resonance C = (Roff − Ron)/Roff has the form [1]:
+C = (1 − 2ηcav)2.
+(1)
+The photon extraction efficiency from the cavity ηcav = κwg/κtot is the ratio of cavity outcoupling rate κwg to the total
+loss κtot = κwg +κint, including internal cavity losses κint. For the device used in this work, we find ηcav = 0.26. The
+grating coupler and waveguide efficiency is estimated by measuring the round-trip optical losses away from the cavity
+resonance. We compute the efficiency as ηGC =
+�
+Pout/Pin, and find ηGC = 0.36 for the device used in this work.
+The transmission through passive optical components (beamsplitters, splices, etc.) is measured independently to be
+ηnet = 0.61 (the HOM interferometer has additional losses, discussed below). The detection efficiency of the SNSPD
+is ηdet = 0.85. This gives a combined predicted photon detection probability of P1 = ηcav ×ηGC ×ηnet ×ηdet = 0.049.
+We lose an additional 7% of the single-ion emission from the finite time required to switch on the SNSPD bias current
+(≈ 900 ns), which is turned off during the optical excitation pulse to avoid saturating the detector. The predicted
+photon detection probability is then P1 = 0.045, close to the measured value of 0.035.
+arXiv:2301.03564v1  [quant-ph]  9 Jan 2023
+
+2
+In the HOM experiment, the two-photon coincidence probability is decreased by the losses of the MZI delay line.
+We measured the optical transmission in the 36 km fiber spool to be T = 0.2, consistent with an attenuation length
+La = 21.7 km. To maximize the coincidence probability and balance the power in the two arms of the interferometer,
+we use a 75 : 25 ratio beamsplitter at the input, such that 0.75 of the power is directed into the long MZI arm,
+while 0.25 is directed into the short one. In the shorter arm, we utilize a variable optical attenuator set to match
+the powers before the final MZI beamsplitter input ports. Accounting for both the losses in the fiber spool in the
+long interferometer arm (matched with a VOA on the short arm), as well as the beamsplitter ratio, the two-photon
+coincidence probability at |t1 − t2| = 0 is PHOM = 0.5 × (0.75P1T)2 = 0.01P 2
+1 . Here, the factor of 0.5 accounts for
+the requirement that the early (late) photon must pass through the long (short) interferometer arm. For P1 = 0.035
+we get PHOM = 1.4 × 10−5, close to the measured value of 8.5 × 10−6.
+II.
+CAWO4 SAMPLE PREPARATION
+The CaWO4 substrates used in this study were procured from SurfaceNet GmbH with a single sided epi polish and
+(100) orientation (i.e., with a surface spanned by a and c axes). According to the vendor, the crystals were grown
+using 99.9999% purity precursors and sold as “high purity CaWO4.” Polished samples were implanted with erbium
+(II-VI Inc.) using an energy of 35 keV which corresponds to a target depth of 10 nm (calculated by Stopping-Range
+of Ions in Matter simulations [2]). Samples were prepared using two different Er3+ concentrations, a high density
+sample used for ensemble spectroscopy utilizing a fluence of 1 × 1012 ions/cm2 and a low density sample used for
+single ion spectroscopy implanted with a fluence of 5 × 109 ions/cm2. Subsequent to implantation, samples were
+annealed in air at a temperature of T = 300 ◦C for 1 hour, with a heating rate of T = 300 ◦C/hour. Annealing
+healed implantation damage and improved site occupation of the S4 point-group symmetry substitutional site.
+Silicon photonic crystal cavities are prepared as described previously [3]. The cavities are oriented on the substrate
+such that the predominant electric field polarization is parallel to the CaWO4 c-axis.
+III.
+SITE-SELECTIVE EXCITATION SPECTROSCOPY
+In order to confirm that implanted Er3+ ions substitute at a site with S4 symmetry we performed a site-selective
+excitation spectroscopy measurement using the high density sample cooled to 4 K. A tunable narrowband laser was
+employed in conjunction with an optical chopper to generate a train of excitation pulses. Using a second chopper,
+fluorescence was collected out of phase with the excitation laser, dispersed using a monochromator, and detected
+with an InGaAs detector array. By performing a fluorescence measurement while sweeping the excitation laser a
+map of site-specific excitation and emission frequencies was obtained (Fig. S1a). Table S1 summarizes the measured
+transition energies of four 4I15/2 and three 4I13/2 levels. The resulting energy level structure is shown in Fig. S1b, and
+found to be in close agreement with the transition energies previously reported for bulk doped samples [4], confirming
+the S4 site assignment. The spectrum does not show any detectable fluorescence at other wavelengths within our
+scan range, suggesting that this is the only Er3+ incorporation site.
+TABLE S1.
+Transition energies of implanted Er3+:CaWO4 determined using site-selective excitation spectroscopy. All values
+are in cm−1. Transitions to Zn and Yn for n greater than what is shown were not observed due to limitations in detection
+sensitivity. The observed transition energies are in close agreement with values obtained for bulk doped samples [4].
+n
+4I15/2Zn
+4I13/2Yn
+1
+0
+6524.4
+2
+20.3
+6532.9
+3
+25.9
+6573.2
+4
+52.0
+*
+We note that compared to the optical excited state lifetime of 6.3 ms the Yn crystal field levels thermalize rapidly,
+such that an identical fluorescence spectrum is obtained independent of which 4I13/2 level is excited (green arrows
+in Fig. S1). Furthermore, while the fluorescence spectrum is dominated by decay from the 4I13/2Y1 level, a small
+fraction of fluorescence originates from the 4I13/2Y2 level. This leads to two sets of fluorescence patterns with identical
+energy splittings (but different intensities), overlaid with an offset given by the 4I13/2Y1-4I13/2Y2 splitting (indicated
+using dashed arrows in Fig. S1).
+
+3
+Y1
+Y2
+Y3
+Z2
+Z3
+Z1
+Z4
+2 3
+1
+2
+3
+1
+1 2 3
+5
+6
+4
+3
+1
+2 & 6
+4
+7
+5
+7
+4I15/2
+4I13/2
+a
+b
+1520
+1522
+1524
+1526
+1528
+1530
+1532
+Excitation wavelength (nm)
+1510
+1515
+1520
+1525
+1530
+1535
+1540
+1545
+1550
+Fluorescence wavelength (nm)
+Y7
+Z8
+FIG. S1. Ensemble spectroscopy of Er3+:CaWO4. a Site-selective excitation spectrum of Er3+:CaWO4. Green arrows
+indicate when the excitation laser is resonant with different excited state crystal-field levels, where the corresponding excitation
+path is labeled using the matching number in (b). Solid orange arrows denote the fluorescence energies due to decay from
+the 4I13/2Y1 level, whereas the dashed arrows denote decay from the 4I13/2Y2 level. The prominent line with unity gradient
+corresponds to laser scatter and has been re-scaled in intensity for clarity. b The inferred energy level structure of Er3+:CaWO4.
+In total, the performed spectroscopy yielded energies of four 4I15/2 and three 4I13/2 levels.
+IV.
+SPIN STATE INITIALIZATION AND READOUT
+To achieve fast and efficient spin state initialization we follow the approach used in Ref. [5]. This initialization
+scheme consists of using optical π pulses resonant with either the A or B transition, each followed by a microwave
+pulse resonant with the excited state spin transition, MWe. For example, to initialize into the |↓g⟩ state we alternate
+π pulses resonant with the optical A and MWe transitions. The number of pulse pairs used for the HOM and spin
+dynamics experiments was 1 and 10, respectively.
+For spin state readout, we used a sequence of n optical π pulses resonant with the A transition each followed by a
+fluorescence collection window. By varying the number of readout pulses we found that the spin state readout fidelity
+is maximized for n = 195, and applying further pulses would reduce the readout fidelity due to optical pumping.
+After optimizing the number of readout pulses, a threshold was set for the total number of photons collected after n
+pulses (Nthresh) to discriminate between the |↓g⟩ and |↑g⟩ states. We found that a threshold of Nthresh = 1 gives the
+highest readout fidelity; that is, if we obtained on average one photon or more after the n pulses then the spin state
+was assigned to |↑g⟩, and if we obtained no photon then the spin state was assigned to |↓g⟩.
+V.
+ENGINEERING AN OPTICAL CYCLING TRANSITION
+As noted in Ref. [6] for Er3+:YSO, the cyclicity of the optical transition depends strongly on the magnetic
+field orientation since changing the field changes the atomic transition dipole moment with respect to the cavity
+polarization. For the case where the cavity linewidth is large compared to the Zeeman splitting, the cyclicity is
+maximized when the spin-flip transitions C, D are orthogonal to the cavity polarization. In this work, by combining
+a larger C-D splitting and a narrower optical cavity than in Ref. [6] we were able to tune the A transition into
+resonance with the cavity while simultaneously keeping the spin-flip transitions C and D detuned from the cavity
+(see Fig. 2b). Consequently we optimize the magnetic field orientation to maximize cyclicity. From this we found
+the optimal field orientation to be in the aa-plane, rotated 22o from the X-axis.
+To probe the cyclicity, the ion was initialized into |↑g⟩ and subsequently read out using 400 pulses. Each readout
+pulse consisted of an optical π pulse resonant with the A transition followed by a fluorescence collection window.
+Due to the finite cyclicity, the fluorescence count rate decays with increasing readout pulse number. This yielded a
+cyclicity of C = ΓA/ΓD = 1030(10) (Fig. S2a).
+To establish the single photon emission we calculate the intensity autocorrelation g(2)(τ) as shown in Fig. S2b. At
+
+4
+zero offset pulse delay, we observe g(2)(0) = 0.018(3), showing strong suppression of multi-photon emission events
+and thus a high purity of single photon emission.
+0
+100
+200
+300
+400
+Readout pulse number
+0.00
+0.01
+0.02
+0.03
+Counts per pulse
+b
+a
+−20
+−10
+0
+10
+20
+Pulse offset n
+10
+−2
+10
+−1
+10
+0
+g(2)(τ)
+FIG. S2. Measuring the cyclicity. a Decay of the A transition fluorescence count rate from optical pumping after initializing
+into |↑g⟩ (blue). The count rate decays as e−n/C revealing a cyclicity of C = 1030(10). The orange trace corresponds to the
+same readout pulse sequence after initializing into |↓g⟩. b Intensity autocorrelation g(2)(τ) of the A transition showing strong
+suppression of the zero-delay peak with g(2)(0) = 0.018(3).
+VI.
+PHOTON ECHO INVESTIGATION OF THE OPTICAL COHERENCE
+We probe the coherence of the optical A transition using the photon echo technique. After initialization of the
+spin state, we utilized an excitation sequence consisting of three optical pulses π/2 - π - π/2 separated by a waiting
+time τ (for a total free evolution time 2τ), followed by a fluorescence collection window. The refocusing π pulse
+in the middle of the sequence removes the slowly varying inhomogeneous dephasing. For each evolution time, we
+swept the phase of the last π/2 pulse and fitted the fluorescence intensity change against the phase angle using a sine
+function. The fitted oscillation amplitude is proportional to the two-level coherence. This yielded a coherence time
+of T2 = 10.2 µs for a Hahn echo sequence (Fig. S3a blue points). To further filter out higher frequency noise, we
+increased the number of refocusing pulses using an XY N dynamical decoupling sequence (see Fig. S3b) and reached a
+radiatively limited coherence time of 18 µs for N = 32 (see Fig. S3a orange points). We note that in this experiment
+the emission lifetime is T1 = 9.1 µs, different from 7.4 µs recorded using time-resolved PLE shown in the main text.
+The results of this experiment implicate slow spectral diffusion as the main source of decoherence in our system. In
+the case of the Hahn echo sequence, the recorded T2 time is approximately a factor of two from the lifetime limit
+T2/(2T1) = 0.56.
+VII.
+HOM FIT FUNCTIONS
+In this section, we derive the two-photon interference functions used to model the HOM histograms in the main
+text. In particular, we consider two dephasing mechanisms, which allow us to estimate the possible bounds on the
+emission linewidth based on the photon visibility determined from the HOM experiment. Finally, we extend the
+model to simulate the reference HOM measurement, where the photons are made distinguishable by periodically
+changing the phase of one of the two interfering photons.
+To model the HOM zero-delay time peak shape, we follow the derivation of Ref [7]. We assume that single photon
+spatio-temporal wave functions have an exponential form:
+ζ(t) = H(t) · exp
+�
+− t
+2T1
+− i[ω(t)t + φ(t)]
+�
+,
+(2)
+where H(t) is the Heaviside function, T1 is the radiative lifetime, ω(t) is the photon frequency and φ(t) is the
+phase. In an ideal case, the frequency and phase of the photon are constant over time, so the photon coherence is
+lifetime limited. A time-dependent frequency and phase noise lead to decoherence. Typically it is assumed that such
+perturbations yield random walks in phase and frequency at two distinct timescales leading to two different physical
+descriptions. The first regime is pure dephasing, caused by fast frequency jumps accumulating phase on a timescale
+
+5
+b
+a
+0
+10
+20
+30
+40
+50
+Free evolution time (μs)
+10
+−1
+10
+0
+Coherence
+0
+5
+10
+15
+20
+25
+30
+ Number of π pulses
+5
+10
+15
+20
+ Coherence time (μs)
+FIG. S3. Optical coherence of Er3+:CaWO4. a An optical Hahn echo measurement (green) reveals an optical coherence
+of T2 = 10.2 µs. Applying XY 32 dynamical decoupling sequence (orange) extends the optical coherence to the radiative limit
+(blue dashed line) T2 = 18 µs, at a field where the optical T1 = 9.1(3) µs. b Optical coherence scaling with the number of
+refocusing pulses of XY dynamical decoupling sequences (red) compared to the lifetime limit (blue).
+shorter than T1. The second regime is described by spectral diffusion, primarily attributed to slow frequency drift on
+a timescale considerably longer than T1. It is worth noting that this time-scale distinction is somewhat artificial, as
+it is known that different noise sources have continuous power spectral densities [8]. Still, we perform this analysis to
+study limiting cases. It can be shown that for single photons passing through an unbalanced MZI with a delay equal
+to the photon generation rate, the HOM histogram peak areas An will be given by: A|i|≥2 = A, A1 = A(1 − R2),
+A−1 = A(1 − T 2) and A0 = A(R2 + T 2 − 2RTVint) [9]. Here, A is the Poissionian peak coincidence level, R/T is the
+reflection/transmission coefficient of the output MZI beamsplitter, and Vint is the emitter visibility. In such cases,
+the HOM two-photon interference coincidence probability function can be described by
+P(τ) =Pdc + A
+N
+�
+k=2
+e−
+|τ±ktrep|
+T1
++ A(1 − R2)e−
+|τ+trep|
+T1
++ A(1 − T 2)e−
+|τ−trep|
+T1
++ Ae− |τ|
+T1 (R2 + T 2 − 2RT · F(τ)),
+(3)
+where Pdc is the coincidence level related to SNSPD dark counts and ambient light counts, and trep is the pulse
+repetition period. Further, F(τ) is an integral defined as [7]
+F(τ) =
+� ∞
+−∞
+dt0 cos [∆ω(τ, t0)τ + ∆φ(τ, t0)] ,
+(4)
+where ∆φ(τ, t0) is a the phase difference and ∆ω(τ, t0) is the frequency difference between two interfering photons.
+If frequency and phase fluctuations are assumed to follow Gaussian distribution functions, F(τ) is given by [7]
+F(τ) = exp
+�
+− 2|τ|
+Tdep
+− σ2τ 2
+�
+,
+(5)
+where Tdep is the dephasing time 1/Tdep = 1/T2 −1/(2T1), and σ is the inhomogeneous linewidth broadening related
+to slow spectral diffusion.
+First, we consider the case where fast spectral diffusion dominates (Lorentzian broadening), for which
+Flor(τ) = exp
+�
+− 2|τ|
+Tdep
+�
+.
+(6)
+This allows for the evaluation of the HOM histogram central peak area
+A0 = A
+� ∞
+−∞
+dτ
+�
+R2 + T 2 − 2RTe
+− 2|τ|
+Tdep
+�
+e− |τ|
+T1 = A
+�
+2T1(R2 + T 2) − 4RT
+T1Tdep
+2T1 + Tdep
+�
+,
+(7)
+
+6
+and the off-center peak area A|i|≥2
+A|i|≥2 = A
+� ∞
+−∞
+dτe− |τ|
+T1 = 2AT1,
+(8)
+giving a visibility of
+V = 1 − 2A0
+A|i|≥2
+= 1 − 2
+�
+R2 + T 2�
++ 4RT
+Tdep
+2T1 + Tdep
+= 1 − 2
+�
+R2 + T 2�
++ 4RT T2
+2T1
+.
+(9)
+Note that this definition of visibility contains information about both the MZI interferometer imperfections (BS R : T)
+and the intrinsic indistinguishability of the emitted photons Vint = T2/(2T1). In our experiment R : T = 0.43 : 0.57
+is determined from the imbalance between A−1 and A1 peak areas in the HOM histogram (see Fig. 3b in the main
+text), contributing to a decrease in visibility of around 4%. By evaluating the integrated counts A0 and A|n|≥2 in
+the HOM experiment, we obtain a visibility of 80% after the dark and ambient light counts are subtracted. This
+allows us to estimate Vint = 0.84, and further T2 = 0.84 × 2T1 = 15.3 µs. Using Eqs. (3) and (6) with T2 = 15.3 µs
+we model the HOM histogram in the main text (Fig. 3b,d). Furthermore, using this model, we can estimate the
+bound on the emission linewidth at the timescale of 100 − 200 µs following
+νL =
+1
+πT2
+=
+1
+2πT1Vint
+,
+(10)
+which yields Lorentzian broadening of νL = 20.6 kHz. Note that in the case of Fourier limited photons, one obtains
+νLF = 1/(2πT1) = 17.3 kHz.
+In the case where slow dynamics dominate decoherence (Gaussian broadening), the function F(τ) is simplified to
+Fgau(τ) = exp
+�
+−σ2τ 2�
+, such that
+A0 = A
+�
+��2T1(R2 + T 2) − 2RT
+√πe
+1
+4σ2T 2
+1 erfc
+�
+1
+2σT1
+�
+σ
+�
+�� .
+(11)
+This leads to a visibility given by
+V = 1 − 2
+�
+R2 + T 2�
++ 2RT
+√πe
+1
+4σ2T 2
+1 erfc
+�
+1
+2σT1
+�
+σT1
+,
+(12)
+where the intrinsic emitter visibility is given by
+Vint =
+√πe
+1
+4σ2T 2
+1 erfc
+�
+1
+2σT1
+�
+2σT1
+.
+(13)
+Using Vint = 0.84 we get σ equal to 0.039 MHz, which corresponds to a Gaussian broadening of νG = 2
+√
+2ln2
+√
+2σ =
+21 kHz. Note that the estimated νG is on the order of the Fourier limit of 17.3 kHz, such that the resultant emission
+line-shape will be described by a Voigt profile with a total width of
+νV = 0.535
+2πT1
++
+�
+0.217
+(2πT1)2 + ν2
+G,
+(14)
+equal to a linewidth of 31.4 kHz. This bounds the emission linewidth to be in the range 20.6 − 31.4 kHz for a
+timescale of 175.2 µs.
+Next, we consider the case of the reference HOM measurement in Fig. 3c. Typically, the distinguishability in
+HOM experiments is tuned by rotating the polarization of one of the photons. However, our SNSPDs have a strongly
+polarization-dependent detection efficiency, which complicates the interpretation of such a measurement. Therefore,
+we instead make the photons distinguishable by artificial spectral broadening, achieved by rapidly modulating the
+path length of one of the interferometer arms with a large amplitude. Specifically, the phase of the photons traveling
+through the shorter MZI arm is modulated using a triangle wave with frequency
+ωm
+2π = 43 kHz and amplitude
+Am = 0.75π. Consequently, the phase difference can be described directly by
+∆φ(τ, t0) = 2Am
+π
+[arcsin (sin (ωm(τ + t0))) − arcsin (sin (ωmt0))] .
+(15)
+
+7
+In this case we can assume that the HOM zero-delay peak area will be dominated by the external phase modulation
+so that we can omit the ∆ω term, and F(τ) will only depend on ∆φ(t0, τ)
+Fmod(τ) =
+� ∞
+−∞
+dt0 cos
+�2Am
+π
+[arcsin (sin (ωm(τ + t0))) − arcsin (sin (ωmt0))]
+�
+.
+(16)
+The periodic modulation of the phase difference with τ will translate into a quantum-beat-like signal with a distinct
+dip at zero detection time difference. To fit the HOM data in Fig. 3c,d of the main text, we evaluate Fmod(τ)
+numerically. The best fit to the experimental data is obtained for a modulation amplitude of 0.73(5)π where the
+modulation frequency was fixed to the value used in the experiment, 43 kHz.
+VIII.
+DISTORTION OF THE MAGNETIC MOMENT TENSOR
+The magnetic moments of individual single ions changed appreciably from ion-to-ion as well as from the previously
+recorded ground state g-tensor [4]. To investigate this, the magnetic response of two single ions (different from the
+single ion investigated in the main text) was fully characterized by probing the optical transition frequencies of the
+four transitions A, B, C, and D for a range of field orientations using a magnetic field magnitude of 50 G, with the
+results shown in Tab. S2.
+TABLE S2. Single-ion g-tensors measured for two separate ions. Due to a small distortion of the S4 point-group site, the
+single-ion magnetic moments do not satisfy gx = gy = g⊥. We note that in this and subsequent sections we use the convention
+where the z axis points along the crystallographic c axis, which is different to the coordinate system used in the main text.
+Ground state
+Excited state
+gx
+gy
+gz
+gx
+gy
+gz
+Ion 1
+8.5
+7.6
+1.7
+7.3
+6.9
+1.8
+Ion 2
+8.6
+7.9
+2.5
+7.6
+6.9
+2.3
+In order to gain a deeper understanding of the g-tensor anisotropy, an effective crystal-field Hamiltonian was used
+to model the intra-4f transitions of Er3+:CaWO4. The complete Hamiltonian has the form
+H = HFI + HCF + HZ,
+(17)
+where HFI corresponds to the free-ion components of the Hamiltonian, HCF accounts for the interaction of the
+valance electrons with the host material, and HZ is the Zeeman Hamiltonian. The free-ion Hamiltonian used is [10]
+HFI = EAVG +
+�
+1,2,3
+F kfk + ζ4fASO + αL(L + 1) + βG(G2) + γG(R7) +
+�
+i=2,3,4,6,7,8
+T iti.
+(18)
+Noting only the most significant contributions, EAVG accounts for the spherically symmetric one-electron component,
+F k are the Slater parameters, f k are the electrostatic repulsion with angular dependence, ζ4f is the spin-orbit coupling
+term, and ASO is the spin-orbit coupling operator. The remaining contributions are higher-order and the reader is
+referred to Ref. [11] for details. To model the contribution of the S4 point-group symmetry crystal-field the following
+Hamiltonian was used
+HCF = B2
+0C(2)
+0
++ B4
+0C(4)
+0
++ B6
+0C(6)
+0
++ B4
+4(C(4)
+4
++ C(4)
+−4) + B6
+4(C(6)
+4
++ C(6)
+−4),
+(19)
+with the coefficients Bk
+q the crystal-field parameters and C(k)
+q
+spherical-tensor operators in Wybourne’s normalization.
+We note that the above Bk
+q are all real with the exception of B6
+4, which is complex. The above Hamiltonian was
+fitted to data obtained using site-selective fluorescence spectroscopy of the 4I13/2 →4I15/2 transitions as well as the
+barycenter of higher-energy transitions up to 4S3/2 obtained from literature [4]. Furthermore, the fit was optimized
+to reproduce the ensemble 4I15/2Z1 and 4I13/2Y1 g-tensors.
+Using the crystal-field parameters obtained via the method outlined above as a starting point, a fit was then
+performed to the g-tensor of both Ion 1 and Ion 2 to determine the crystal-field environment local to each ion. We
+hypothesize that the perturbation of the S4 point-group symmetry was caused by a single dominant defect, potentially
+due to the substrate surface or some form of remote charge compensation. Consequently, an axial term ¯B2
+0 oriented
+at an arbitrary angle with respect to the crystal-field quantization axis was introduced, and both the magnitude as
+
+8
+well as the orientation were varied to reproduce the observed ground and excited state g-tensors. This assumed that
+the crystal-field potential due to the defect is superposable with the nominal crystal-field potential [12].
+For Ion 1, the observed g-tensor was reproduced by an axial term ¯B2
+0 = 9.3 cm−1, rotated by an Euler rotation
+from the quantization axis with α = 90.4◦, β = 265◦ and γ = 0◦, following the Euler angle convention of Messiah [13].
+Similarly, Ion 2 required an axial term ¯B2
+0 = 10.7 cm−1, rotated by an Euler rotation from the quantization axis with
+α = 270.2◦, β = 98.0◦ and γ = 0◦. We note that the unperturbed rank 2 crystal-field parameter has a magnitude
+of B2
+0 = 578 cm−1, such that the observed change in the rank 2 crystal-field potential consisted of around ∼ 2% for
+both of the ions. This amounts to a frequency shift of 3.5 GHz for Ion 1 and 3.9 GHz for Ion 2 for the 4I15/2Z1
+to 4I13/2Y1 transition when compared to the crystal-field model not including any perturbation, which is somewhat
+larger than, but comparable to, the observed inhomogeneous linewidth for single ions coupled to the nanophotonic
+device. Therefore, the observed g-value anisotropy is consistent with a small amount of strain near the ion, potentially
+due to proximity to the surface or a charge compensation mechanism.
+In order to perform the initial crystal-field Hamiltonian fit we independently measured the 4I15/2Z1 and 4I13/2Y1
+ensemble magnetic moments using optical spectroscopy by utilizing the high density implanted sample. From this we
+obtain g⊥ = 8.6 and g∥ = 1.4 for the ground state, and g⊥ = 7.6 and g∥ = 1.3 for the excited state. The fitted ground
+state g-values deviate from the literature values measured by electron-spin resonance [4]; however, the deviation is
+within the uncertainty of our vector magnet calibration due to magnetic field gradients within the sample space. We
+note that the above conclusion about the mechanism for the anisotropy of single ion g-values does not depend on the
+precise magnetic moments assumed for bulk Er3+:CaWO4. Additionally, in the above fit we have constrained that
+gx = gy, and our data was also consistent with a fit for which gx ̸= gy at the 5% level. This suggests that some of
+the observed distortion of the magnetic moment tensor may also be present in an ensemble average.
+IX.
+DEPENDENCE OF T1 ON MAGNETIC FIELD
+To understand the observed spin lifetime, we performed a lifetime measurement at a second magnetic field strength,
+|B| = 950 G, obtaining T1 = 0.393 s. At low temperatures, Raman and Orbach processes are slow and the spin
+relaxation rate is dominated by the direct process. This has the following functional form with respect to an applied
+magnetic field [14]:
+T −1
+1
+= Ad
+�gµBB
+h
+�5
+coth
+�gµBB
+2kbT
+�
+.
+(20)
+Fitting the above equation to the observed spin lifetime data (Fig. S4) yielded a direct process constant Ad =
+6.4(2) × 10−6 s−1 GHz−5, with the expected T1 ∝ 1/B5 scaling. The spin T1 has previously been measured for
+CaWO4 at low temperatures, and the zero-temperature extrapolated lifetime was found to be 4.8 s at a frequency
+of 7.881 GHz [15]. This corresponds to Ad = 7.2 × 10−6 s−1GHz−5, approximately consistent with our value.
+X.
+SPIN COHERENCE MODELING
+In order to explain the observed spin coherence, we consider the magnetic environment of the Er3+ ion in the
+CaWO4 host crystal, consisting of the 183W nuclear spin bath and paramagnetic impurities. While the concentration
+and dynamics of paramagnetic impurities is not well known, the dynamics of the nuclear spin bath under decoupling
+sequences can be understood using standard CCE (Cluster Correlation Expansion) techniques [16]. First, we describe
+the Er3+ spin coupling to the nuclear spin bath and explain features observed in the Hahn experiment at short times.
+Then, we apply our understanding of the nuclear spin bath to find the expected decoherence rates for the Hahn and
+Ramsey experiments and observe that it cannot explain the observed rates in either. This leads us to conjecture the
+existence of an appreciable concentration of paramagnetic impurities, which we discuss in the next section.
+To form the nuclear spin bath, we generate random configurations of nuclear spins by allowing each W atom to
+be an 183W isotope (I = 1/2) with probability 14.3%. Under the secular approximation for the electron spin, this
+bath can be described by the following Hamiltonian in the rotating frame of the Er3+ spin:
+H = 2Sz
+�
+i
+(A(i)
+|| Ii
+z + A(i)
+⊥ Ii
+x) +
+�
+i
+ωL,W Ii
+z +
+�
+i,j
+H(i,j)
+nn ,
+(21)
+where Ii
+z/x are the nuclear spin operators of the ith spin, A(i)
+||
+and A(i)
+⊥ are the parallel and perpendicular hyperfine
+interaction terms, ωL,W is the Larmor frequency of the W nuclear spin (107.7 kHz at |B| = 600 G) and H(i,j)
+nn
+is the
+dipolar interaction Hamiltonian between W nuclear spins.
+
+9
+10
+2
+10
+3
+|B| (G)
+10
+−1
+10
+0
+10
+1
+10
+2
+Spin T1 (s)
+FIG. S4. Spin lifetime as a function of magnetic field strength. Solid line is a fit to Eq.(20) as is predicted for the
+spin-lattice relaxation time.
+This implies that there can be considerable variation in ESEEM (Electron Spin Echo Envelope Modulation) features
+observed for an Er3+ ion, depending on whether a nearby W nuclear spin is present. We observe such features as
+dips in coherence in the Hahn experiment (Fig. 4d). In contrast to decay envelopes, these sharp features occur at
+particular pulse spacings and indicate coherent coupling to a W nuclear spin occupying one of the nearest sites.
+Since we are working under the assumption that there is only one nearby W nuclear spin, we do not allow another
+nuclear spin within the first ten nearest W sites when generating the random nuclear spin baths. We find hyperfine
+parameters of the strongly coupled W nuclear spin by minimizing over the following cost function:
+C(A||, A⊥) =
+�
+k
+�
+j
+(Ssim,k(τj) − Sexp(τj))2;
+Ssim,k(τj) = e−(2τj/T2)n · Sbath,k(τj) · S(τj, A||, A⊥).
+(22)
+Here, τj are the time units that were sampled in the Hahn experiment, Ssim,k(τj) is the simulated signal for the
+kth bath and Sexp(τj) is the experimentally measured contrast for the experiment. Ssim,k(τj) is calculated by taking
+a product of three factors: a stretched exponential decay, with decay constants T2 = 44 µs and n = 1.4 used in
+Fig. 4d, the CCE simulation for the nuclear spin bath, Sbath,k(τj), and simulation for a single W nuclear spin coupled
+to Er3+, S(τj, A||, A⊥). The form of Ssim,k(τj) is justified as a first order CCE expansion, which does not take
+interactions between constituents of the bath into account. The envelope e−(2τj/T2)n is assumed to come from a
+source independent of the nuclear spin bath.
+The minimization yields the hyperfine parameters, (A||, A⊥) = (25.2, 31.7) kHz. These values are within range of
+expected interaction strengths for nearby W nuclear spin. In particular, for the W nuclear spin coordinates given as
+rW = ±a/2 + c/2, where a = aˆx and c = cˆz are CaWO4 lattice vectors, we calculate (A||, A⊥) = (15.5, 30.5) kHz at
+our field orientation. The discrepancy could be due to uncertainty in the field alignment or the Er3+ spin g-tensor,
+which can lead to rotations as discussed in Sec. VIII.
+Finally, ignoring the contribution from the phenomenological decay, we simulate longer time delays to extract the
+W bath limited coherence times for the Hahn and Ramsey experiments. We perform a second order CCE simulation
+for the simulation of the Hahn experiment, which takes into account the dipolar coupling between nuclear spins.
+Noting that ESEEM features will persist as observed in Fig. 4d, we find that the interaction between nuclear spin
+pairs leads to an envelope which decays in 22.6 ms (Fig. S5a). We also perform a Ramsey simulation and show
+that the expected T ∗
+2 decoherence due to the W-bath is about 4 µs (Fig. S5b). Both of these coherence values are
+significantly longer than the observed values for T2 and T ∗
+2 . We attribute the difference to paramagnetic impurities
+and explore the expected concentration in the next section.
+
+10
+a
+b
+|  g
+Free evolution time 2τ (µs)
+ 
+ 
+Population 
+0
+5
+10
+15
+20
+25
+30
+0
+0.5
+1
+Population |  g
+ 
+Free evolution time τ (µs)
+0
+10
+20
+30
+40
+50
+60
+70
+80
+0
+0.5
+1
+FIG. S5. W bath limited coherence. a The second order contribution to the CCE simulation for the Hahn experiment
+for 10 random W-bath configurations, where we assume that the nearest W nuclear spin is located at rW. Fitting each of
+the curves to a stretched exponential yields T2 = 22.7(4) ms with n = 2.7(1). We note that this is only the envelope and
+faster ESEEM features, obtained from the first order CCE simulation, persist as seen in Fig. 4d. This simulation considers
+W nuclear spins within an 11 nm radius of the Er3+ spin. b CCE simulation of Ramsey experiment for the same W bath
+configurations. Fitting each of the curves to a Gaussian decay yields T ∗
+2 = 4.0(4) µs. Both simulations are performed at our
+experimental field configuration.
+XI.
+ESTIMATING CONCENTRATION OF PARAMAGNETIC IMPURITIES
+In order to estimate the concentration of paramagnetic impurities, we use the Ramsey experiment as a probe of
+the static magnetic noise experienced by the Er3+ ion. As stated in the previous sections, the W bath limited T ∗
+2
+(Fig. S5b) is significantly longer than the measured T ∗
+2 of 247 ns. This indicates that the measured T ∗
+2 is limited
+by paramagnetic impurities. Therefore, we can use the measured T ∗
+2 to roughly estimate the concentration of these
+impurities.
+Without loss of generality, we can assume that the interaction between the Er3+ spin and the paramagnetic impurity
+will be of the Ising form under the secular approximation, forbidding exchanges that do not conserve magnetization.
+Under these assumptions, we can write down the Hamiltonian concerning the Er3+ spin and the paramagnetic bath
+in the frame rotating at Er3+ and the impurity frequency for a single impurity species
+H = Sz
+�
+i
+J(i)
+I Si
+z +
+�
+i,j
+J(i,j)
+I
+Si
+zSj
+z + J(i,j)
+S,k (Si
+xSj
+x + Si
+ySj
+y) +
+�
+i
+∆iSi
+z.
+(23)
+Here, J(i)
+I
+is the Ising interaction strength between the Er3+ spin and the impurity, while J(i,j)
+I
+and J(i,j)
+S
+are the Ising
+and Symmetric interaction strengths between the two impurities and ∆i is the disorder in the precession frequency
+of each impurity. While the bath interaction and disorder terms become important for decoupling sequences, these
+processes do not contribute on the timescale of the Ramsey experiment, which is dominated by the static noise
+represented by the first term. Based on this understanding, we can compute the net frequency, in the rotating frame,
+of the Er3+ spin given the state of the bath. For the purposes of this calculation, we take this bath to consist of
+S = 1/2 electrons with g = 2 and represent its state by the bitstring k of length N. We can then write down the
+Hamiltonian projected by the state k of the bath
+Hk = ⟨k|Sz
+�
+i
+J(i)
+I Si
+z|k⟩ = Sz
+�
+i
+J(i)
+I
+(−1)ki
+2
+= ωkSz;
+ωk =
+�
+i
+(−1)kiJ(i)
+I
+2
+.
+(24)
+Here, ωk is the precession frequency of the Er3+ spin given the state k of the bath. We can then calculate the Ramsey
+
+11
+coherence as T ∗
+2 = π/∆ω, where ∆ω2 is the variance over the set {ωk}:
+∆ω2 = 1
+2N
+�
+k
+ω2
+k = 1
+2N
+�
+k
+�
+i
+�(−1)kiJ(i)
+I
+2
+�2
++ 1
+2N
+�
+k
+�
+i<j
+(−1)ki+kjJ(i)
+I J(j)
+I
+=
+�
+i
+�J(i)
+I
+2
+�2
+,
+(25)
+where we have used the fact that the distribution is centered at zero and the summations can be reordered. We
+observe that the frequency standard deviation can be interpreted as the norm of the vector of Ising interaction
+strengths.
+In order to estimate the concentration based on this expression, we generate 2000 instances of randomly distributed
+electron spin baths across a range of concentrations, calculate the resultant T ∗
+2 of each configuration and use Bayes
+rule to infer a probability density, P(ρ|T ∗
+2 ), for the concentration, ρ, given our observation of T ∗
+2
+P(ρ|T ∗
+2 ) =
+P(T ∗
+2 |ρ)
+� ∞
+0
+P(T ∗
+2 |ρ′)dρ′ .
+(26)
+Here, P(T ∗
+2 |ρ′) is the probability to obtain a given T ∗
+2 for the bath concentration ρ. In practice, we calculate this
+expression by counting the number of instances for each concentration that yields a value of T ∗
+2 within 3 standard
+deviations of the observed value. We perform this estimate for both a 3D geometry of electron spins located in the
+bulk and a 2D geometry of spins located on the sample surface, estimated to be 10 nm away. We obtain concentrations
+of 1.6 − 5.7 × 1016 cm−3 and 0.5 − 1.3 nm−2 for the bulk and surface concentration estimates respectively with 70%
+confidence (Fig. S6). As both of these are plausible concentrations, we note that both may be playing a non-negligible
+role. We are not able to make a calculation of how this concentration of paramagnetic impurities affects the Hahn
+coherence because it depends on the dynamics of this paramagnetic impurity bath, which is not well known. Doing
+this calculation would require further information such as the species and spin-lifetime of impurities, as well as the
+disorder of the bath.
+a
+b
+0
+5
+10
+15
+20
+0.0
+0.1
+0.2
+Prob. density (10-16 cm3)
+0.0
+0.5
+1.0
+1.5
+0.0
+1.0
+2.0
+3.0
+Prob. density (nm2)
+ρ (1016 cm-3)
+ρΑ (nm-2)
+FIG. S6. Probability density of paramagnetic impurity concentration. a Assuming a 3D uniform distribution of
+impurities, we estimate that the bath concentration is in the range 1.6×1016 – 5.7×1016 cm−3 with 70% confidence, with the
+likeliest concentration at 3.7×1016 cm−3. b Assuming a 2D distribution on the surface of our crystal, assumed to be located
+10 nm away from the Er3+ spin, we estimate an area concentration in the range of 0.5 – 1.3 nm−2 with 70% confidence, with
+the likeliest concentration at 0.77 nm−2. Dashed lines indicate the confidence ranges for the impurity concentrations.
+
+12
+∗ These authors contributed equally to this work.
+† jdthompson@princeton.edu
+[1] Dibos, A. M., Raha, M., Phenicie, C. M. & Thompson, J. D. Atomic Source of Single Photons in the Telecom Band.
+Physical Review Letters 120, 243601 (2018).
+[2] Ziegler, J. F., Ziegler, M. D. & Biersack, J. P.
+SRIM – The stopping and range of ions in matter (2010).
+Nuclear
+Instruments and Methods in Physics Research Section B: Beam Interactions with Materials and Atoms 268, 1818–1823
+(2010).
+[3] Chen, S. et al. Hybrid microwave-optical scanning probe for addressing solid-state spins in nanophotonic cavities. Optics
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+[4] Enrique, B. G. Optical spectrum and magnetic properties of Er3+ in CaWO4. The Journal of Chemical Physics 55,
+2538–2549 (1971).
+[5] Chen, S., Raha, M., Phenicie, C. M., Ourari, S. & Thompson, J. D. Parallel single-shot measurement and coherent control
+of solid-state spins below the diffraction limit. Science 370, 592–595 (2020).
+[6] Raha, M. et al. Optical quantum nondemolition measurement of a single rare earth ion qubit. Nature Communications
+11, 1605 (2020).
+[7] Kambs, B. & Becher, C. Limitations on the indistinguishability of photons from remote solid state sources. New Journal
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+[8] Kuhlmann, A. V. et al. Transform-limited single photons from a single quantum dot. Nature Communications 6, 8204
+(2015).
+[9] Loredo, J. C. et al. Scalable performance in solid-state single-photon sources. Optica 3, 433 (2016).
+[10] Carnall, W. T., Goodman, G. L., Rajnak, K. & Rana, R. S. A systematic analysis of the spectra of the lanthanides doped
+into single crystal LaF3. The Journal of Chemical Physics 90, 3443–3457 (1989).
+[11] Wybourne, B. G. Spectroscopic properties of rare earths (Interscience Publishers, 1965).
+[12] Newman, D. Theory of lanthanide crystal fields. Advances in Physics 20, 197–256 (1971).
+[13] Messiah, A. Quantum mechanics (Dover Publications, 1961).
+[14] Abragam, A. & Bleaney, B. Electron Paramagnetic Resonance of Transition Ions (OUP Oxford, 1970).
+[15] LeDantec, M. et al. Twenty-three–millisecond electron spin coherence of erbium ions in a natural-abundance crystal.
+Science Advances 7, eabj9786 (2021).
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+Physical Review B 79, 115320 (2009).
+

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+Pylon: Table Union Search through Contrastive Representation
+Learning
+Tianji Cong
+University of Michigan
+Ann Arbor, Michigan, USA
+congtj@umich.edu
+H. V. Jagadish
+University of Michigan
+Ann Arbor, Michigan, USA
+jag@umich.edu
+ABSTRACT
+The large size and fast growth of data repositories, such as data
+lakes, has spurred the need for data discovery to help analysts find
+related data. The problem has become challenging as (i) a user
+typically does not know what datasets exist in an enormous data
+repository; and (ii) there is usually a lack of a unified data model to
+capture the interrelationships between heterogeneous datasets from
+disparate sources. The common practice in production is to provide
+a keyword search interface over the metadata of datasets but users
+often have discovery needs that cannot be precisely expressed by
+keywords. In this work, we address one important class of discovery
+needs: finding union-able tables.
+The task is to find tables in the repository (or on the web) that
+can be unioned with a given query table. The challenge is to rec-
+ognize union-able columns that may be represented differently. In
+this paper, we propose a data-driven learning approach: specifically,
+an unsupervised representation learning and embedding retrieval
+task. Our key idea is to exploit self-supervised contrastive learn-
+ing to learn an embedding model that produces close embeddings
+for columns with semantically similar values while pushing apart
+columns with semantically dissimilar values. We then find union-
+able tables based on similarities between their constituent columns
+in embedding space. On a real-world dataset, we demonstrate that
+our best-performing model achieves significant improvements in
+precision (16% ↑), recall (17% ↑), and query response time (7x faster)
+compared to the state-of-the-art.
+CCS CONCEPTS
+• Information systems → Information integration.
+KEYWORDS
+data discovery, data integration, table union search, contrastive
+learning, embeddings
+Permission to make digital or hard copies of all or part of this work for personal or
+classroom use is granted without fee provided that copies are not made or distributed
+for profit or commercial advantage and that copies bear this notice and the full citation
+on the first page. Copyrights for components of this work owned by others than ACM
+must be honored. Abstracting with credit is permitted. To copy otherwise, or republish,
+to post on servers or to redistribute to lists, requires prior specific permission and/or a
+fee. Request permissions from permissions@acm.org.
+Conference acronym ’XX, June 03–05, 2018, Woodstock, NY
+© 2018 Association for Computing Machinery.
+ACM ISBN 978-1-4503-XXXX-X/18/06...$15.00
+https://doi.org/XXXXXXX.XXXXXXX
+ACM Reference Format:
+Tianji Cong and H. V. Jagadish. 2018. Pylon: Table Union Search through
+Contrastive Representation Learning. In Proceedings of Make sure to en-
+ter the correct conference title from your rights confirmation emai (Confer-
+ence acronym ’XX). ACM, New York, NY, USA, 13 pages. https://doi.org/
+XXXXXXX.XXXXXXX
+1
+INTRODUCTION
+Recent years have witnessed a vast growth in the amount of data
+available to the public, particularly from data markets, open data
+portals, and data communities (e.g., Wikidata and Kaggle) [7]. To
+benefit from the many new opportunities for data analytics and
+data science, the user first usually has to find related datasets in
+a large repository (e.g., data lakes). The challenge for a system is
+to support users with varying discovery needs, without the help
+of a unified data model capturing the interrelationships between
+datasets.
+In response to the challenge, there are many ongoing efforts
+under the umbrella of data discovery. One task of the interest in
+data discovery is to find union-able tables [1, 5, 10, 27] with the
+aim of adding additional relevant rows to a user-provided table.
+Figure 1 shows an example of two tables union-able over four pairs
+of attributes. In general, the literature considers two tables union-
+able if they share attributes from the same domain and assumes the
+union-ability of two attributes can be implied by some notion of
+similarity. We refer to the problem of finding union-able tables as
+table union search (termed in [27]) in the rest of the paper.
+The typical solution path is to first identify union-able attributes
+(or columns in the tables) and then aggregate column-level results
+to obtain candidate union-able tables. To uncover the union-ability
+of attributes, both syntactic and semantic methods have been em-
+ployed in the literature. Syntactic methods are the easiest, and have
+been used the longest. While they are robust at catching small
+changes, such as capitalization or the use of a hyphen, they are
+unable to address the use of common synonyms. Semantic methods
+offer the possibility of finding union-able columns of semantically
+similar values despite their syntactic dissimilarity (e.g., the "venue"
+column and the "platform" column in figure 1). [10, 27] link cell val-
+ues to entity classes in an external ontology and compare similarity
+of entity sets. [1, 27] use off-the-shelf word embeddings to measure
+semantics. Both methods have notable limitations. [27] observed
+that only 13% of attribute values of their collected Open Data ta-
+bles can be mapped to entities in YAGO [32], one of the largest
+and publicly available ontologies. Although word embeddings can
+provide more semantic coverage of attributes, they are subject to
+the training text corpus and may not generalize well to textual data
+in tables [18, 23].
+arXiv:2301.04901v1  [cs.DB]  12 Jan 2023
+
+Conference acronym ’XX, June 03–05, 2018, Woodstock, NY
+Tianji Cong and H. V. Jagadish
+id
+title
+authors
+venue
+year
+671167
+A Database System for Real-Time Event Aggregat...
+Jerry Baulier, Stephen Blott, Henry F. Korth, ...
+Very Large Data Bases
+1998
+672964
+Integrating a Structured-Text Retrieval System...
+Tak W. Yan, Jurgen Annevelink
+Very Large Data Bases
+1994
+872823
+Evaluating probabilistic queries over imprecis...
+Reynold Cheng, Dmitri V. Kalashnikov, Sunil Pr...
+International Conference on Management of Data
+2003
+...
+...
+...
+...
+...
+Title
+Authors
+Platform
+Cited_url
+Cited_count
+Year
+Fg-index: towards verification-free query proc...
+J Cheng, Y Ke, W Ng, A Lu
+Proceedings of the 2007 ACM SIGMOD internation...
+https://scholar.google.com/scholar?oi=bibs&hl=...
+286
+2007
+Efficient query processing on graph databases
+J Cheng, Y Ke, W Ng
+ACM Transactions on Database Systems (TODS) 34...
+https://scholar.google.com/scholar?oi=bibs&hl=...
+83
+2009
+Context-aware object connection discovery in l...
+J Cheng, Y Ke, W Ng, JX Yu
+2009 IEEE 25th International Conference on Dat...
+https://scholar.google.com/scholar?oi=bibs&hl=...
+66
+2009
+...
+...
+...
+...
+...
+...
+Figure 1: An example of two tables union-able over four pairs of attributes: title - Title, authors - Authors, venue - Platform,
+and year - Year.
+Instead of relying on low-coverage ontologies or pre-trained
+word embeddings, a data-driven learning approach seems more
+promising to capture semantics as shown in many data manage-
+ment tasks such as entity resolution [24, 25], data cleaning [34], and
+table interpretation [11]. A large part of their success requires la-
+beled data for supervised learning. However, there is no large-scale
+labeled dataset for table union search. The only publicly available
+benchmark [27] with table- and column-level ground truth con-
+tains limited number of tables synthesized from only 32 base tables,
+which is far from being representative for training purposes. Ad-
+ditionally, labeling new datasets could be very laborious and time
+consuming as curators need to examine every pair of columns in
+every pair of tables in a collection. Even if the training data prob-
+lem were resolved, we would only be able to determine column
+matches pairwise. It would still be very inefficient to exhaustively
+consider every query column and every column in the corpus pair-
+wise to predict union-ability. In short, the inherent search nature
+of the problem makes it unsuitable to formulate it as a supervised
+classification problem.
+In this work, we overcome the aforementioned difficulties by
+casting table union search as an unsupervised representation learn-
+ing and embedding retrieval task. Our goal is to learn column-level
+embeddings into a high-dimensional feature space. Locality search
+in this feature space can then directly be used for union-able table
+search. To achieve this goal, our key idea is to exploit self-supervised
+contrastive learning to learn an embedding model that produces
+close embeddings for columns with semantically similar values
+and pushes away columns with semantically dissimilar values. We
+propose Pylon, a novel contrastive learning framework that learns
+column representations for tabular data and serves the table union
+search problem without relying on labeled data.
+There are two main challenges in the development of Pylon,
+specifically, on how to adapt contrastive learning for tabular data.
+(1) How to create training data without human labeling? The
+self-supervised contrastive learning technique requires con-
+structing positive and negative examples from the data itself.
+In the field of computer vision where contrastive learning
+first took off, [9] applies a series of random data augmen-
+tation of crop, flip, color jitter, and grayscale to generate
+stochastic views of an image. These views preserve the se-
+mantic class label of the image and so make positive exam-
+ples for training. They further consider any two views not
+from the same image as a negative example. However, the
+tabular data modality is dramatically different from images
+and it remains unclear how to create different views of tables
+while keeping the semantics.
+(2) What is a feasible feature encoder for tabular data? Another
+key component in contrastive learning is a pre-trained base
+encoder that gives initial embeddings for raw data. Both
+computer vision (CV) and natural language processing (NLP)
+communities have widely recognized models for feature ex-
+tractions (e.g., ResNet [19] in CV and BERT [12] in NLP).
+On the contrary, there is no generally accepted feature ex-
+traction model for tables despite the recent progress in Web
+table modeling (which we discuss in subsection 2.2).
+In summary, we make the following contributions:
+• We formulate semantic table union search as an unsuper-
+vised representation learning and embedding retrieval prob-
+lem, and propose to use self-supervised contrastive learning
+to avoid the labeling issue.
+• We present Pylon, to the best of our knowledge, the first con-
+trastive learning framework for learning semantic column
+representations from large collections of tables. We also ex-
+plore the design of each component in contrastive learning
+and take an initiative in adapting contrastive learning for
+tabular data.
+• We empirically show that our embedding approach is both
+more effective and efficient than existing embedding meth-
+ods on a self-curated real-world dataset and a synthetic pub-
+lic benchmark. On the real-world dataset, two of our model
+variants outperform their corresponding baseline version by
+14% and 6% respectively on both precision and recall. We
+also observe that they speed up the query response time by
+2.7x and 9x respectively. We (plan to) open-source the new
+benchmark for future research study.
+• We demonstrate that our embedding approach can be further
+augmented by syntactic measures and that our best ensemble
+model has significant advantages over the state-of-the-art
+(namely, 𝐷3𝐿 [1]), more than 15% improvement in precision
+and recall, and 7x faster in query response time.
+We give a formal problem setup and background about embed-
+ding models in Section 2. We describe our framework Pylon includ-
+ing embedding training and search in Section 3. Section 4 reports
+experiments that validate our approach. We discuss related work
+in Section 5 and conclude in Section 6.
+
+Pylon: Table Union Search through Contrastive Representation Learning
+Conference acronym ’XX, June 03–05, 2018, Woodstock, NY
+2
+PROBLEM DEFINITION & BACKGROUND
+In this section, we start by describing the formal problem setup
+in section 2.1, and then provide an overview of existing embed-
+ding models for tabular data. We also discuss the challenges of
+representation learning for table union search.
+2.1
+Table Union Search
+The table union search problem [27] is motivated by the need to
+augment a (target) table at hand with additional data from other
+tables containing similar information. For example, starting with
+a table about traffic accidents in one state for a particular year, an
+analyst may wish to find similar traffic accident data for other states
+and years. Ideally, these tables would have the same schema (e.g.
+data from the same state agency for two different years) so that
+we could simply union the row-sets. However, this is typically not
+the case for data recorded independently (e.g. data from different
+states). We consider two tables union-able if they share attributes
+from the same domain. Also, as in prior work on this topic, we
+assume the union-ability of attributes can be quantified by some
+notion of similarity.
+Definition 1 (Attribute Union-ability). Given two attributes 𝐴 and
+𝐵, the attribute union-ability U𝑎𝑡𝑡𝑟 (𝐴, 𝐵) is defined as
+U𝑎𝑡𝑡𝑟 (𝐴, 𝐵) = M(T (𝐴), T (𝐵))
+where T (·) is a feature extraction technique that transforms raw
+columns (attribute names, attribute values, or both) to a feature
+space and M(·, ·) is a similarity measure between two instances in
+the feature space.
+With the definition of attribute union-ability, we can define table
+uniona-bility as a bipartite graph matching problem where the
+disjoint sets of vertices are attributes of the target table and the
+source table respectively, and edges can be defined by attribute
+union-ability. In this paper, we restrict ourselves to the class of
+greedy solutions. Therefore, we formalize the definition table union-
+ability as a greedy matching problem as follows:
+Definition 2 (Union-able Tables). A source table 𝑆 with attributes
+B = {𝐵𝑗 }𝑛
+𝑗=1 is union-able to a target table 𝑇 with attributes A =
+{𝐴𝑖}𝑚
+𝑖=1 if there exists a one-to-one mapping 𝑔 : A′(≠ ∅) ⊆ A →
+B′ ⊆ B such that
+(1) |A′| = |B′|;
+(2) ∀𝐴𝑖 ∈ A′, U𝑎𝑡𝑡𝑟 (𝐴𝑖,𝑔(𝐴𝑖)) ≥ 𝜏 where
+𝑔(𝐴𝑖) = arg max
+𝐵𝑗
+{U𝑎𝑡𝑡𝑟 (𝐴𝑖, 𝐵𝑗) : 1 ≤ 𝑗 ≤ 𝑛}
+and 𝜏 is a pre-defined similarity threshold.
+Definition 3 (Table Union-ability). Following notations in Defini-
+tion 2, the table union-ability U(𝑆,𝑇) is defined as
+U(𝑆,𝑇) =
+�𝑙
+𝑖=1 𝑤𝑖 · U𝑎𝑡𝑡𝑟 (𝐴𝑖,𝑔(𝐴𝑖))
+�𝑙
+𝑖=1 𝑤𝑖
+where 𝑙 is the number of union-able attribute pairs between the
+target table 𝑇 and a source table 𝑆, and 𝑤𝑖 weights the contribution
+of the attribute pair (𝐴𝑖,𝑔(𝐴𝑖)) to the table union-ability.
+Considering the scale of the dataset repository, we also follow
+the common practice[1, 10, 27] of performing top-𝑘 search. The
+table union search problem is formally defined as below.
+Definition 4 (Top-𝑘 Table Union Search). Given a table corpus
+S, a target table 𝑇, and a constant 𝑘, find up to 𝑘 candidate tables
+𝑆1,𝑆2, ...,𝑆𝑘 ∈ S in descending order of table union-ability with
+respect to the query table 𝑇 such that 𝑆1,𝑆2, ...,𝑆𝑘 are most likely
+to be union-able with 𝑇.
+2.2
+Embedding Models for Tabular Data
+The advance of language modeling in the field of NLP has sparked
+its adoption in many applications of data management such as
+semantic queries in relational databases [3, 17], entity resolution [24,
+25], and data cleaning [34]. We give an (non-exhaustive) overview
+of embedding models that have been used for tabular data.
+Word Embeddings. Word embeddings are vector representa-
+tions of words in a low-dimension space where words that share the
+common context are close to each other. Most popular word embed-
+ding models include Word2Vec [26], GloVe [30], and fastText [2].
+Unlike Word2Vec and GloVe that learn embeddings directly for
+words, fastText represents a word as an n-gram of characters (i.e.,
+subwords) and generate word embeddings based on subwords. In
+this way, fastText is able to handle out-of-vocabulary words that
+do not appear in the training corpus. In the context of table union
+search, both [27] and [1] employed off-the-shelf fastText embed-
+dings to measure the semantic relatedness between two columns,
+which implies union-ability. One issue with pre-trained word em-
+beddings is that the text value distribution in tables is different
+from what models capture in the training corpus consisting of un-
+structured documents. To generate embeddings for textual values
+in tables, [18] serialized tables to sequences of tokens and trained
+a fastText model on a text corpus extracted from Web tables. The
+resulting web table embedding models demonstrated better perfor-
+mance as compared to off-the-shelf fastText embeddings in a task
+of ranking union-able columns pairs.
+Transformer-based Language Models. Another well-known
+issue with word embedding models is that they assign a fixed em-
+bedding for each word regardless of various meanings a word could
+have and different linguistic contexts in which a word could appear.
+This (partly) motivates the development of contextual language
+models (LMs) such as BERT [12]. The underlying Transformer ar-
+chitecture empowered by the attention mechanism [35] enables
+LMs to represent any word relative to all other words in the context
+(i.e., surrounding words in the same sentence). Note that these LMs
+are first pre-trained on a large text corpus with a general-purpose
+objective (e.g., masked language model (MLM), which predicts ran-
+domly masked words based on their contexts) and fine-tuned with
+supervision (labeled data) for downstream tasks. This pre-training
+and fine-tuning paradigm has become the de facto practice in many
+NLP tasks.
+The tremendous success of Transformer-based LMs has inspired
+their counterparts on tabular data. TAPAS [20] extended the BERT
+architecture to answer questions over tables by pre-training the
+model on text-table pairs using the MLM objective and fine-tuning
+it on downstream task datasets in a weakly supervised manner.
+In addition to MLM, TURL [11] proposed a new Masked Entity
+
+Conference acronym ’XX, June 03–05, 2018, Woodstock, NY
+Tianji Cong and H. V. Jagadish
+Recovery objective for pre-training on entity-rich relational tables
+from Web. Their pre-trained contextualized representations were
+shown to generalize well to six downstream table understanding
+tasks with fine-tuning. TaBERT [38] jointly learned semi-structured
+Web tables and their surrounding texts in pre-training and was fine-
+tuned for semantic parsing tasks. TUTA [36] devised a tree-based
+Transformer and expanded pre-training to generally structured
+tables including entity and matrix tables, and spreadsheet tables.
+They fine-tuned the model for cell-type classification and table-type
+classification tasks.
+2.3
+Challenges
+Representation learning for tables has achieved excellent results
+for many table-centric tasks. We hypothesize that the table union
+search problem can also benefit from advances in table modeling.
+However, several challenges remain to be addressed.
+(1) To the best of our knowledge, no prior work has taken the
+learning approach for table union search. We argue that this
+is mainly because neither the supervised learning setting nor
+the popular pre-training and fine-tuning paradigm is directly
+applicable for the problem. It is inefficient to formulate the
+underlying search of union-able columns as a classification
+problem. In a supervised learning setting, one can attempt to
+train a classifier to predict whether two columns are union-
+able, but it will quickly become computationally prohibitive
+in the search phase to classify every pair of target column
+in a query table with every column in a large corpus.
+(2) The scarcity of table union search datasets is another severe
+bottleneck of applying a learning approach and studying
+the problem in general. The only publicly available bench-
+mark [27] with table- and column-level ground truth is syn-
+thesized from only 32 base tables, which is barely enough for
+evaluation. It is also very laborious and time consuming to
+label such datasets, as curators need to examine every pair
+of columns for every pair of tables in a collection.
+(3) How to encode non-Web tables? Transformer-based mod-
+els surveyed above are primarily designed for Web tables
+and assume access to abundant metadata such as table cap-
+tions, surrounding text, and topic entities. In contrast, non-
+Web tables like Open Data tables and tables extracted from
+GitHub [21] do not have such information available in gen-
+eral. We can reasonably assume access to data values and
+table headers but not much more. Even informative schema
+is not always available.
+In the next section, we present our design that contributes a
+representation learning approach to table union search while effec-
+tively mitigating the challenges we point out here.
+3
+PYLON: A SELF-SUPERVISED
+CONTRASTIVE LEARNING FRAMEWORK
+FOR TABULAR DATA
+Our key idea is to leverage self-supervised contrastive learning
+that provides a feasible training objective for learning effective
+column representations for the table union search problem while
+not requiring any labeled data (corresp. to challenge 1 and 2). Within
+the framework of contrastive learning, we propose two strategies
+that arithmetically construct training data from unlabeled data to
+tackle challenge 2. We also experiment with several encoders to
+gain empirical insights into challenge 3.
+3.1
+Contrastive Learning
+The high-level goal of contrastive learning is to learn to distin-
+guish (so called "contrast") between pairs of similar and dissimilar
+instances. Ideally, in the learned representation space, similar in-
+stances stay close to each other whereas dissimilar ones are pushed
+far away. A pair of instances is considered similar and labeled a
+positive example in training if it comprises different views of the
+same object; otherwise, they are considered dissimilar and make a
+negative example. Contrastive learning has been used extensively
+in computer vision [9], where a positive example consists of a pair
+of augmented images transformed from the same image (e.g., by
+applying cropping or color distortion).
+We introduce, Pylon, our self-supervised contrastive learning
+framework for learning representations from large collections of
+tables. As table union search begins by finding union-able columns,
+Pylon is designed to generate a vector representation for each
+column of input tables where columns containing semantically
+similar values have embeddings closer to one another.
+3.2
+Pylon Workflow
+Figure 2 shows the training workflow of the framework that consists
+of the following major components.
+Training Data Construction. Without labeled data, the suc-
+cess of contrastive learning hinges on the construction of positive
+and negative examples from the data itself. To make positive ex-
+amples, it requires an operation to transform a data instance in a
+way that introduces variations while preserving the semantics. As
+table union search builds on union-able column search, we propose
+two strategies to construct positive and negative examples at the
+column level.
+(1) Online sampling strategy. Consider a training batch of 𝑁
+tables {𝑇𝑖}𝑁
+𝑖=1 where each table 𝑇𝑖 has 𝑚𝑖 columns {𝐶𝑖
+𝑗 }𝑚𝑖
+𝑗=1,
+giving 𝑀 = �𝑁
+𝑖=1 𝑚𝑖 columns in total. We obtain a positive
+example of column pairs (𝑥𝑘,𝑥𝑘+𝑀) (1 ≤ 𝑘 ≤ 𝑀) by ran-
+domly sampling values from each column 𝐶𝑖
+𝑗 of each table𝑇𝑖.
+Since both 𝑥𝑘 and 𝑥𝑘+𝑀 are samples from the same column
+of the same table, we consider they share semantics and
+make a positive example. The sampling process yields 2𝑀
+column instances, and we treat the other 2(𝑀 − 1) samples
+as negatives with respect to 𝑥𝑘. In other words, considering
+(𝑥𝑘,𝑥𝑘+𝑀) and (𝑥𝑘+𝑀,𝑥𝑘) as distinct positive examples, we
+construct 2𝑀 positive examples and 2𝑀(𝑀 − 1) negative
+examples from each training batch.
+(2) Offline approximate matching strategy. An alternative is to
+construct positive examples ahead of training. Instead of
+relying on ad-hoc sampling, we can leverage existing ap-
+proaches to find a union-able candidate for each column,
+which in turn makes positive examples in training. Based
+on the observation that top-𝑘 union-able column search of
+existing techniques is highly precise when 𝑘 is small (e.g.,
+
+Pylon: Table Union Search through Contrastive Representation Learning
+Conference acronym ’XX, June 03–05, 2018, Woodstock, NY
+r1
+r2
+r3
+r4
+r5
+r6
+Online Training
+Data 
+Construction
+Base Encoder f & 
+Projection Head g 
+f
+g
+f
+g
+e1
+e2
+e3
+e1+M
+e2+M
+e3+M
+Projected column embeddings 
+Table Samples
+c1 c2 c3
+r1
+r2
+r5
+r2
+r3
+r6
+Figure 2: Training workflow of Pylon (with online training data construction).
+𝑘 = 1), we are able to use this approximate matching without
+human involvement. We find that it produces valid results
+and does not suffer the issue of false positives.
+Base Encoder & Projection Head. We pass column instances
+{𝑥𝑘}2𝑀
+𝑘=1 through a base encoder 𝑓 (·) to get initial column embed-
+dings {𝑒𝑘}2𝑀
+𝑘=1. Note that our contrastive learning framework is
+flexible about the choice of the base encoder. The encoder can give
+embeddings at token/cell/column level, and if necessary, we can
+apply aggregation (e.g, average or max) to obtain column-level
+embeddings. Our framework has the flexibility to benefit from the
+advance of modeling techniques in NLP over time without being
+tied to a specific model. We describe the choices of 𝑓 (·) we experi-
+ment with in subsection 3.3.
+Following the encoder, a small multi-layer neural network 𝑔(·),
+called projection head, maps the representations from the encoder
+to a latent space through linear transformations and non-linear
+activation in between. Note that unlike the practice in CV which
+discards projection head in inference and uses encoder outputs for
+downstream tasks, we preserve projection head and use projected
+embeddings for table union search. This is because we found pro-
+jected embeddings yield better performance in initial experiments,
+and for encoders like word embedding models, only projection head
+is trainable and has to be preserved for inference. For simplicity,
+we keep using the notations {𝑒𝑘}2𝑀
+𝑘=1 for projection outputs.
+Contrastive Loss. One common setting of contrastive learning
+defines a prediction task of identifying positive examples from the
+training batch. Given embedded columns {𝑒𝑘}2𝑀
+𝑘=1, the model learns
+to predict 𝑒𝑘+𝑀 as the most similar one to 𝑒𝑘 and vice versa for
+each 𝑒𝑘 (1 ≤ 𝑘 ≤ 𝑀). The similarity between any two instances 𝑒𝑖
+and 𝑒𝑗 is measured by their cosine similarity as
+𝑠𝑖𝑚(𝑖, 𝑗) =
+𝑒𝑇
+𝑖 𝑒𝑗
+∥𝑒𝑖 ∥∥𝑒𝑗 ∥
+and the loss is calculated as
+𝑙(𝑘,𝑘 + 𝑀) = − log
+exp (𝑠𝑖𝑚(𝑘,𝑘 + 𝑀) / 𝜏)
+�2𝑀
+𝑙=1,𝑙≠𝑘 exp (𝑠𝑖𝑚(𝑘,𝑙) / 𝜏)
+where 𝜏 > 0 is a scaling hyper-parameter called temperature. Mini-
+mizing 𝑙(𝑘,𝑘 + 𝑀) is equivalent to maximizing the probability of
+𝑒𝑘+𝑀 being the most similar to 𝑒𝑘 among all the embedded columns
+except 𝑒𝑘 itself.
+Finally, the loss over all the 2𝑀 positive column pairs in a training
+batch is computed as
+𝐿 =
+1
+2𝑀
+𝑀
+∑︁
+𝑘=1
+[𝑙(𝑘,𝑘 + 𝑀),𝑙(𝑘 + 𝑀,𝑘)]
+This loss formulation is called InfoNCE loss [28] (also known as
+the normalized temperature-scaled cross entropy loss [9]), which
+approximately maximizes the mutual information (i.e., a measure
+of how dependent two random variables are to each other) between
+two views of the same object.
+3.3
+Choices of the Base Encoder
+Although we expect the input to the contrastive loss function to be
+column embeddings, the base encoder does not necessarily need to
+give column embeddings directly. It is possible for the encoder
+model to generate embeddings at different granularity (i.e., to-
+ken/cell/column) because we can apply aggregation if necessary.
+We describe the basic encoding process of embedding models we
+experimented with in section 4.
+Word Embedding Models (WEM). As a WEM assigns a fix
+representation to a token, WEM-based encoders treat each column
+independently as a document where a standard text parser tokenizes
+data values in a column. With a fastText embedding model, we first
+get cell embeddings by averaging token embeddings in each cell and
+then aggregate cell embeddings to get a column embedding. More
+
+Conference acronym ’XX, June 03–05, 2018, Woodstock, NY
+Tianji Cong and H. V. Jagadish
+interestingly, web table embedding models [18] consider each cell as
+a single token (they concatenate tokens in a cell with underscores)
+and output embeddings at cell level. Nevertheless, we aggregate
+cell embeddings to derive the column embedding.
+Language Models (LM). Since a table is a cohesive structure
+for storing data, considering values in neighboring columns could
+integrate context into the embeddings and help mitigate ambiguity
+in unionable column search. For example, encoding column "year"
+in figure 1 individually loses the context that this column refers to
+the publication year of research papers. In this case, the embeddings
+of "Year" columns in the corpus are less distinguishable (in terms of
+cosine similarity) even though they may refer to different concepts
+of year such as the birth year of people or the release year of movies.
+With context provided by other columns like "Title" and "Venue", it
+is more likely that "Year" columns appearing in tables about papers
+are more close to each other than "Year" columns in tables about
+other topics, which helps find more related tables.
+We leverage LMs to derive contextual column embeddings. We
+first serialize each row in 𝑇𝑖 as a sequence by concatenating tok-
+enized cell values. For example, the first row of the table at the top
+in Figure 1 will be encoded as follows
+[𝐶𝐿𝑆] title | A Database ... [SEP] authors | Jerry... [SEP] ...[𝐸𝑁𝐷]
+The sequence is annotated with special tokens in the LM where
+[𝐶𝐿𝑆] token indicates the beginning of the sequence, [𝐸𝑁𝐷] token
+indicates the end, and [𝑆𝐸𝑃] tokens separate cell values in different
+columns. Then the LM takes in each sequence and generates a con-
+textual representation for each token in the sequence (essentially
+taking into account the relation between values in the same row).
+We apply mean pooling to tokens in the same cell and get cell em-
+beddings. To consider the relation of values in the same column, we
+adopt the vertical attention mechanism in [38] to have weighted
+column embeddings by attending to all of the sampled cells in the
+same column.
+Word embedding models have previously been used to find
+union-able tables. Two state-of-the-art choices are fastText and
+WTE (web table embeddings [18]). Language models have not thus
+far been used for the union-ability problem. BERT[12] is a lead-
+ing language model used for many purposes today. We develop
+three versions of Pylon, one for each of these three encoder choices:
+fastText, WTE, and a BERT-based language model, and refer to the
+derived models as Pylon-fastText, Pylon-WTE, Pylon-LM respectively.
+We evaluate the effect of encoder choices in subsection 4.5.
+3.4
+Embedding Indexing and Search
+To avoid exhaustive comparisons of column embeddings over a
+large corpus at query time, we use locality-sensitive hashing (LSH) [22]
+for approximate nearest neighbor search and treat union-able col-
+umn search as an LSH-index lookup task [1, 27]. LSH utilizes a
+family of hash functions that maximize collisions for similar inputs.
+The result of LSH indexing is that similar inputs produce the same
+hash value and are bucketed together whereas dissimilar inputs are
+ideally placed in different buckets. Algorithm 1 gives the indexing
+procedure. For approximate search with respect to the cosine simi-
+larity, we index all column embeddings in a random projection LSH
+index [8]. The idea of random projection is to separate data points
+Algorithm 1: Embedding Inference and Indexing
+Input
+:
+S, a corpus of tables;
+𝑔 ◦ 𝑓 , a Pylon model;
+𝑝, a list of LSH index parameters.
+Output:
+I, a random projection LSH index.
+1 I ← create_index(𝑝);
+2 for 𝑡 ∈ S do
+3
+𝑡_𝑠𝑒𝑟𝑖𝑎𝑙𝑖𝑧𝑒𝑑 ← preprocess(𝑡);
+4
+𝑐𝑜𝑙𝑢𝑚𝑛_𝑒𝑚𝑏𝑒𝑑𝑑𝑖𝑛𝑔𝑠 ← 𝑔 ◦ 𝑓 (𝑡_𝑠𝑒𝑟𝑖𝑎𝑙𝑖𝑧𝑒𝑑);
+5
+for 𝑒 ∈ 𝑐𝑜𝑙𝑢𝑚𝑛_𝑒𝑚𝑏𝑒𝑑𝑑𝑖𝑛𝑔𝑠 do
+6
+I.insert(𝑒);
+7
+end
+8 end
+9 return I;
+Algorithm 2: Top-𝑘 Table Union Search
+Input
+:
+I, a LSH index;
+𝑄, a query table;
+𝑘, a constant.
+Output:
+top-𝑘 union-able tables.
+1 𝑐𝑜𝑙𝑢𝑚𝑛_𝑐𝑎𝑛𝑑𝑖𝑑𝑎𝑡𝑒𝑠 ← {};
+2 𝑐𝑜𝑙𝑢𝑚𝑛_𝑠𝑐𝑜𝑟𝑒𝑠 ← {};
+3 for 𝑐 ∈ 𝑄.𝑐𝑜𝑙𝑢𝑚𝑛𝑠 do
+4
+𝑐_𝑐𝑎𝑛𝑑𝑖𝑑𝑎𝑡𝑒𝑠,𝑐_𝑠𝑐𝑜𝑟𝑒𝑠 ← I.lookup(𝑐);
+5
+𝑐𝑜𝑙𝑢𝑚𝑛_𝑐𝑎𝑛𝑑𝑖𝑑𝑎𝑡𝑒𝑠[𝑐].add(𝑐_𝑐𝑎𝑛𝑑𝑖𝑑𝑎𝑡𝑒𝑠);
+6
+𝑐𝑜𝑙𝑢𝑚𝑛_𝑠𝑐𝑜𝑟𝑒𝑠[𝑐].add(𝑐_𝑠𝑐𝑜𝑟𝑒𝑠);
+7 end
+8 𝑡𝑎𝑏𝑙𝑒_𝑐𝑎𝑛𝑑𝑖𝑑𝑎𝑡𝑒𝑠 ← group_by(𝑐𝑜𝑙𝑢𝑚𝑛_𝑐𝑎𝑛𝑑𝑖𝑑𝑎𝑡𝑒𝑠);
+9 𝑟𝑎𝑛𝑘𝑒𝑑_𝑡𝑎𝑏𝑙𝑒_𝑐𝑎𝑛𝑑𝑖𝑑𝑎𝑡𝑒𝑠 ←
+cmpt_table_unionability(𝑡𝑎𝑏𝑙𝑒_𝑐𝑎𝑛𝑑𝑖𝑑𝑎𝑡𝑒𝑠,𝑐𝑜𝑙𝑢𝑚𝑛_𝑠𝑐𝑜𝑟𝑒𝑠);
+10 return 𝑟𝑎𝑛𝑘𝑒𝑑_𝑡𝑎𝑏𝑙𝑒_𝑐𝑎𝑛𝑑𝑖𝑑𝑎𝑡𝑒𝑠[: 𝑘];
+in a high-dimensional vector space by inserting hyper-planes. Em-
+beddings with high cosine similarity tend to lie on the same side of
+many hyper-planes.
+Algorithm 2 summarizes the top-𝑘 table union search. Following
+Definition 1, we instantiate the union-ability of two attributes as
+the cosine similarity of their embeddings (𝑐_𝑠𝑐𝑜𝑟𝑒𝑠 in line 4 of
+Algorithm 2). Line 8 groups retrieved column candidates across
+query columns by their table sources. To decide on the table union-
+ability from Definition 3 (𝑐𝑚𝑝𝑡_𝑡𝑎𝑏𝑙𝑒_𝑢𝑛𝑖𝑜𝑛𝑎𝑏𝑖𝑙𝑖𝑡𝑦 in line 9), we
+use the same weighting strategy as [1] over query attributes and
+corresponding matching attributes in candidate tables. For a target
+attribute 𝐴, let 𝑅𝐴 denote the distribution of all similarity (union-
+ability) scores between 𝐴 and any attribute 𝐵 returned by the LSH
+index. The weight𝑤 of a similarity score U𝑎𝑡𝑡𝑟 (𝐴, 𝐵) is given by the
+cumulative distribution function of 𝑅𝐴 evaluated at U𝑎𝑡𝑡𝑟 (𝐴, 𝐵):
+𝑤 = Pr(U𝑎𝑡𝑡𝑟 (𝐴, 𝐵′) ≤ U𝑎𝑡𝑡𝑟 (𝐴, 𝐵)), ∀ U𝑎𝑡𝑡𝑟 (𝐴, 𝐵′) ∈ 𝑅𝐴
+In other words, a similarity score is weighted by its percentile in
+the distribution. This weighting scheme helps balance between a
+
+Pylon: Table Union Search through Contrastive Representation Learning
+Conference acronym ’XX, June 03–05, 2018, Woodstock, NY
+candidate table with a few union-able attributes of high similarity
+scores and another candidate table with more union-able attributes
+but of lower similarity scores.
+Using the same index and search structure as previous works
+makes it transparent to compare our embedding approach with
+theirs in effectiveness and efficiency.
+3.5
+Integrating Syntactic Methods
+Thus far, we have focused purely on semantic methods to unify
+similar attributes. It makes sense to prefer semantic methods to
+syntactic ones because of their potential robustness to many differ-
+ent types of variation. However, we note that syntactic methods
+are based on measures of similarity very different from semantic
+methods. Intuitively, one should expect to be able to do better if we
+can integrate the two.
+Indeed, some previous work [1, 27] has made this observation
+as well, and shown that an ensemble of semantic and syntactic
+methods can do better than either on its own. The Pylon semantic
+method permits the use of an additional complementary syntactic
+method. As in [1], we independently obtain scores from the two
+methods and then use the average of the two as our final score.
+4
+EXPERIMENTS
+We first evaluate the effectiveness and efficiency of three model vari-
+ants from our contrastive learning framework and compare them
+with their corresponding base encoders. We then demonstrate that
+our embedding approach is orthogonal to existing syntactic mea-
+sures, which can further improve the results. We finally compare
+our best model with the state-of-the-art 𝐷3𝐿 [1].
+4.1
+Datasets and Metrics
+TUS Benchmark1. [27] compiled the first benchmark with ground
+truth out of Canadian and UK Open Data. They synthesized around
+5, 000 tables from 32 base tables by performing random projection
+and selection. They also generated a smaller benchmark consisting
+of around 1, 5002 tables from 10 base tables. We refer to them as
+TUS-Large and TUS-Small respectively.
+Pylon Benchmark. We create a new dataset from GitTables [21],
+a corpus of 1.7𝑀 tables extracted from CSV files on GitHub3. The
+benchmark comprises 1,746 tables including union-able table sub-
+sets under topics selected from Schema.org [16]: scholarly article,
+job posting, and music playlist. We end up with these three topics
+since we can find a fair number of union-able tables of them from
+diverse sources in the corpus (we can easily find union-able tables
+from a single source but they are less interesting for table union
+search as simple syntactic methods can identify all of them because
+of the same schema and consistent value representations).
+Cleaning and Construction. We download three largest sub-
+sets of GitTables ("object", "thing", and "whole") and preprocess them
+by removing HTML files, tables without headers, rows with foreign
+languages, and finally small tables with fewer than four rows or
+four columns. We cluster the resulting tables by their schema and
+1TUS benchmark can be accessed from https://github.com/RJMillerLab/table-union-
+search-benchmark.
+2We export 1, 530 tables from the small benchmark although the paper and the website
+claim ∼1, 300 tables.
+3GitTables 1.7𝑀 is available from https://zenodo.org/record/4943312#.Ylm2ftPMJxZ.
+perform a keyword search over schema with keywords related to
+three topics. We manually select 35 union-able tables of topic schol-
+arly article, 41 tables of topic job posting, and 48 tables of topic
+music playlist. We then randomly sample 100,000 tables4 for train-
+ing, 5,000 tables for validation, and put the rest of tables as noise5
+in a pool with union-able table subsets for the search evaluation.
+Table 1 provides an overview of basic statistics of tables in each
+evaluation dataset.
+Table 1: Basic statistics of evaluation datasets.
+Pylon
+TUS-Small
+TUS-Large
+# Tables
+1,746
+1,530
+5,043
+# Base Tables
+1,746
+10
+32
+Avg. # Rows
+115
+4,466
+1,915
+Avg. # Columns
+10
+10
+11
+# Queries
+124
+1,327
+4,296
+Avg. # Answers
+42
+174
+280
+Metrics. For effectiveness, we report both precision and recall
+of top-𝑘 search with varying 𝑘. At each value of 𝑘, we average the
+precision and recall numbers over all the queries. We consider a
+table candidate as a true positive with respect to the target table as
+long as it is in the corresponding ground truth. We do not require
+perfect attribute pair matching as it is a more challenging setting
+and requires laborious column-level labeling.
+As to efficiency, we report indexing time (i.e., total amount of
+time in minutes to index all columns in a dataset) and query re-
+sponse time (i.e., average amount of time in seconds for the LSH
+index to return results over all queries in a dataset).
+In evaluation, we randomly sample 1000 queries from TUS-Large
+for efficient experiment purposes. The query subset has an average
+answer size of 277, which is very close to that of the full query set
+(i.e., 280). We use all the queries in Pylon and TUS-Small datasets.
+4.2
+Baselines
+We consider two embedding methods and one full approach as
+baselines for comparison.
+fastText. Many data discovery tasks [15, 27] not limited to table
+union search have adopted fastText in their approach, which is a
+popular word embedding model trained on Wikipedia documents.
+WTE. [18] devised a word embedding-based technique to rep-
+resent text values in Web tables. They generated text sequences
+from tables for training by serializing tables in two different ways
+that capture row-wise relations and relations between schema and
+data values respectively. It is reported that the model using both
+serialization obtained the best performance in a task of ranking
+unionable columns. We use this model6 in comparison and refer to
+it as WTE (for web table embeddings).
+4We noticed that a few schemas have an overwhelming number of tables (because
+some GitHub repositories publish hundreds and thousands of tables with the same
+schema). In sampling, we take at most 200 tables from each schema to increase the
+diversity of the training set.
+5we filtered these tables using their schema to reduce the chance of them being union-
+able to selected tables in the union-able subsets (i.e., true noise).
+6𝑊𝑐𝑜𝑚𝑏𝑜 150dim: https://github.com/guenthermi/table-embeddings/tree/main#pre-
+trained-models
+
+Conference acronym ’XX, June 03–05, 2018, Woodstock, NY
+Tianji Cong and H. V. Jagadish
+D3L. [1] proposed a distance-based framework 𝐷3𝐿 that uses
+five types of evidence to decide on column unionability: (i) attribute
+name similarity; (ii) attribute extent overlap; (iii) word-embedding
+similarity; (iv) format representation similarity; (v) domain distri-
+bution similarity for numerical attributes. Their aggregated ap-
+proach is shown to be more effective and efficient than previous
+work [13, 27] on the TUS benchmark and another self-curated
+dataset of Open Data tables. To the best of our knowledge, 𝐷3𝐿 is
+the current state-of-the-art of the table union search problem.
+4.3
+Comparisons of Interest
+We have 5 variants of Pylon to compare against baseline systems for
+both effectiveness and efficiency in identifying union-able tables
+using semantic similarity methods: 3 variants from the online train-
+ing data construction strategy and 2 variants from the offline data
+construction strategy. In addition, we have 3 syntactic similarity
+measures that could be used to augment each of these 5 variants.
+Finally, we have 3 baselines, two of which are semantic word embed-
+ding based, and hence could also be augmented with the syntactic
+similarity measures. The third baseline (D3L) already integrates
+both syntactic and semantic similarity, and hence does not benefit
+from additional augmentation with syntactic techniques.
+Since there are a very large number of alternatives to compare,
+we break up the comparisons into four sets, as follows, and present
+the results for each set separately. For the first three sets, we restrict
+ourselves to the online training data construction strategy for Pylon.
+We refer to the derived models as Pylon-fastText, Pylon-WTE, Pylon-
+LM respectively based on the corresponding encoder choice. Results
+for the offline data construction strategy show generally similar
+trends, and the most interesting are shown in the fourth set.
+The first set of comparisons look purely at semantic methods,
+considering the 3 variants of Pylon and comparing them to the first
+two baselines. We leave out D3L because it already incorporates
+syntactic methods as well. The second set of comparisons look
+purely at the benefit obtained when semantic methods are enhanced
+with syntactic measures. We do so for all methods evaluated in the
+first set. Finally, we bring everything together by comparing the
+best methods of the second set with the best integrated baseline,
+D3L. This is the final top line "take away" from the experiments,
+eliding details from the first two sets of comparisons.
+4.4
+Experiment Details
+As to model training, we train Pylon-fastText for 50 epochs with a
+batch size of 16 on 2 NVIDIA GeForce RTX 2080 Ti GPUs; Pylon-
+WTE for 20 epochs with a batch size of 32 on a single NVIDIA
+Tesla P100 GPU; Pylon-LM for 20 epochs with a batch size of 8
+on 4 NVIDIA Tesla P100 GPUs from Google Cloud Platform. As
+seen in table 2, the training is especially efficient for simple word
+embedding encoders (as only parameters in projection head are
+updated) and the offline data construction strategy (as embeddings
+are pre-computed before training). We save the models with the
+smallest validation loss. The model training is implemented in
+PyTorch [29] and PyTorch Lightning7.
+For evaluation of table union search, we set the similarity thresh-
+old of LSH index to 0.7 in all experiments and use the default hash
+7https://www.pytorchlightning.ai/
+Table 2: Model training time (min / epoch) where each model
+is defined by the encoder choice and the training data con-
+struction strategy.
+Online Sampling
+Offline Approximate Matching
+Pylon-fastText
+6.5
+0.42
+Pylon-WTE
+0.99
+0.13
+Pylon-LM
+33
+-
+size (a MinHash size of 256 and a random projection size of 1024) as
+D3L. We run all evaluation on a Ubuntu 20.04.4 LTS machine with
+128 GiB RAM and Intel(R) Xeon(R) Bronze 3106 CPU @ 1.70GHz.
+4.5
+Results
+As Pylon is an embedding-based approach, we first evaluate Pylon
+model variants against embedding baselines fastText and WTE, and
+inspect what effects contrastive learning have on them.
+Experiment 1(a): Comparison of effectiveness between Py-
+lon model variants and their corresponding base encoders.
+Figure 3 shows the precision and recall of each embedding measure
+on the Pylon dataset. Both Pylon-WTE and Pylon-fastText outper-
+form their corresponding base models with a notable margin. When
+𝑘 = 40, around the average answer size, Pylon-WTE is 6% better
+than WTE on both metrics, and Pylon-fastText performs better than
+fastText by 15% on precision and 14% on recall.
+Overall, our Pylon-WTE model consistently achieves the highest
+precision and recall as 𝑘 increases. We also note that Pylon-LM
+has strong performance up until 𝑘 = 30 but degrades after that.
+This is because Pylon-LM only samples 10 rows from each table to
+construct embeddings (for indexing efficiency) while other word-
+embedding methods can afford to encode the entire table at low
+indexing time, which we demonstrate in experiment 1(b).
+Experiment 1(b): Comparison of efficiency between Pylon
+model variants and their corresponding base encoders. In fig-
+ure 4, we see both embedding baselines are very efficient in index
+construction and it takes less than 2 minutes to index the entire
+Pylon dataset. Unlike fixed embeddings, our models need to infer
+embeddings at runtime. For Pylon-fastText and Pylon-WTE, since
+the encoder is fixed, the inference cost is exclusively from projec-
+tion head. It takes both less than 3.5 minutes to build the index. In
+contrast, the runtime inference cost of Pylon-LM is more expensive
+as the language model has much more complex architecture and has
+130M parameters versus 35.8K parameters in projection head. We
+also acknowledge the less efficient implementation of embedding
+inference at this point (e.g., run inference for each column with-
+out batch predictions). Nevertheless, indexing time, as a one-time
+overhead, can be amortized among queries.
+On the other hand, all of our models are considerably more
+efficient in query response time. Pylon-fastText is 2.7x faster than
+fastText and Pylon-WTE is 9x faster than WTE. The significant
+speedup of query response time is attributed to contrastive learning
+where embeddings of attribute values occurring in the same context
+are pushed close to each other whereas embeddings of two random
+columns are pushed apart. As the embedding similarity between
+two random columns is suppressed, this dramatically reduces the
+chance of two random columns sharing many LSH buckets. In
+
+Pylon: Table Union Search through Contrastive Representation Learning
+Conference acronym ’XX, June 03–05, 2018, Woodstock, NY
+10
+20
+30
+40
+50
+60
+70
+80
+90
+100
+k
+0.0
+0.1
+0.2
+0.3
+0.4
+0.5
+0.6
+0.7
+0.8
+0.9
+1.0
+Precision
+fastText
+Pylon-fastText
+WTE
+Pylon-WTE
+Pylon-LM
+10
+20
+30
+40
+50
+60
+70
+80
+90
+100
+k
+0.0
+0.1
+0.2
+0.3
+0.4
+0.5
+0.6
+0.7
+0.8
+0.9
+1.0
+Recall
+fastText
+Pylon-fastText
+WTE
+Pylon-WTE
+Pylon-LM
+Figure 3: Top-k precision and recall of embedding mea-
+sures on the Pylon dataset.
+fastText
+Pylon-fastText
+WTE
+Pylon-WTE
+Pylon-LM
+Model
+0
+5
+10
+15
+20
+25
+30
+Indexing Time (min)
+1.8
+2.4
+1.3
+3.4
+23.8
+fastText
+Pylon-fastText
+WTE
+Pylon-WTE
+Pylon-LM
+Model
+0
+5
+10
+15
+20
+25
+Query Response Time (s / query)
+15.8
+4.3
+24.7
+2.4
+3.0
+Figure 4: Indexing time and query response time on the
+Pylon dataset.
+0
+20 40 60 80 100
+k
+0.0
+0.1
+0.2
+0.3
+0.4
+0.5
+0.6
+0.7
+0.8
+0.9
+1.0
+Precision
+Pylon-fastText
+Pylon-fastText-NVF
+Pylon-fastText-NV
+0
+20 40 60 80 100
+k
+0.0
+0.1
+0.2
+0.3
+0.4
+0.5
+0.6
+0.7
+0.8
+0.9
+1.0
+Precision
+Pylon-WTE
+Pylon-WTE-NVF
+Pylon-WTE-NV
+0
+20 40 60 80 100
+k
+0.0
+0.1
+0.2
+0.3
+0.4
+0.5
+0.6
+0.7
+0.8
+0.9
+1.0
+Precision
+Pylon-LM
+Pylon-LM-NVF
+Pylon-LM-NV
+0
+20 40 60 80 100
+k
+0.0
+0.1
+0.2
+0.3
+0.4
+0.5
+0.6
+0.7
+0.8
+0.9
+1.0
+Recall
+0
+20 40 60 80 100
+k
+0.0
+0.1
+0.2
+0.3
+0.4
+0.5
+0.6
+0.7
+0.8
+0.9
+1.0
+Recall
+0
+20 40 60 80 100
+k
+0.0
+0.1
+0.2
+0.3
+0.4
+0.5
+0.6
+0.7
+0.8
+0.9
+1.0
+Recall
+(a) Pylon dataset
+10
+50
+90
+130
+170
+210
+250
+290
+k
+0.0
+0.1
+0.2
+0.3
+0.4
+0.5
+0.6
+0.7
+0.8
+0.9
+1.0
+Precision
+Pylon-fastText
+Pylon-fastText-NVF
+Pylon-fastText-NV
+10
+50
+90
+130
+170
+210
+250
+290
+k
+0.0
+0.1
+0.2
+0.3
+0.4
+0.5
+0.6
+0.7
+0.8
+0.9
+1.0
+Precision
+Pylon-WTE
+Pylon-WTE-NVF
+Pylon-WTE-NV
+10
+50
+90
+130
+170
+210
+250
+290
+k
+0.0
+0.1
+0.2
+0.3
+0.4
+0.5
+0.6
+0.7
+0.8
+0.9
+1.0
+Precision
+Pylon-LM
+Pylon-LM-NVF
+Pylon-LM-NV
+10
+50
+90
+130
+170
+210
+250
+290
+k
+0.0
+0.1
+0.2
+0.3
+0.4
+0.5
+0.6
+0.7
+0.8
+0.9
+1.0
+Recall
+10
+50
+90
+130
+170
+210
+250
+290
+k
+0.0
+0.1
+0.2
+0.3
+0.4
+0.5
+0.6
+0.7
+0.8
+0.9
+1.0
+Recall
+10
+50
+90
+130
+170
+210
+250
+290
+k
+0.0
+0.1
+0.2
+0.3
+0.4
+0.5
+0.6
+0.7
+0.8
+0.9
+1.0
+Recall
+(b) TUS-Small dataset
+Figure 5: Precision and recall (w.r.t. varying 𝑘) of the ensemble of Pylon embedding models and syntactic measures.
+other words, LSH index can process much fewer candidates at the
+configured similarity threshold.
+To illustrate the suppression effect of contrastive learning, we
+compare heatmaps of pairwise cosine similarity of column em-
+beddings encoded by WTE and Pylon-WTE respectively. Consider
+the three text columns of the first table in Figure 1. As shown in
+Figure 6(a), the pairwise cosine similarity of WTE embeddings is
+mostly above 0.5. There is a very high similarity (0.87) between the
+"title" column and the "venue" column and they will be mistakenly
+viewed as unionable. But this is not an issue for Pylon-WTE embed-
+dings as shown in Figure 6(b) where the pairwise similarity between
+different columns are much lower (below 0.51) and the LSH index
+will not return the "venue" column as a unionable candidate of the
+"title" column.
+Figure 6: Pairwise cosine similarity of column embeddings:
+(a) WTE embeddings; (b) Pylon-WTE embeddings.
+
+title
+authors
+venue
+title
+authors
+venue
+1.0
+1.0
+0.5203
+0.8675
+1.0
+0.0182
+0.5095
+title -
+title -
+0.8
+0.6
+0.5203
+1.0
+0.492
+0.0182
+1.0
+authors - 
+0.0063
+authors -
+0.4
+0.2
+0.8675
+0.492
+1.0
+0.5095
+0.0063
+1.0
+venue
+venue -
+0.0Conference acronym ’XX, June 03–05, 2018, Woodstock, NY
+Tianji Cong and H. V. Jagadish
+In the next set of experiments, we consider three syntactic mea-
+sures used by D3L and evaluate how much they can augment our
+embedding measures.
+(1) Name (𝑁): Jaccard similarity between q-gram sets of at-
+tribute names.
+(2) Value (𝑉 ): Jaccard similarity between the TF-IDF sets of
+attribute values.
+(3) Format (𝐹): Jaccard similarity between regular-expression
+sets of attribute values.
+Experiment 2: Effectiveness of the ensemble of Pylon model
+variants and syntactic measures. Figure 5(a) and (b) show the
+precision and recall of the ensemble of Pylon embedding models and
+syntactic measures on Pylon and TUS-Small datasets respectively.
+We consistently observe from both datasets that adding syntactic
+measures can further enhance the performance. In particular, name
+(𝑁) and value (𝑉 ) similarity are most effective syntactic measures.
+Around the average answer size of the Pylon dataset (𝑘 = 40), 𝑁
+and 𝑉 together raise up the precision and recall of Pylon-fastText
+by nearly 20%, of Pylon-WTE by 10%, and of Pylon-LM by over 5%.
+Similarly, around the average answer size of the TUS-Small dataset
+(𝑘 = 170), there is an increase of about 10% in both precision and
+recall for Pylon-fastText, about 5% for Pylon-WTE, and more than
+10% for Pylon-LM.
+We also observe that adding additional format measure (𝐹) hurts
+the performance (notably on the Pylon dataset and slightly on TUS-
+small). This is because tables in the Pylon dataset are mostly from
+disparate sources and so the value format tends to be inconsistent
+across tables whereas tables in TUS-Small are synthesized from
+only 8 base tables and it is much more likely for many tables to share
+format similarity. Even worse, including format index imposes non-
+trivial runtime cost (see figure 7). For example, compared to model
+Pylon-WTE-NV, the query response time of Pylon-WTE-NVF (with
+the extra format measure) surges by 66.7% on the Pylon dataset and
+by 32.2% on TUS-Small.
+Pylon-fastText
+Pylon-WTE
+Pylon-LM
+0
+1
+2
+3
+4
+5
+6
+7
+Query Response Time (s / query)
+7.2
+3.9
+4.2
+4.7
+2.3
+3.3
+NVF
+NV
+(a) Pylon dataset
+Pylon-fastText
+Pylon-WTE
+Pylon-LM
+0
+5
+10
+15
+20
+25
+30
+35
+Query Response Time (s / query)
+34.9
+27.3
+16.2
+28.7
+20.6
+11.1
+NVF
+NV
+(b) TUS-Small dataset
+Figure 7: Comparison of query response time between in-
+cluding and excluding the format measure.
+Finally, we compare our best-performing model Pylon-WTE-NV
+with the state-of-the-art D3L. As Pylon-WTE-NV does not use for-
+mat and domain measures in D3L, for fair comparison, we consider
+three versions of D3L. We refer to the full version of D3L as D3L-5,
+the one without the format measure as D3L-4, and the one without
+format and domain measures as D3L-3.
+Experiment 3: Comparison of effectiveness and efficiency
+between our best model and D3L. Figure 8 shows the perfor-
+mance of Pylon-WTE-NV and three D3L variants on Pylon , TUS-
+Small, TUS-Large datasets respectively. Around the average answer
+size (𝑘 = 40) of the Pylon dataset, Pylon-WTE-NV is around 15%
+better than the strongest D3L instance (i.e., D3L-3) in both precision
+and recall. Pylon-WTE-NV performs much better than D3L in this
+case because our embedding model using contrastive learning was
+trained on a dataset of a distribution similar to the test set and can
+capture more semantics than the off-the-shelf fastText embedding
+model used in D3L.
+On TUS-Small and TUS-Large, we observe all instances have
+relatively competitive performance while Pylon-WTE-NV performs
+marginally better compared to all D3L variants. On TUS-Small,
+around the average answer size (𝑘 = 170), Pylon-WTE-NV is 2%
+better than D3L-3 and 5% better than D3L-5 in both precision and
+recall. On TUS-Large, around the average answer size (𝑘 = 290),
+Pylon-WTE-NV is more than 2% better than D3L variants in both
+metrics. The small performance gap is due to the synthetic nature
+of TUS benchmark where most of union-able tables are generated
+from the same base table and share common attribute names and
+many attribute values. So syntactic measures (𝑁 and𝑉 ) can capture
+most of similarity signals and obtain high precision and recall even
+without support of semantic evidence.
+Additional to the performance gain, the biggest advantage of
+Pylon-WTE-NV is the fast query response time. On the Pylon dataset,
+our model is nearly 9x faster than the full version D3L-5 and 7x
+faster than D3L-3. Even on TUS-Small and TUS-Large, which are
+datasets of a different data distribution (Open Data tables), we still
+save runtime by 44% and 32% respectively compared to D3L-5, and
+by 35.5% and 21.9% respectively compared to D3L-3.
+Experiment 4: Effectiveness and efficiency of Pylon model
+variants from the offline training data construction strategy.
+Figure 10 shows the precision and recall of 4 Pylon variants from
+two training data construction strategies and their baselines. On the
+Pylon dataset, around the average answer size (𝑘 = 40), two Pylon
+models from the alternative data construction strategy, Pylon-WTE-
+offline and Pylon-fastText-offline, retain strong performance and
+outperform the corresponding baseline by 3% and 9% respectively.
+Note that Pylon models derived from the sampling data construc-
+tion strategy have consistently better performance as 𝑘 increases.
+We also observe a similar trend on the TUS benchmark while the
+performance gap of all instances is smaller.
+As shown in figure 11, both new models are efficient in index-
+ing time and query response time. Compared to the correspond-
+ing baseline, Pylon-WTE-offline is 12x faster and Pylon-fastText-
+offline is 14.5x faster in query response time. Again, this signifi-
+cant speedup demonstrates the distinguishing power of contrastive
+learning, which enables the LSH index to work more efficiently
+with embeddings.
+4.6
+Discussion
+Although this paper mainly focuses on the novel learning approach
+for the table union search problem, we also leave and discuss a few
+clues for future extensions.
+
+Pylon: Table Union Search through Contrastive Representation Learning
+Conference acronym ’XX, June 03–05, 2018, Woodstock, NY
+0
+20
+40
+60
+80
+100
+k
+0.0
+0.1
+0.2
+0.3
+0.4
+0.5
+0.6
+0.7
+0.8
+0.9
+Precision
+Pylon-WTE-NV
+D3L-3
+D3L-4
+D3L-5
+0
+20
+40
+60
+80
+100
+k
+0.0
+0.1
+0.2
+0.3
+0.4
+0.5
+0.6
+0.7
+0.8
+0.9
+Recall
+Pylon-WTE-NV
+D3L-3
+D3L-4
+D3L-5
+(a) Pylon dataset
+10
+50
+90 130 170 210 250 290
+k
+0.5
+0.6
+0.7
+0.8
+0.9
+1.0
+Precision
+Pylon-WTE-NV
+D3L-3
+D3L-4
+D3L-5
+10
+50
+90 130 170 210 250 290
+k
+0.0
+0.1
+0.2
+0.3
+0.4
+0.5
+0.6
+0.7
+0.8
+0.9
+1.0
+Recall
+Pylon-WTE-NV
+D3L-3
+D3L-4
+D3L-5
+(b) TUS-Small dataset
+10
+50
+90 130 170 210 250 290
+k
+0.5
+0.6
+0.7
+0.8
+0.9
+1.0
+Precision
+Pylon-WTE-NV
+D3L-3
+D3L-4
+D3L-5
+10
+50
+90 130 170 210 250 290
+k
+0.0
+0.1
+0.2
+0.3
+0.4
+0.5
+0.6
+0.7
+0.8
+0.9
+1.0
+Recall
+Pylon-WTE-NV
+D3L-3
+D3L-4
+D3L-5
+(c) TUS-Large dataset
+Figure 8: Comparison of precision and recall between D3L instances and our best model Pylon-WTE-NV.
+D3L-5
+D3L-4
+D3L-3
+Pylon-WTE-NV
+Model
+0.0
+2.5
+5.0
+7.5
+10.0
+12.5
+15.0
+17.5
+20.0
+Query Response Time (s / query)
+19.6
+17.1
+16.1
+2.3
+(a) Pylon dataset
+D3L-5
+D3L-4
+D3L-3
+Pylon-WTE-NV
+Model
+0
+5
+10
+15
+20
+25
+30
+35
+40
+Query Response Time (s / query)
+41.2
+42.2
+32
+20.6
+(b) TUS-Small dataset
+D3L-5
+D3L-4
+D3L-3
+Pylon-WTE-NV
+Model
+0
+20
+40
+60
+80
+100
+Query Response Time (s / query)
+110
+110.5
+95.6
+74.6
+(c) TUS-Large dataset
+Figure 9: Comparison of query response time between D3L instances and Pylon-WTE-NV.
+10 20 30 40 50 60 70 80 90 100
+k
+0.0
+0.1
+0.2
+0.3
+0.4
+0.5
+0.6
+0.7
+0.8
+0.9
+1.0
+Precision
+Pylon-WTE
+Pylon-WTE-offline
+WTE
+Pylon-fastText
+Pylon-fastText-offline
+fastText
+10 20 30 40 50 60 70 80 90 100
+k
+0.0
+0.1
+0.2
+0.3
+0.4
+0.5
+0.6
+0.7
+0.8
+0.9
+1.0
+Recall
+Pylon-WTE
+Pylon-WTE-offline
+WTE
+Pylon-fastText
+Pylon-fastText-offline
+fastText
+Figure 10: Top-k precision and recall of 6 embedding
+measures on the Pylon dataset.
+fastText
+Pylon-fastText
+Pylon-fastText-offline
+WTE
+Pylon-WTE
+Pylon-WTE-offline
+Model
+0
+1
+2
+3
+4
+Indexing Time (min)
+1.8
+2.4
+2.2
+1.3
+3.4
+1.7
+fastText
+Pylon-fastText
+Pylon-fastText-offline
+WTE
+Pylon-WTE
+Pylon-WTE-offline
+Model
+0
+5
+10
+15
+20
+25
+Query Response Time (s / query)
+15.8
+4.3
+1.2
+24.7
+2.4
+1.7
+Figure 11: Indexing time and query response time on the
+Pylon dataset.
+Alternative Contrastive Loss. While InfoNCE (used in this
+project) is a popular and effective loss function, it is not the only
+feasible training objective for self-supervised contrastive learning.
+For example, triplet loss [31] considers a triplet (𝑥,𝑥+,𝑥−) as a
+training example where 𝑥 is an input, 𝑥+ is a positive sample (be-
+longing to the same class as 𝑥 or semantically similar to 𝑥) and
+
+Conference acronym ’XX, June 03–05, 2018, Woodstock, NY
+Tianji Cong and H. V. Jagadish
+𝑥− is a negative sample. Additionally, what considers as negative
+examples and "hardness" of negative examples are also interesting
+perspectives to explore.
+Verification of Column Union-ability. Besides quantitative
+evaluation, we also manually inspect results of a few queries for
+each dataset. We observe that even in correct table matches, there
+are false positives of union-able column candidates. To mitigate
+this issue, we believe that progress in column semantic type pre-
+diction [33, 39] can be beneficial for verifying the union-ability of
+columns as a post-processing step.
+5
+RELATED WORK
+Our work is most related to data integration in the Web context
+and data discovery over enterprise and Open Data repositories.
+Web Table Search. [4] presents OCTOPUS that integrate rel-
+evant data tables from relational sources on the Web. OCTOPUS
+includes operators that perform a search-style keyword query over
+extracted relations and their context, and cluster results into groups
+of union-able tables using column-to-column mean string length
+similarity and TF-IDF cosine similarity. [37] defines three infor-
+mation gathering tasks on Web tables: augmentation by attribute
+names, augmentation by example, and attribute discovery. The task
+of augmentation by example essentially involves finding union-able
+tables that can be used to fill in the missing values in a given table.
+Their Infogather system leverages indirectly matching tables in
+addition to directly matching ones to augment a user input. [10]
+formalizes the problem of detecting related Web tables. At the log-
+ical level, the work considers two tables related to each other if
+they can be viewed as results to queries over the same (possibly
+hypothetical) original table. In particular, one type of relatedness
+they define is Entity Complement where two tables with coherent
+and complementary subject entities can be unioned over the com-
+mon attributes. This definition requires each table to have a subject
+column of entities indicating what the table is about and that the
+subject column can be detected. Following the definition, the work
+captures entity consistency and expansion by measuring the relat-
+edness of detected sets of entities with signals mined from external
+ontology sources. Finally, they perform schema mapping of two
+complement tables by computing a schema consistency score made
+up of the similarity in attribute names, data types, and values.
+Data Discovery in the Enterprise. [14] identifies data discov-
+ery challenges in the enterprise environment. The position paper
+describes a data discovery system including enrichment primitives
+that allow a user to perform entity and schema complement opera-
+tions. Building on top of the vision in [14], [13] presents AURUM,
+a system that models syntactic relationships between datasets in
+a graph data structure. With a two-step process of profiling and
+indexing data, AURUM constructs a graph with nodes representing
+column signatures and weighted edges indicating the similarity be-
+tween two nodes (e.g., content and schema similarity). By framing
+queries as graph traverse problems, AURUM can support varied
+discovery needs of a user such as keyword search and similar con-
+tent search (which can be used for finding union-able columns
+and tables). [15] further employs word embeddings in AURUM to
+identify semantically related objects in the graph.
+Data Discovery over Open Data Repositories. [27] defines
+the table union search problem on open data and decomposes it as
+finding union-able attributes. They propose three statistical tests to
+determine the attribute union-ability: (1) set union-ability measure
+based on value overlap; (2) semantic union-ability measure based
+on ontology class overlap; and (3) natural language union-ability
+measure based on word embeddings, where union-ability is the esti-
+mated probability that the text values contained in two columns are
+drawn from the same domain. A synthesized benchmark consisting
+of original tables from Canadian and UK Open Data shows that nat-
+ural language union-ability works best for larger 𝑘 in top-𝑘 search.
+In the meantime, set union-ability is decent when 𝑘 = 1 for each
+query but vulnerable to value overlap in attributes of non-unionable
+tables, and semantic union-ability stays competitive to find some
+union-able tables for most queries despite incomplete coverage of
+external ontologies. The ensemble of three measures further im-
+proves the evaluation metrics. [1] adopts more types of similarity
+measures based on schema- and instance-level fine-grained features.
+Without relying on any external sources, their D3L framework is
+shown effective and efficient on Open Data Lakes. EMBDI [6] pro-
+poses a graph model to capture relationships across relational tables
+and derives training sequences from random walks over the graph.
+They further take advantage of embedding training algorithms like
+fastText to construct embedding models. Their relational embed-
+dings demonstrate promising results for data integration tasks such
+as schema matching and entity resolution.
+For a broader overview of the literature, we refer readers to the
+survey of dataset search [7].
+6
+CONCLUSION
+In this work, we present Pylon, a self-supervised contrastive learn-
+ing framework for learning semantic column representations from
+large collections of tables. We demonstrate that contrastive learning
+is a feasible way of learning effective representations for the table
+union search problem without relying on labeled data or being
+restricted to off-the-shelf embedding models. In comparison with
+embedding baselines and the state-of-the-art, we observe that (i)
+on the real-world dataset of a data distribution similar to the train-
+ing data, our models consistently achieve significant gain in both
+effectiveness and efficiency; (ii) on the synthetic benchmark of a
+different data distribution, our models have marginal performance
+improvement while staying more efficient.
+We hypothesize that the contrastive learning paradigm can also
+benefit other data discovery and table understanding problems that
+do not fit into the classification formulation or lack large scale
+of labeled data (e.g., join-path discovery). It is also worth noting
+that contrastive learning does not contradict supervision. It will be
+interesting to see if contrastive learning can also enhance existing
+supervised learning solutions for entity resolution and many table
+understanding tasks such as semantic column type annotation.
+ACKNOWLEDGMENTS
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