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1863f7e00243930ce9d7647848dc7c467a0d7961 | subsection | 64 | 98 | Extensions | The following section outlines different ideas on how to extend the generalized SART-approach to an even more versatile tool for devising efficient Kaczmarz-type reconstruction methods. | {
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6fc90ba97c7ece8e0874aaa8b508ad3ced85740a | subsection | 65 | 98 | Box constraints | Analogously as in other Kaczmarz-methods, box constraints f_{\min } \le f \le f_{\max } on the admissible values of the object f may be incorporated in GenSART-schemes simply by settingf_{k+1} \leftarrow \max \big \lbrace \min \lbrace f_{k+1}, f_{\max } \rbrace , f_{\min } \big \rbrace .after each iteration. This appro... | {
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6772439ce26a380776748f4279322e621efc4c7d | subsection | 66 | 98 | Additional quadratic regularizer | So far, the penalization in the considered Kaczmarz-iterations was always with respect to the preceding iterate f_k. In addition, it might be desirable to impose a static regularizer such that the total penalty is given by\mathcal {R}_k(f) = \alpha _1 \Vert T(f - f_k) \Vert _Z^2 + \alpha _2 \Vert T(f - f_{\textup {ref}... | {
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1965f9eb2342d473d7dc88b8c1007e25b082c997 | subsection | 67 | 98 | Kaczmarz-type splitting and primal-dual methods | Variational reconstruction methods seek to minimize terms of the form \mathcal {S}_{\textup {tot}}\left( P_{\textup {tot}}(f) \right) + \mathcal {R}(f) with \mathcal {S}_{\textup {tot}}(p_1, \ldots , p_{N_{\textup {proj}}}) = \sum _{j=1}^{N_{\textup {proj}}}\mathcal {S}(p_j), compare §REF . Often, this is achieved by s... | {
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b36d4a342a99dd0469d73fae66efde4102a61872 | subsection | 68 | 98 | Numerical examples | All of the GenSART-schemes from § have been successfully implemented as numerical algorithms.
In the following, exemplary results are presented. | {
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} | 1803.04726 | Generalized SART Methods for Tomographic Imaging | [
"Simon Maretzke"
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435673478de4fc191484358178d02164639e0b68 | subsection | 69 | 98 | Implementation | In previous studies, Kaczmarz-type reconstruction methods have usually been derived for a discretized tomographic model. On the contrary, the theory in this work relies on properties of the parallel- or cone-beam projectors P\in \lbrace {P}, {D}\rbrace that are valid only in continuous space. In particular, while the g... | {
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fca446fbe6c9c01764a663bfb1875b0db87e95a9 | subsection | 70 | 98 | Implementation | If {P}\in \mathbb {R}^{m\times n}, {f}\in \mathbb {R}^{n}, S, {u}\in \mathbb {R}^{m} are suitable discretizations of P, f, \mathcal {S}, u_P, then this would look as follows for L^2-penalized Kaczmarz:f_{\textup {new}} &\in \operatornamewithlimits{argmin}_{f \in L^2(\Omega )} \mathcal {S}( P( f ) ) + \alpha \Vert f - f... | {
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} | 1803.04726 | Generalized SART Methods for Tomographic Imaging | [
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92402131b6173840c177c05077a0640377bf6448 | subsection | 71 | 98 | Implementation | For example, L^q-norms are then identified with q-norms in \mathbb {R}^n, i.e.\int _{\Omega } |f({x})|^q \, = \Vert f \Vert _{L^q}^q \sim \Vert {f} \Vert _q^q = \sum _{i = 1}^n |{f}_i|^q \;\;\;\;\;\text{if}\;\;\;\;\; {f}= ({f}_i)_{i =1}^n \text{ discretizes } f.Discrete and continuous quantities can be related via samp... | {
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dd66d859bb99c57cd6030529b8592d424fd7f2d5 | subsection | 72 | 98 | Unit-projections and precomputations: | Recall that all of the generalized SART formulas in § and § involve (weighted) unit-projections u_{j} or {\tilde{u}_{j}}. Clearly, discrete approximations {u}_{j}, \tilde{{u}}_{j} of these are needed for numerical computations. For general object-domains \Omega , these may be precomputed via {u}_{j} = {P}_j(1_{\mathbb ... | {
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709823ca3377c5ed7fd97fa1c4ad60f909939870 | subsection | 73 | 98 | Robust tomography test case | As a first numerical example, we consider the application of robust reconstrution from tomographic projection data, as introduced in REF . | {
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96854f238278499a6fb4a28977a17d02d9aaab45 | subsection | 74 | 98 | Robust tomography test case | To this end, we compare L^2-regularized Tikhonov-regularization and Kaczmarz-iterationsf^{\textup {Tik}} &= \operatornamewithlimits{argmin}_{f \in L^2(\Omega ) } \mathcal {S}_{{\textup {tot}}}\left( g_{\textup {tot}}^{\textup {obs}}; \, P_{{\textup {tot}}}(f)\right) + \alpha _{\textup {Tik}} \Vert f \Vert _{L^2}^2 \\
f... | {
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} | 1803.04726 | Generalized SART Methods for Tomographic Imaging | [
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56fcb3208c17184a74b0be752a36a37c95980d7e | subsection | 75 | 98 | Robust tomography test case | On the other hand, we design a generalized SART-analogue, alg:RobustSART, by discretizing the update-formula (). Note that the discrete analogue of the optimization problem in () factorizes into a family of scalar problems just like in the continuous setting, see §REF . | {
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} | 1803.04726 | Generalized SART Methods for Tomographic Imaging | [
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e75f0832e35f3fd3c6457ae9fe95796d0bcb5a57 | subsection | 76 | 98 | Robust tomography test case | This enables a highly efficient implementation of this step regardless of the choice s \in \lbrace s_{L^2}, s_{L^1_{\textup {H}}, \nu }, s_{\textup {s-t}, \nu }\rbrace .Robust Tikhonov reconstructionData {g}^{\textup {obs}}_{\textup {tot}}\in \mathbb {R}^{{m_{\textup {proj}}}{N_{\textup {proj}}}}, projector {P}_{{\text... | {
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897705ae2d5063f73f04087e165f9af5bb7eed94 | subsection | 77 | 98 | Robust tomography test case | (b) Simulated parallel-beam data. (c) Plot of the different data-fidelity functions. (f) FBP-reconstruction. (d),(e) Tikhonov-reconstruction (alg:RobustTikh) with L^2- and L^1-Huber-data-fidelity. (g)–(i) SART-reconstruction (alg:RobustSART) with L^2- and L^1-Huber- and Student's-t data-fidelity. The linear color scale... | {
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d97f652ec31c5f5aa5d970f8e74ea8ec94cd374a | subsection | 78 | 98 | Robust tomography test case | The GenSART-results for all(!) data-fidelities are plotted in fig:RobustSART(f)-(h). For comparison, fig:RobustSART(c) also shows a reconstruction by filtered back-projection (FBP), computed using an implementation from the ASTRA-toolbox , with default-parameters.As expected, the results for L^2-data fidelities (fig:Ro... | {
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e2c95c0e8f4f9beb1cbbb7805c4a6137cbd8dc9a | subsection | 79 | 98 | Newton-Kaczmarz-GenSART for experimental XPCT-data | For a second and somewhat more involved numerical test case, we implement the Newton-Kaczmarz-iterations for X-ray phase contrast tomography (XPCT) from §REF . To this end, the obtained GenSART-formula () is discretized, where the gradients \nabla are replaced by finite-difference operators {\nabla }\in \mathbb {R}^{M_... | {
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7166e09c50f6e2d7c2091e55f35f69934279482b | subsection | 80 | 98 | Newton-Kaczmarz-GenSART for experimental XPCT-data | The {U}_{{\nabla }, j} \in \mathbb {R}^{M_{\textup {grad}} \times M_{\textup {grad}}} are diagonal, positive-semidefinite matrices that implement a discrete analogue {\nabla }( {p}) \mapsto {U}_{{\nabla }, j} {\nabla }( {p}) of the multiplication \nabla (p) \mapsto u_j \cdot \nabla (p) for gradients of continuous proje... | {
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0d853146bd9655b57255eaa5f58b429606d1f544 | subsection | 81 | 98 | Newton-Kaczmarz-GenSART for experimental XPCT-data | The obtained Newton-Kaczmarz-GenSART method is summarized in alg:PCTSART, which is notably not limited to XPCT but may be adapted for a wide range of other image-formation operators F.Newton-Kaczmarz-GenSART (for X-ray phase contrast tomography)
Data {g}^{\textup {obs}}_j \in \mathbb {R}^{{m_{\textup {data}}}}, paral... | {
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b795e20a8cc0ea2ee45d3299998be1d018a739dd | subsection | 82 | 98 | Newton-Kaczmarz-GenSART for experimental XPCT-data | The 3D tomographic data, measured at the synchrotron light-source PETRAIII (see for experimental details), is visualized by orthoslices in fig:PCTSART(a).
[Figure: X-ray phase contrast tomography test case: (a) Orthoslice-plot of the 3D tomographic data {g}^{{\textup {obs}}} = ({g}^{{\textup {obs}}}_1, \ldots , {g}^{{\... | {
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0593945e253fe556ba42ceb1e0ea4cb5b04b6a89 | subsection | 83 | 98 | Conclusions | In this work, efficient solution formulas have been proposed for the computation of regularized Kaczmarz-iterations (also known as “Tikhonov-Kaczmarz” or “incremental proximal iterations”) for tomographic reconstruction. By their structural analogy and similar computational efficiency to classical SART-iterations , the... | {
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"doi": "10.1016/0161-7346(84)90008-7",
"end": 375,
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"raw": "A. H. Andersen and A. C. Kak, Simultaneous algebraic reconstruction technique (SART): a superior implementation of the ART algorithm, Ultr... | 1803.04726 | Generalized SART Methods for Tomographic Imaging | [
"Simon Maretzke"
] | [
"math.NA",
"physics.med-ph"
] | 2,018 | en | Mathematics | [
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119b26d52a7eb5e4d0706681541684eb48ff210f | subsection | 84 | 98 | Geometry of the Projectors | [Proof of thm:AdjProjClosedRange:]
We show that B_{\textup {iso}}: {L^2({\mathbb {D}\!_{P}})}\rightarrow L^2(\Omega ); \; p \mapsto {w_P}\cdot {\tilde{P}}^{\textup {B}}({\tilde{u}_P}^{-1/2} \cdot p) is well-defined and isometric.
Let p \in {L^2({\mathbb {D}\!_{P}})} be arbitrary. For P= {P}, we have\Vert B_{\textup {is... | {
"cite_spans": []
} | 1803.04726 | Generalized SART Methods for Tomographic Imaging | [
"Simon Maretzke"
] | [
"math.NA",
"physics.med-ph"
] | 2,018 | en | Mathematics | [
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e7e5dbf66e98ee7a1b84d9a97b598b04a495b083 | subsection | 85 | 98 | Geometry of the Projectors | \underbrace{ \Big ( {\textstyle \int }_{{L^{P}_{{x}_\perp }}} 1 \, z̥ \Big )}_{= P(1_\Omega ) ({x}_\perp ) = u_{P} ({x}_\perp ) } \!\!\!\! _\perp \\&= \int _{\Omega _{P}} \left( u_{P}({x}_\perp )/ \tilde{u}_{P}({x}_\perp ) \right) \cdot |p ({x}_\perp )|^2 \, _\perp \stackrel{\tilde{u}_{P} = u_{P}}{=} \Vert p \Vert _{{L... | {
"cite_spans": []
} | 1803.04726 | Generalized SART Methods for Tomographic Imaging | [
"Simon Maretzke"
] | [
"math.NA",
"physics.med-ph"
] | 2,018 | en | Mathematics | [
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da0549b7b252c6b1370b4a096e1e7d5e1db726ae | subsection | 86 | 98 | Geometry of the Projectors | Hence, P_{\textup {iso}}= B_{\textup {iso}}^\ast :L^2(\Omega ) \rightarrow {L^2({\mathbb {D}\!_{P}})} is bounded with norm \Vert P_{\textup {iso}} \Vert = \Vert B_{\textup {iso}} \Vert = 1. By the isometry-property, P_{\textup {iso}}^\ast = B_{\textup {iso}} has closed range. According to the closed range theorem, the ... | {
"cite_spans": []
} | 1803.04726 | Generalized SART Methods for Tomographic Imaging | [
"Simon Maretzke"
] | [
"math.NA",
"physics.med-ph"
] | 2,018 | en | Mathematics | [
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6dcaad9ed58a089295988c22454d563c7bf80017 | subsection | 87 | 98 | Projectors and Gradients | [Proof of lem:ProjGradient:]
Let f \in {{C}^\infty _{\textup {c}} ( \Omega ) } be smooth and compactly supported. Then f can be identified with a function in {{C}^\infty _{\textup {c}} ( \mathbb {R}^3 ) } by simply extending it with 0 outside \Omega (notably, this would not be true if we only assumed f \in {{C}^\infty ... | {
"cite_spans": []
} | 1803.04726 | Generalized SART Methods for Tomographic Imaging | [
"Simon Maretzke"
] | [
"math.NA",
"physics.med-ph"
] | 2,018 | en | Mathematics | [
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8fd3766ffef2bfce3ad0b60a4822184356e96fc4 | subsection | 88 | 98 | Projectors and Gradients | For the cone-beam case P= {D}, the gradient can be expressed in polar coordinates: if f^{(\textup {p})} is defined by f^{(\textup {p})}( {\varphi }, t ) := f( t {\varphi }), then\nabla f (t {\varphi }) &= t^{-1} \nabla \!_{{\varphi }} f^{(\textup {p})} ({\varphi }, t) + {\varphi }\partial _t f^{(\textup {p})} ({\varphi... | {
"cite_spans": []
} | 1803.04726 | Generalized SART Methods for Tomographic Imaging | [
"Simon Maretzke"
] | [
"math.NA",
"physics.med-ph"
] | 2,018 | en | Mathematics | [
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3655a27251f6b691679df50faa4de287f38e4034 | subsection | 89 | 98 | Projectors and Gradients | Since P_{\textup {iso}}: L^2(\Omega ) \rightarrow {L^2({\mathbb {D}\!_{P}})};\; p \mapsto {\tilde{u}_P}^{-1/2} \cdot P(p) is bounded with norm 1 according to thm:AdjProjClosedRange, this furthermore implies that\big \Vert {\tilde{u}_P}^{-1/2}\cdot \nabla _{\mathbb {D}} P(f) \big \Vert _{L^2} &= \big \Vert {\tilde{u}_P}... | {
"cite_spans": []
} | 1803.04726 | Generalized SART Methods for Tomographic Imaging | [
"Simon Maretzke"
] | [
"math.NA",
"physics.med-ph"
] | 2,018 | en | Mathematics | [
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f5343f3e3e382559893869e2e3b6a0d6f307767c | subsection | 90 | 98 | Projectors and Gradients | By inserting the expressions for \nabla \!_P in the parallel- and cone-beam geometry, this yields for almost all ({x}_\perp , z), t{\varphi }\in \Omega\nabla \big ( {P}^{\textup {B}}(p) \big ) ({x}_\perp , z) &= \nabla \!_P\big ( {P}^{\textup {B}}(p) \big ) ({x}_\perp , z) \stackrel{(\ref {eq:DefBackProj})}{=} \nabla _... | {
"cite_spans": []
} | 1803.04726 | Generalized SART Methods for Tomographic Imaging | [
"Simon Maretzke"
] | [
"math.NA",
"physics.med-ph"
] | 2,018 | en | Mathematics | [
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b3e5e1d5769f9cc4d69b2176e731af1693851659 | subsection | 91 | 98 | Projectors and Gradients | The equality (REF ) furthermore shows the equivalencesP^{\textup {B}}(p) \in W^{1,2}(\Omega ) \; \Leftrightarrow \; \nabla \left( P^{\textup {B}}(p) \right) \in L^2(\Omega ) \; \Leftrightarrow \; u_P^{1/2} \cdot \nabla _{\mathbb {D}} (p) \in {L^2({\mathbb {D}\!_{P}})}.By continuity of \nabla : W^{1,2}(\Omega ) \rightar... | {
"cite_spans": []
} | 1803.04726 | Generalized SART Methods for Tomographic Imaging | [
"Simon Maretzke"
] | [
"math.NA",
"physics.med-ph"
] | 2,018 | en | Mathematics | [
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c92ce66697bb993c94f7c3877e9aef145484c4d7 | subsection | 92 | 98 | Projectors and Gradients | Using the expressions for \nabla \big ( P^{\textup {B}}(p) \big ) derived above, we obtain&\big \langle \nabla \left( P^{\textup {B}}(p)\right) , \nabla f \big \rangle _{L^2} = \big \langle \nabla \!_P\left( P^{\textup {B}}(p)\right) , \nabla f \big \rangle _{L^2} = \big \langle \nabla \!_P\left( P^{\textup {B}}(p)\rig... | {
"cite_spans": []
} | 1803.04726 | Generalized SART Methods for Tomographic Imaging | [
"Simon Maretzke"
] | [
"math.NA",
"physics.med-ph"
] | 2,018 | en | Mathematics | [
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c38ca8f52ce02270fdfc370260aa7d6e6e85ec06 | subsection | 93 | 98 | Admissibility of | [Proof of thm:admissibleLq:]
Let p \in {L^2({\mathbb {D}\!_{P}})} and f_0 \in \operatornamewithlimits{kern}({P}_{\textup {iso}}) be arbitrary. If \mathcal {R}(f_{\textup {ref}} + {P}_{\textup {iso}}^\ast (p) + f_0) = \Vert {P}_{\textup {iso}}^\ast (p) + f_0 \Vert _{L^q}^q = \infty , then (REF ) trivially holds true. He... | {
"cite_spans": []
} | 1803.04726 | Generalized SART Methods for Tomographic Imaging | [
"Simon Maretzke"
] | [
"math.NA",
"physics.med-ph"
] | 2,018 | en | Mathematics | [
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153b42c4b992503217b319f84151672e75bd11a5 | subsection | 94 | 98 | Admissibility of | Since f = {P}_{\textup {iso}}^\ast (p) + f_0 with f_0 \in \operatornamewithlimits{kern}({P}_{\textup {iso}}), this implies\int _{{L^{P}_{{x}_\perp }}} \left| f({x}_\perp , z) \right|^q \, z̥ \ge u_{P}({x}_\perp )^{1-q} \left| {P}(f) ({x}_\perp ) \right|^q = u_{P}({x}_\perp )^{1-q} \big | {P}{P}_{\textup {iso}}^\ast (p)... | {
"cite_spans": []
} | 1803.04726 | Generalized SART Methods for Tomographic Imaging | [
"Simon Maretzke"
] | [
"math.NA",
"physics.med-ph"
] | 2,018 | en | Mathematics | [
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7ee6fbca56c8d149ea2bfbcefde3de3720b336bc | subsection | 95 | 98 | Admissibility of | Since {P}{P}_{\textup {iso}}^\ast (p) = u_{P}^{1/2} \cdot {P}_{\textup {iso}}{P}_{\textup {iso}}^\ast (p) = u_{P}^{1/2} \cdot p by thm:AdjProjClosedRange, (REF ) furthermore yields\mathcal {R}(f_{\textup {ref}} + {P}_{\textup {iso}}^\ast (p)) &= \int _{\mathbb {R}^2} \int _{L^{P}_{{x}_\perp }}|{P}_{\textup {iso}}^\ast ... | {
"cite_spans": []
} | 1803.04726 | Generalized SART Methods for Tomographic Imaging | [
"Simon Maretzke"
] | [
"math.NA",
"physics.med-ph"
] | 2,018 | en | Mathematics | [
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194777a4ca5448b59dbeb6dcad3f51485c8be2d4 | subsection | 96 | 98 | Poisson-noise-adapted data fidelity | In the following, it is shown that the log-likelihood for Poisson-noisy data given in (REF ) can be approximated by the functional in (REF ) if variations of the true data g_j within the supports of the \omega _i are negligible. Specifically, we assume that g_j is “constant enough” in \operatornamewithlimits{supp}(\ome... | {
"cite_spans": []
} | 1803.04726 | Generalized SART Methods for Tomographic Imaging | [
"Simon Maretzke"
] | [
"math.NA",
"physics.med-ph"
] | 2,018 | en | Mathematics | [
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3b9e8aa2c85e5c2e91307e54f37e8999ab915d46 | subsection | 97 | 98 | Poisson-noise-adapted data fidelity | Inserting this approximation into (REF ) yields\mathcal {S}^{\textup {Poi}} \left( g^{\textup {obs}}_j; g_j \right) &\approx \sum _{i = 1}^{m_{\textup {proj}}}\int _{\mathbb {D}} t g_j \omega _i \, x̥ - g^{\textup {obs}}_{ji} \cdot \left( \frac{\int _{\mathbb {D}} \ln \left( g_j \right) t \omega _i \, x̥}{ \int _{\math... | {
"cite_spans": []
} | 1803.04726 | Generalized SART Methods for Tomographic Imaging | [
"Simon Maretzke"
] | [
"math.NA",
"physics.med-ph"
] | 2,018 | en | Mathematics | [
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124e5fa0426dcc3a223fbae727a04c5ece5128c6 | abstract | 0 | 25 | Abstract | Heretofore, global burned area (BA) products are only available at coarse
spatial resolution, since most of the current global BA products are produced
with the help of active fire detection or dense time-series change analysis,
which requires very high temporal resolution. In this study, however, we focus
on automated... | {
"cite_spans": []
} | 1805.02579 | 30m resolution Global Annual Burned Area Mapping based on Landsat images
and Google Earth Engine | [
"Tengfei Long",
"Zhaoming Zhang",
"Guojin He",
"Weili Jiao",
"Chao Tang",
"Bingfang Wu",
"Xiaomei Zhang",
"Guizhou Wang",
"Ranyu Yin"
] | [
"cs.CV"
] | 2,018 | en | Computer Science | [
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e797272e3ee6806e2486ea4557f4e69508342653 | subsection | 1 | 25 | Introduction | Accurate and complete data on fire locations and burned areas (BA) are important for a variety of applications including quantifying trends and patterns of fire occurrence and assessing the impacts of fires on a range of natural and social systems, e.g. simulating carbon emissions from biomass burning . Remotely sensed... | {
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and Google Earth Engine | [
"Tengfei Long",
"Zhaoming Zhang",
"Guojin He",
"Weili Jiao",
"Chao Tang",
"Bingfang Wu",
"Xiaomei Zhang",
"Guizhou Wang",
"Ranyu Yin"
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"cs.CV"
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30cd11725378cdea52f6ea96a25ba5875b981759 | subsection | 2 | 25 | Introduction | One of the difficulties to produce Landsat based burned area products is that the traditional approaches successfully applied to extract global burned area from MODIS, VEGETATION, etc. don’t work well due to the limited temporal resolution of the Landsat sensors. Moreover, the analysis of post-fire reflectance may be e... | {
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"raw": "Alonso-Canas, I., Chuvieco, E., 2015. Global burned area mapping from ENVISAT-MERIS and MODIS active fire data. Remote Sensing of Environment... | 1805.02579 | 30m resolution Global Annual Burned Area Mapping based on Landsat images
and Google Earth Engine | [
"Tengfei Long",
"Zhaoming Zhang",
"Guojin He",
"Weili Jiao",
"Chao Tang",
"Bingfang Wu",
"Xiaomei Zhang",
"Guizhou Wang",
"Ranyu Yin"
] | [
"cs.CV"
] | 2,018 | en | Computer Science | [
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e125aed813592a9fd923fd988b367e0977509518 | subsection | 3 | 25 | Sampling design | The spectral characteristics of burned areas vary in complex ways for different ecosystems, fire regimes and climatic conditions. In terms of guaranteeing the accuracy of global burned area map and also the completeness of quality assessment, a stratified random sampling method, , was used to generate two sets of sites... | {
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"arxiv_id": "",
"doi": "10.1016/j.rse.2015.01.005",
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and Google Earth Engine | [
"Tengfei Long",
"Zhaoming Zhang",
"Guojin He",
"Weili Jiao",
"Chao Tang",
"Bingfang Wu",
"Xiaomei Zhang",
"Guizhou Wang",
"Ranyu Yin"
] | [
"cs.CV"
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587e5971528d07c055003aa4be296f902cad1a39 | subsection | 4 | 25 | Training dataset | In terms of analyzing the characteristics of burned areas in Landsat images, 120 Landsat-8 image scenes were chosen according to the WRS-II frames generated by stratified random sampling in subsec:sampling. All the Landsat-8 images used in this study were acquired from datasets of USGS Landsat-8 Surface Reflectance Tie... | {
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"arxiv_id": "",
"doi": "10.1016/j.rse.2016.04.008",
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"raw": "Vermote, E., Justice, C., Claverie, M., Franch, B., 2016. Preliminary analysis of the performance of the landsat 8/OLI land surface reflectan... | 1805.02579 | 30m resolution Global Annual Burned Area Mapping based on Landsat images
and Google Earth Engine | [
"Tengfei Long",
"Zhaoming Zhang",
"Guojin He",
"Weili Jiao",
"Chao Tang",
"Bingfang Wu",
"Xiaomei Zhang",
"Guizhou Wang",
"Ranyu Yin"
] | [
"cs.CV"
] | 2,018 | en | Computer Science | [
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dd9d95b5fa80ad5226196ba46e48ee90e72814ef | subsection | 5 | 25 | Sensitive features for burned surfaces | Figure REF shows the statistical mean reflectance (with standard deviations) of burned samples in Landsat 8 bands.
[Figure: Means and standard deviations of land surface reflectance of burned Landsat-8 pixels in different bands.]Burned areas are characterized by deposits of charcoal, ash and fuel, and the reflectance o... | {
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and Google Earth Engine | [
"Tengfei Long",
"Zhaoming Zhang",
"Guojin He",
"Weili Jiao",
"Chao Tang",
"Bingfang Wu",
"Xiaomei Zhang",
"Guizhou Wang",
"Ranyu Yin"
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ee98a2d6b9c9431f448cda158f1ea829264295a1 | subsection | 6 | 25 | Burned area mapping via GEE | In this work, annual burned area map is defined as spatial extent of fires that occurs within a whole year and not of fires that occurred in previous years. Therefore, global 30-meter resolution annual burned areas mapping needs to utilize dense time-series Landsat images, and the pipeline of annual burned area mapping... | {
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} | 1805.02579 | 30m resolution Global Annual Burned Area Mapping based on Landsat images
and Google Earth Engine | [
"Tengfei Long",
"Zhaoming Zhang",
"Guojin He",
"Weili Jiao",
"Chao Tang",
"Bingfang Wu",
"Xiaomei Zhang",
"Guizhou Wang",
"Ranyu Yin"
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1b6efd1fac4d4cbec4ec6aa24997da7e8b18b0e3 | subsection | 7 | 25 | Model Training | The random forest (RF) algorithm provided by GEE were applied to train a decision forest classifier, and the global training data consisted of 6735 burned and 6146 unburned samples which were manually collected from 120 Landsat scenes generated by stratified random sampling (in subsec:sampling and subsec:training). Ran... | {
"cite_spans": []
} | 1805.02579 | 30m resolution Global Annual Burned Area Mapping based on Landsat images
and Google Earth Engine | [
"Tengfei Long",
"Zhaoming Zhang",
"Guojin He",
"Weili Jiao",
"Chao Tang",
"Bingfang Wu",
"Xiaomei Zhang",
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"Ranyu Yin"
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f4f6e5d6c4630004dab130e2de6e60f7ec1ec187 | subsection | 8 | 25 | Per-pixel Processing | In this step, Landsat surface reflectance collections from GEE, which consist of all the available Landsat scenes, were employed for dense time-series processing. At a pixel, the occurrence of a single Landsat satellite could be more than 20 or 40 times (considering the overlap between adjacent paths) within a year, an... | {
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and Google Earth Engine | [
"Tengfei Long",
"Zhaoming Zhang",
"Guojin He",
"Weili Jiao",
"Chao Tang",
"Bingfang Wu",
"Xiaomei Zhang",
"Guizhou Wang",
"Ranyu Yin"
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19f45c573398010f8616a576373e8c27590a4aa1 | subsection | 9 | 25 | Per-pixel Processing | This constraint is useful to exclude false detections with periodic variation of NBR and NDVI, such as mountain shadows, burned-like soil in deciduous season, snow melting and flooding.
t_1>t_2 or t_2-t_1>T_{DAY}, the most flourishing date of vegetation should be earlier than the burning date, or the lagged days shoul... | {
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and Google Earth Engine | [
"Tengfei Long",
"Zhaoming Zhang",
"Guojin He",
"Weili Jiao",
"Chao Tang",
"Bingfang Wu",
"Xiaomei Zhang",
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"Ranyu Yin"
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19b5d5f394389f44f5a89463a337fab5fecca62a | subsection | 10 | 25 | Burned Area Shaping | In this step, a region growing process was employed to shape the burned areas. Region growing has proved to be necessary for BA mapping in many studies , , , because spectral based methods sometimes give ambiguous evidence (i.e. spectral overlapping between burned areas and unrelated phenomena with similar spectral cha... | {
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and Google Earth Engine | [
"Tengfei Long",
"Zhaoming Zhang",
"Guojin He",
"Weili Jiao",
"Chao Tang",
"Bingfang Wu",
"Xiaomei Zhang",
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"Ranyu Yin"
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a93508cd4b3116b711b95466bb70ae1fe4510948 | subsection | 11 | 25 | Product description | Employing the proposed approach, we produced the global annual burned area map of 2015 (GABAM 2015), which was projected in a Geographic (Lat/Long) projection at 0.00025^\circ (approximately 30 meters) resolution, with the WGS84 horizontal datum and the EGM96 vertical datum. The result consists of 10x10 degree tiles sp... | {
"cite_spans": []
} | 1805.02579 | 30m resolution Global Annual Burned Area Mapping based on Landsat images
and Google Earth Engine | [
"Tengfei Long",
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"Weili Jiao",
"Chao Tang",
"Bingfang Wu",
"Xiaomei Zhang",
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"Ranyu Yin"
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72f39fff8152c90d478a6d35a667b889b226dc65 | subsection | 12 | 25 | Data preparing | As 30m resolution global burned area products are currently not available, we made a comparison between GABAM 2015 and the Fire_cci version 5.0 products (spatial resolution is approximately 250 meters) , which are based on MODIS on board the Terra satellite.
The monthly Fire_cci pixel BA products of 2015 were composite... | {
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"raw": "Pettinari, M., Chuvieco, E., 2018. Esa cci ecv fire disturbance: D3.3.3 product user guide - modis, version 1.0. ESA Fire-CCI project (http://www.esa-firecci.org/documents) .",
"source_ref_id"... | 1805.02579 | 30m resolution Global Annual Burned Area Mapping based on Landsat images
and Google Earth Engine | [
"Tengfei Long",
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"Guojin He",
"Weili Jiao",
"Chao Tang",
"Bingfang Wu",
"Xiaomei Zhang",
"Guizhou Wang",
"Ranyu Yin"
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761a4cc06d14f86bdcb0191811fd090d8e17926e | subsection | 13 | 25 | Visually comparing | Figure REF shows an example of the two annual pixel BA products, and it can be seen that both products correctly detected the BAs in Landsat image (Figure REF ), yet the BAs in Figure REF occupy more pixels than those in Figure REF . Due to the limitation in spatial resolution of the input sensor of the Fire_cci BA pro... | {
"cite_spans": []
} | 1805.02579 | 30m resolution Global Annual Burned Area Mapping based on Landsat images
and Google Earth Engine | [
"Tengfei Long",
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"Guojin He",
"Weili Jiao",
"Chao Tang",
"Bingfang Wu",
"Xiaomei Zhang",
"Guizhou Wang",
"Ranyu Yin"
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1bf354d8c95081bedb9aac58fc6b423c52436678 | subsection | 14 | 25 | Global grid map | Figure REF illustrates the GABAM and Fire_cci annual grid composition of BA, consisting of percentage of burned pixels in each 0.25^{\circ }\times 0.25^{\circ } grid. Figure REF and Figure REF show similar global distributions of BA density.
[Figure: Global distribution of burned area density (percentage of burned pixe... | {
"cite_spans": []
} | 1805.02579 | 30m resolution Global Annual Burned Area Mapping based on Landsat images
and Google Earth Engine | [
"Tengfei Long",
"Zhaoming Zhang",
"Guojin He",
"Weili Jiao",
"Chao Tang",
"Bingfang Wu",
"Xiaomei Zhang",
"Guizhou Wang",
"Ranyu Yin"
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867f3afba81b950305d12e335dc63a0ce904fe07 | subsection | 15 | 25 | Regression analysis | Figure REF shows the proportion of BA in 0.25^{\circ }\times 0.25^{\circ } grids of different land cover categories in Table REF , for Fire_cci product (x-axis) and GABAM 2015 (y-axis), and regression analysis was also performed between the two products, providing a regression line (expressed as the slope and the inter... | {
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"raw": "Chuvieco, E., Padilla, M., Hantson, S., Theis, R., Snadow, C., 2011. Esa cci ecv fire disturbance-product validation plan (v3. 1). ESA Fire-CCI project (http://www. esa-fire-cci. org/) .",
"s... | 1805.02579 | 30m resolution Global Annual Burned Area Mapping based on Landsat images
and Google Earth Engine | [
"Tengfei Long",
"Zhaoming Zhang",
"Guojin He",
"Weili Jiao",
"Chao Tang",
"Bingfang Wu",
"Xiaomei Zhang",
"Guizhou Wang",
"Ranyu Yin"
] | [
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] | 2,018 | en | Computer Science | [
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2d32c7dca3888433c90da83c31ee43aba7f47dc8 | subsection | 16 | 25 | Data sources | Accuracy assessment was carried out according to the 80 validation sites which were created in subsec:sampling, and the reference data were selected in these sites from multiple data sources, including fire perimeter datasets and satellite images. Commonly, when satellite data are used as reference data, they should ha... | {
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"raw": "Boschetti, L., Roy, D., Justice, C., 2009. International global burned area satellite product validation protocol (part i–production and standardization of validation reference data), in: CEOS-CalVa... | 1805.02579 | 30m resolution Global Annual Burned Area Mapping based on Landsat images
and Google Earth Engine | [
"Tengfei Long",
"Zhaoming Zhang",
"Guojin He",
"Weili Jiao",
"Chao Tang",
"Bingfang Wu",
"Xiaomei Zhang",
"Guizhou Wang",
"Ranyu Yin"
] | [
"cs.CV"
] | 2,018 | en | Computer Science | [
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a25faa1e5bd097c130ea7c23a18bacbaa7f9e47e | subsection | 17 | 25 | Reference data generation | In each validation site, all the available image scenes (LC8https://earthexplorer.usgs.gov
, CB4http://www.dgi.inpe.br/catalogo/ or GF1http://218.247.138.119:7777/DSSPlatform/productSearch.html) acquired in 2015 were used. LC8 images were ortho-rectified surface reflectance products, CB4 images were ortho products, and... | {
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and Google Earth Engine | [
"Tengfei Long",
"Zhaoming Zhang",
"Guojin He",
"Weili Jiao",
"Chao Tang",
"Bingfang Wu",
"Xiaomei Zhang",
"Guizhou Wang",
"Ranyu Yin"
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54b876071dcf68afd7e0de2f11442113fb5aff40 | subsection | 18 | 25 | Validation results | To assess the accuracy of GABAM 2015, a cross tabulation between the pixels assigned by in our BA product and in the reference data was computed to produce the confusion matrix for each validation site. Afterwards, the global cross tabulation (Table REF ) was generated by averaging all the cross tabulations.
[Table: C... | {
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"raw": "Pontius, R.G., Millones, M., 2011. Death to kappa: birth of quantity disagreement and allocation disagreement for accuracy assessment. Int... | 1805.02579 | 30m resolution Global Annual Burned Area Mapping based on Landsat images
and Google Earth Engine | [
"Tengfei Long",
"Zhaoming Zhang",
"Guojin He",
"Weili Jiao",
"Chao Tang",
"Bingfang Wu",
"Xiaomei Zhang",
"Guizhou Wang",
"Ranyu Yin"
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d92f88578129ea0c5827b2a7fb65fdf57b3f8ce3 | subsection | 19 | 25 | Discussion | Different from the satellite images of coarse spatial resolution, the temporal resolution of Landsat images is not high enough to capture the short-term events on the earth. Specifically, the general revisit period of Landsat image is more than 10 days, hence active fire will be observed by Landsat satellite with proba... | {
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and Google Earth Engine | [
"Tengfei Long",
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"Guojin He",
"Weili Jiao",
"Chao Tang",
"Bingfang Wu",
"Xiaomei Zhang",
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3aedb0f57c56017a79f5689a6a2804d4b0dc6798 | subsection | 20 | 25 | BA in Agriculture land | It is difficult to detect BA in cropland with high confidence (low commission error and low omission error) from satellite images:A lot of croplands have comparable spectral characteristics to burned areas when harvested or ploughed.
The temporal behaviour of harvest or burning of cropland is similar to that of grassl... | {
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} | 1805.02579 | 30m resolution Global Annual Burned Area Mapping based on Landsat images
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6bd03376a24d9ca7f5687d51176c9ec712009ab9 | subsection | 21 | 25 | Omission of observations | Using Landsat images as input data for GABAM, the number of valid observations is a limiting factor for detecting fires, since the active- or post-fire evidence may be omitted or weaken due to the temporal gaps caused by temporal resolution as well as cloud contamination. Especially in Tropical regions, where vegetatio... | {
"cite_spans": []
} | 1805.02579 | 30m resolution Global Annual Burned Area Mapping based on Landsat images
and Google Earth Engine | [
"Tengfei Long",
"Zhaoming Zhang",
"Guojin He",
"Weili Jiao",
"Chao Tang",
"Bingfang Wu",
"Xiaomei Zhang",
"Guizhou Wang",
"Ranyu Yin"
] | [
"cs.CV"
] | 2,018 | en | Computer Science | [
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96a9a9eca5b7a47af9471d9f757774f7bc22d14e | subsection | 22 | 25 | Validation | For satellite data product validation, a commonly used method is to employ higher spatial resolution satellite data. For example, in order to validate MODIS derived data product (1 km spatial resolution), Landsat satellite data is commonly used. In this study, however, Landsat images were used as the main reference sou... | {
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and Google Earth Engine | [
"Tengfei Long",
"Zhaoming Zhang",
"Guojin He",
"Weili Jiao",
"Chao Tang",
"Bingfang Wu",
"Xiaomei Zhang",
"Guizhou Wang",
"Ranyu Yin"
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2ec955884e1c6d54a214d6bdb5875fbe8bc89c8e | subsection | 23 | 25 | Conclusions | An automated pipeline for generating 30m resolution global-scale annual burned area map utilizing Google Earth Engine was proposed in this study. Different from the previous coarse resolution global burned area products, GABAM 2015, a novel 30-m resolution global annual burned area map of 2015 year, was derived from al... | {
"cite_spans": []
} | 1805.02579 | 30m resolution Global Annual Burned Area Mapping based on Landsat images
and Google Earth Engine | [
"Tengfei Long",
"Zhaoming Zhang",
"Guojin He",
"Weili Jiao",
"Chao Tang",
"Bingfang Wu",
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fd69420c83489e51fb50bcd7639c2b292ee407da | subsection | 24 | 25 | Examples of validation sites | Figure REF –REF show some examples of site validation, and Table REF summarizes the information of these validation sites, including the location, source of reference data, commission error, omission error and overall accuracy.
[Table: Information of site validation examples.][Figure: Example of validation using GF-1 i... | {
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} | 1805.02579 | 30m resolution Global Annual Burned Area Mapping based on Landsat images
and Google Earth Engine | [
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"Chao Tang",
"Bingfang Wu",
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0107f92617fa279303e97dde002c37baf644720f | abstract | 0 | 23 | Abstract | An adjoint-based optimization is applied to study the thrust performance of a
pitching-rolling ellipsoidal plate in a uniform stream at Reynolds number 100.
To achieve the highest thrust, the optimal kinematics of pitching-rolling
motion is sought in a large control space including the pitching amplitude, the
rolling a... | {
"cite_spans": []
} | 1809.04100 | Adjoint-based optimization for thrust performance of a three-dimensional
pitching-rolling plate | [
"Min Xu",
"Mingjun Wei",
"Chengyu Li",
"Haibo Dong"
] | [
"physics.flu-dyn"
] | 2,018 | en | Physics | [
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9a5d412ee472c51031b613719cc4557749188e5d | subsection | 1 | 23 | Nomenclature | @l @ = l@
\mathbf {q} primary variable\mathbf {u} velocityp pressure\rho densityt timex Cartesian coordinate\Omega fluid domain\mathcal {S} solid boundary\Gamma _\infty far-field boundaryn normal direction\mathbf {S} solid boundary location\mathbf {V} velocity at the solid boundary\mathbf {U} velocity at the far-f... | {
"cite_spans": []
} | 1809.04100 | Adjoint-based optimization for thrust performance of a three-dimensional
pitching-rolling plate | [
"Min Xu",
"Mingjun Wei",
"Chengyu Li",
"Haibo Dong"
] | [
"physics.flu-dyn"
] | 2,018 | en | Physics | [
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e739827e3e2d0270d06105e57f9316ffb8fe6da9 | subsection | 2 | 23 | Introduction | The mechanism of flapping motion provides an energy-efficient way for bio-inspired propulsion and is the most common way that has been adopted by flying and swimming animals, such as insects, birds, and fishes. In comparison with conventional man-made designs, flapping propulsors used by natural flyers/swimmers show ma... | {
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pitching-rolling plate | [
"Min Xu",
"Mingjun Wei",
"Chengyu Li",
"Haibo Dong"
] | [
"physics.flu-dyn"
] | 2,018 | en | Physics | [
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e5ad22458a82a794e9edbdf8c746bf4e26c92b86 | subsection | 3 | 23 | Introduction | In the current work, we take the continuous approach, considering its advantage of simplicity and clarity in the governing equation over the discrete approach , . Jameson used a continuous adjoint approach to optimize aerodynamic shape designs in both inviscid and viscous compressible steady flow in a fixed-domain setu... | {
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"doi": "10.2514/6.2000-667",
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"raw": "Nadarajah, S., and Jameson, A., “A comparison of the continuous and discrete adjoint approach to automatic aerodynamic optimization,” AIAA Paper 200... | 1809.04100 | Adjoint-based optimization for thrust performance of a three-dimensional
pitching-rolling plate | [
"Min Xu",
"Mingjun Wei",
"Chengyu Li",
"Haibo Dong"
] | [
"physics.flu-dyn"
] | 2,018 | en | Physics | [
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f086ccecede5f8aa2392a50f58439204fe39c523 | subsection | 4 | 23 | Methodology | Non-cylindrical calculus is applied to formulate the adjoint equation system in a morphing domain for the current study. The basics of theoretical derivation and numerical implementation of the approach are provided here, while more details may be referred to earlier works , , , . | {
"cite_spans": [
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pitching-rolling plate | [
"Min Xu",
"Mingjun Wei",
"Chengyu Li",
"Haibo Dong"
] | [
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] | 2,018 | en | Physics | [
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292a340638fd210f802d4fb817143efaae804dba | subsection | 5 | 23 | Governing equation and cost function | The flow is described by the incompressible Navier-Stokes equations, where all the variables are non-dimensionalized accordingly by
the spanwise wing length, incoming velocity, and fluid density, as\begin{aligned}\mathcal {N}({\mathbf {q}}) &= \mathbf {0} \qquad \mathrm {in}\;\; \Omega ,\\
\mathbf {u} &= \mathbf {V} \q... | {
"cite_spans": []
} | 1809.04100 | Adjoint-based optimization for thrust performance of a three-dimensional
pitching-rolling plate | [
"Min Xu",
"Mingjun Wei",
"Chengyu Li",
"Haibo Dong"
] | [
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a87318c54e0fbcb5fdca326819863835ec84651c | subsection | 6 | 23 | Governing equation and cost function | The wing motion is prescribed by flapping motion with a set of control parameters \phi , therefore solid boundary location and velocity are functions of control parameters and can be expressed as: S_i=S_i(\phi , t) and V_i=V_i(\phi , t).To optimize the thrust performance, the negative of thrust coefficient is chosen to... | {
"cite_spans": []
} | 1809.04100 | Adjoint-based optimization for thrust performance of a three-dimensional
pitching-rolling plate | [
"Min Xu",
"Mingjun Wei",
"Chengyu Li",
"Haibo Dong"
] | [
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16ccf911adc251200c0e98b68aac79bca4dadeec | subsection | 7 | 23 | Linearized perturbation equation and perturbed cost function | It has been demonstrated that non-cylindrical calculus has great advantages in efficiency and simplicity in the derivation of an adjoint equation in continuous form for moving boundary problems , , . Following the same derivation, we can easily derive the linearized perturbation equation for the Navier-Stokes equations... | {
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pitching-rolling plate | [
"Min Xu",
"Mingjun Wei",
"Chengyu Li",
"Haibo Dong"
] | [
"physics.flu-dyn"
] | 2,018 | en | Physics | [
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af05f9577ab8fdb7954e7641e0013f8f3ac4d9e9 | subsection | 8 | 23 | Adjoint equation and gradient calculation | Adjoint variables \mathbf {q}=[p^\ast \; \mathbf {u}^\ast ]^T are introduced as Lagrange multipliers to impose the flow equations, so that we obtain the derivative of the enhanced cost function,\begin{aligned}\mathcal {J}^\prime = -\frac{1}{T D_0} \left(\int _0^T \int _{\mathcal {S}} \left( {\sigma _1}^\prime \cdot {\m... | {
"cite_spans": []
} | 1809.04100 | Adjoint-based optimization for thrust performance of a three-dimensional
pitching-rolling plate | [
"Min Xu",
"Mingjun Wei",
"Chengyu Li",
"Haibo Dong"
] | [
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c07db1f6b6894f1fc33d473377e618cb16fa4b1c | subsection | 9 | 23 | Adjoint equation and gradient calculation | \int _{\Omega } u^\ast _j u^\prime _j \text{d}\Omega \right|_{t=0}^{t=T} + b_{\infty } + \int _0^T \int _{\Omega } (u^\ast _i+\delta _{1i})\sigma ^\prime _{ij}n_j \text{d}s\text{d}t\\
&- \int _0^T \int _{\Omega } u_i^\prime (\sigma _{ij}^\ast n_j+u_j^\ast u_j n_i) \text{d}s\text{d}t + \int _0^T \int _{\Omega } Z_{k} \f... | {
"cite_spans": []
} | 1809.04100 | Adjoint-based optimization for thrust performance of a three-dimensional
pitching-rolling plate | [
"Min Xu",
"Mingjun Wei",
"Chengyu Li",
"Haibo Dong"
] | [
"physics.flu-dyn"
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b8ff91ab8f12b2fb93a8f5e4809f5820707f8a49 | subsection | 10 | 23 | Adjoint equation and gradient calculation | When both the flow and adjoint solutions are periodic, the gradient of cost function \mathcal {J} with respect to controls \mathbf {\phi } is then given by,\begin{aligned}g_l= \frac{\partial \mathcal {J}}{\partial \phi _l} = -\frac{1}{TD_0} \int _0^T \int _{\mathcal {S}} \left[ Z_{k,l} \frac{\partial \sigma _{1j}}{\par... | {
"cite_spans": []
} | 1809.04100 | Adjoint-based optimization for thrust performance of a three-dimensional
pitching-rolling plate | [
"Min Xu",
"Mingjun Wei",
"Chengyu Li",
"Haibo Dong"
] | [
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] | 2,018 | en | Physics | [
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66b4f50052077ce9975683aa4474253dda8ecea1 | subsection | 11 | 23 | Numerical algorithm | For both the forward (flow) simulation and the backward (adjoint) simulation, we used immersed boundary method to treat moving boundaries. This immersed-boundary-method-based simulations have been widely used to simulate the bio-inspired flapping locomotion , , . A staggered Cartesian mesh with local refinement through... | {
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pitching-rolling plate | [
"Min Xu",
"Mingjun Wei",
"Chengyu Li",
"Haibo Dong"
] | [
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362a3af9f3550c67a1b9d88c26a22c6536abf117 | subsection | 12 | 23 | Thrust Study of a Three-Dimensional Pitching-Rolling Plate | In this section, we first use the adjoint-based optimization approach to investigate the thrust production of a rigid pitching-rolling plate. The control parameters include the pitching amplitude, the rolling amplitude, and the phase delay between the pitching and rolling motion. To reveal the underlying flow physics o... | {
"cite_spans": []
} | 1809.04100 | Adjoint-based optimization for thrust performance of a three-dimensional
pitching-rolling plate | [
"Min Xu",
"Mingjun Wei",
"Chengyu Li",
"Haibo Dong"
] | [
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91ac8f985438667fcba2a620686ce4a9b78fe09a | subsection | 13 | 23 | Kinematics and computational setup | An ellipsoidal plate is used in the current study, with the non-dimensional span length of l=1 after non-dimensionalization, the mid-chord length c=0.5 and thickness h=0.05. The plate is oriented based on a fixed point at the root, as shown in Figure REF . The rolling motion of the plate is along the global x axis and ... | {
"cite_spans": []
} | 1809.04100 | Adjoint-based optimization for thrust performance of a three-dimensional
pitching-rolling plate | [
"Min Xu",
"Mingjun Wei",
"Chengyu Li",
"Haibo Dong"
] | [
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3d9fb5458692afe604ec713b5816c978354217e1 | subsection | 14 | 23 | Solid and fluid meshes | As shown in Figure REF , the surface of the ellipsoidal plate is discretized by 4536 unstructured triangle mesh. A Cartesian mesh, stretched in x and y directions and uniform in z direction, is used for an overall Eulerian description of the combined fluid and solid domain, where uniform grid is adopted in z direction ... | {
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pitching-rolling plate | [
"Min Xu",
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ac1dead61a29dbc1088b44c3743ec19845503fcf | subsection | 15 | 23 | Optimization results | The control parameters of the pitching and rolling motions include pitching amplitude (a_x), rolling amplitude (a_z), and the phase delay (\varphi _z) between the pitching and the rolling motion. The control,\phi =(a_x,a_z,\varphi _z),is optimized to improve the propulsive force, with the parameters being optimized in ... | {
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pitching-rolling plate | [
"Min Xu",
"Mingjun Wei",
"Chengyu Li",
"Haibo Dong"
] | [
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3d0d343b49d5d9c9724271231c27937a1fb7fbd8 | subsection | 16 | 23 | Optimization results | The iso-surface contours are color coded by the streamwise vorticity (\omega _x).][Figure: The iso-surfaces of (a) adjoint velocity magnitude at |u^\ast |=0.3 and (b) adjoint pressure magnitude at |p^\ast |=0.3 for the pitching-rolling plate with the initial control at t/T=0.75, t/T=0.5, and t/T=0.25.][Table: The contr... | {
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"raw": "Xu, M., and Wei, M., “Using adjoint-based optimization to study kinematics and deformation of flapping wings,” J. Fluid Mech., Vol. 799, 2016, pp... | 1809.04100 | Adjoint-based optimization for thrust performance of a three-dimensional
pitching-rolling plate | [
"Min Xu",
"Mingjun Wei",
"Chengyu Li",
"Haibo Dong"
] | [
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] | 2,018 | en | Physics | [
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72b7f620ab784ab0bcd550bd768eb37b8981e3e8 | subsection | 17 | 23 | Optimization results | The phase delay between pitching and rolling is increased from 90^\circ to 122.6^\circ during the optimization. Table REF shows that the thrust coefficient drops down from 2.390 to 1.991 by 16.7%, if we change the phase delay in the optimal case back to the initial value. The phase delay changes the timing between pitc... | {
"cite_spans": []
} | 1809.04100 | Adjoint-based optimization for thrust performance of a three-dimensional
pitching-rolling plate | [
"Min Xu",
"Mingjun Wei",
"Chengyu Li",
"Haibo Dong"
] | [
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] | 2,018 | en | Physics | [
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dbb98ae711a3fb8f3e3d158e5c0dfcfe0fa3e00a | subsection | 18 | 23 | Comparison of vortex structures | Figure REF compares the wake topology of each cases. During the pitching-rolling motion, a pair of vortex rings are produced from each flapping cycle and form a bifurcated wake pattern in the downstream. For the initial control (Figure REF a), the downstream vortex rings gradually become weaker and annihilate quickly ... | {
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pitching-rolling plate | [
"Min Xu",
"Mingjun Wei",
"Chengyu Li",
"Haibo Dong"
] | [
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] | 2,018 | en | Physics | [
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ec4dc14014ab54d9d2b0cf1966c01f733f6878a9 | subsection | 19 | 23 | Comparison of vortex structures | The circulation is calculated based on the spanwise vorticity (\omega _{z^\prime } ) contours, and then normalized by U_\infty c: we first identify a closed contour line around the vortex with a specified level (\omega _{z^\prime }=34), and the circulation (\Gamma ) is then calculated by integrating along this line. Al... | {
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} | 1809.04100 | Adjoint-based optimization for thrust performance of a three-dimensional
pitching-rolling plate | [
"Min Xu",
"Mingjun Wei",
"Chengyu Li",
"Haibo Dong"
] | [
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a37dbb6b57a18aa58358015d3b785e12e04ce4f3 | subsection | 20 | 23 | Conclusion | The propulsion performance of a pitching-rolling plate has been investigated using an adjoint-based optimization approach. The rolling amplitude, the pitching amplitude, and the phase delay between the pitching and rolling motion are chosen as control parameters to be optimized for thrust performance. After five main d... | {
"cite_spans": []
} | 1809.04100 | Adjoint-based optimization for thrust performance of a three-dimensional
pitching-rolling plate | [
"Min Xu",
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"Chengyu Li",
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a5c47149d9470d33cdcca512efd6dbcde47a3810 | subsection | 21 | 23 | Appendix | Assuming that the linearized equation is governed by linearized Navier-Stokes equation with infinitesimal body force or infinitesimal mass source \mathbf {f} as the source term, and the control is the infinitesimal body force or mass source instead of solid motion, then the linearized equation can be formulated as\begi... | {
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} | 1809.04100 | Adjoint-based optimization for thrust performance of a three-dimensional
pitching-rolling plate | [
"Min Xu",
"Mingjun Wei",
"Chengyu Li",
"Haibo Dong"
] | [
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ccf9145922352b0d80bd1ffecb0cc3ca409f4655 | subsection | 22 | 23 | Appendix | \end{aligned}When the adjoint equation (REF ) is satisfied, the derivative of the cost function reduces to\begin{aligned}\mathcal {J}^\prime = \frac{1}{T D_0}\left( \int _0^T \int _{\Omega } {\mathbf {q}}^\ast \cdot {\mathbf {f}}^\prime \text{d}t\right).
\end{aligned}This reveals the physical meaning of the adjoint vel... | {
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} | 1809.04100 | Adjoint-based optimization for thrust performance of a three-dimensional
pitching-rolling plate | [
"Min Xu",
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"Haibo Dong"
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8e43b937314aaf86a05ff020fa3f04e75ccb9857 | abstract | 0 | 57 | Abstract | This is a collection of notes that are about spectral form factors of
standard ensembles in the random matrix theory, written for the practical usage
of current study of late time quantum chaos. More precisely, we consider
Gaussian Unitary Ensemble (GUE), Gaussian Orthogonal Ensemble (GOE), Gaussian
Symplectic Ensemble... | {
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} | 10.1103/PhysRevD.98.086026 | 1806.05316 | Spectral form factors and late time quantum chaos | [
"Junyu Liu"
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6e9948960e19749e1d9efc859396c2ad6531a24b | subsection | 1 | 57 | Overview | The theory of quantum chaos, and its connection to random matrix theory, have several new developments recently on understanding novel behaviors of condensed matter system and the quantum nature of black hole physics. The definition of quantum chaos has various versions. Following the pioneer works done by Wigner and D... | {
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fabcb4c11fbb30c62096dfa26e353a37bfe70e2d | subsection | 2 | 57 | Random matrix theory overview | We consider GUE, the Gaussian Unitary Ensemble in this section. The ensemble is defined by introducing the following distribution function over space of Hermitian matrices L\times L,P(H) \propto \exp ( - \frac{L}{2}{\rm {Tr(}}{H^2}{\rm {)}})which means that, for a Hermitian matrices H, the off-diagonal elements are ind... | {
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} | 10.1103/PhysRevD.98.086026 | 1806.05316 | Spectral form factors and late time quantum chaos | [
"Junyu Liu"
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afe27c0e8503b5a1170d87e99071086188da032d | subsection | 3 | 57 | Random matrix theory overview | From random matrix theory, people find that the n point function could be determined by a kernel K{\rho ^{(n)}}({\lambda _1}, \ldots ,{\lambda _n}) = \frac{{(L - n)!}}{{L!}}\det (K({\lambda _i},{\lambda _j}))_{i,j = 1}^nwhere the kernel K, in the large L limit, behaves asK({\lambda _i},{\lambda _j}) \equiv \left\lbrace... | {
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} | 10.1103/PhysRevD.98.086026 | 1806.05316 | Spectral form factors and late time quantum chaos | [
"Junyu Liu"
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0c11758acc3ca7e8a094930b2378e371e4a26a35 | subsection | 4 | 57 | The disconnected piece | We start to compute the two point form factor \mathcal {R}_2,&{{\cal R}_2}(t) = \sum \limits _{i,j} {\int {d{\lambda _i}d{\lambda _j}{\rho ^{(2)}}({\lambda _i},{\lambda _j})} {e^{i({\lambda _i} - {\lambda _j})t}}} \\
&= L + L(L - 1)\int {d{\lambda _1}d{\lambda _2}{\rho ^{(2)}}({\lambda _1},{\lambda _2}){e^{i({\lambda _... | {
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} | 10.1103/PhysRevD.98.086026 | 1806.05316 | Spectral form factors and late time quantum chaos | [
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fd9dfab48ce7854730803671ec63fda8eacb86f1 | subsection | 5 | 57 | The connected piece: box approximation | Now let us discuss the connected piece, which is defined as{\cal R}_2^{{\rm {conn}}}(t) = {{\cal R}_2}(t) - {\cal R}_2^{{\rm {disc}}}(t) = L -{L^2}\int {d{\lambda _1}d{\lambda _2}\frac{{{{\sin }^2}(L({\lambda _1} - {\lambda _2}))}}{{{{(L\pi ({\lambda _1} - {\lambda _2}))}^2}}}{e^{i({\lambda _1} - {\lambda _2})t}}}Howev... | {
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b6607812d2e7d16927b3760fafa8aebbf93d2f9d | subsection | 6 | 57 | The connected piece: box approximation | Let us assume that this cutoff space is symmetric around the origin, [-\text{cut},\text{cut}], then the result is given by{L^2}\int {d{\lambda _1}d{\lambda _2}\frac{{{{\sin }^2}(L({\lambda _1} - {\lambda _2}))}}{{{{(L\pi ({\lambda _1} - {\lambda _2}))}^2}}}{e^{i({\lambda _1} - {\lambda _2})t}}} = \frac{{2{\rm {cut}} \t... | {
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} | 10.1103/PhysRevD.98.086026 | 1806.05316 | Spectral form factors and late time quantum chaos | [
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502ab76c1bf23e89e74f2cf3f3a34043ec7bef77 | subsection | 7 | 57 | The connected piece: box approximation | The plateau value \mathcal {R}_2^\text{conn}(t_p=2L))=L, is fixed by the long time average interpretation of definition of the form factor (which means that the damping e^{(i(\lambda _1-\lambda _2)t)} for \lambda _1\ne \lambda _2 will be cancelled after long time averaging, and the only constant piece with \lambda _1=\... | {
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} | 10.1103/PhysRevD.98.086026 | 1806.05316 | Spectral form factors and late time quantum chaos | [
"Junyu Liu"
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c0f90b1015266b7a2220bd58adeb2b61a5cceda2 | subsection | 8 | 57 | The connected piece: an improvement | Now we introduce an improvement which is more refined than the box cutoff. In this part, we will try to use the short distance kernel\widetilde{K}({\lambda _i},{\lambda _j}) = L\,\frac{{\sin (\pi L({\lambda _i} - {\lambda _j})\rho (({\lambda _i} + {\lambda _j})/2))}}{{\pi L({\lambda _i} - {\lambda _j})}}where this kern... | {
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c426829fa77226a86928252ec8c63f26aeab3124 | subsection | 9 | 57 | The connected piece: an improvement | Suppose that we are now at the center u_2, and the interval has the range [-\Omega _0/2,\Omega _0/2], then performing the integral, in the large L limit, we have&{L^2}\int _{ - {\Omega _0}/2}^{{\Omega _0}/2} {d{u_1}\frac{{{{\sin }^2}(\pi L{u_1}\rho ({u_2}))}}{{{{(\pi L{u_1})}^2}}}{e^{i{u_1}t}}} \\
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} | 10.1103/PhysRevD.98.086026 | 1806.05316 | Spectral form factors and late time quantum chaos | [
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2129dc8b7fc91d590e4dc24456a634395fc4474a | subsection | 10 | 57 | The connected piece: an improvement | \\
&= \max \left( {L\rho ({u_2}) - \frac{t}{{2\pi }},0} \right)Here an assumption we are making is that we are extending the range from an L amplified interval to infinity, regardless of the fact that the exponent will be \mathcal {O}(1) even if u_1 could scale as \mathcal {O}(L).Now, we sum over the all intervals, whi... | {
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... | 10.1103/PhysRevD.98.086026 | 1806.05316 | Spectral form factors and late time quantum chaos | [
"Junyu Liu"
] | [
"hep-th",
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0b03287fd95fe6d81c134270bc58b1c81c85fa12 | subsection | 11 | 57 | Higher point form factor: theorem | Higher point form factor calculations are based on multi-variable Fourier transforms of determinant of sine kernels. We will derive some generic results to establish the framework of computing higher point form factors in general based on the box approximation, and compute a four-point example.
Our starting point will ... | {
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"raw": "M. Mehta. Random matrices. Second edition.",
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... | 10.1103/PhysRevD.98.086026 | 1806.05316 | Spectral form factors and late time quantum chaos | [
"Junyu Liu"
] | [
"hep-th",
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] | 2,018 | en | Physics | [
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8e4e27238fecbc310450a57c91218aaa53cc9f29 | subsection | 12 | 57 | Higher point form factor: theorem | Thus we obtain& \int {\prod \nolimits _{i=1}^{m}{d{{y}_{i}}}\exp (2\pi i\sum \limits _{j=1}^{m}{{{k}_{j}}{{y}_{j}}})s({{y}_{1}}-{{y}_{2}})}s({{y}_{2}}-{{y}_{3}})\ldots s({{y}_{m-1}}-{{y}_{m}})s({{y}_{m}}-{{y}_{1}}) \\
& =\int {\prod \nolimits _{i=1}^{m}{d{{u}_{i}}}\exp (2\pi i\sum \limits _{l=1}^{m}{{{k}_{l}}\sum \limi... | {
"cite_spans": []
} | 10.1103/PhysRevD.98.086026 | 1806.05316 | Spectral form factors and late time quantum chaos | [
"Junyu Liu"
] | [
"hep-th",
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222d6248aa8773d7970447814270b8436510e0fa | subsection | 13 | 57 | Higher point form factor: theorem | Now introduce a new variable u, which iss(\sum \limits _{j=1}^{m-1}{{{u}_{j}}})=s(-\sum \limits _{j=1}^{m-1}{{{u}_{j}}})=\int {du}s(u)\delta (u+\sum \limits _{j=1}^{m-1}{{{u}_{j}}})and then, replace the delta function by exponential functions(\sum \limits _{j=1}^{m-1}{{{u}_{j}}})=\int {du}dks(u)\exp (2\pi ik(u+\sum \li... | {
"cite_spans": []
} | 10.1103/PhysRevD.98.086026 | 1806.05316 | Spectral form factors and late time quantum chaos | [
"Junyu Liu"
] | [
"hep-th",
"cond-mat.str-el",
"quant-ph"
] | 2,018 | en | Physics | [
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27ee5775b54f15e807ae5c335493b3d15ee650c6 | subsection | 14 | 57 | Higher point form factor: theorem | Now it is obvious to generalize this claim to large but finite L. We have&\int {\prod \limits _{i = 1}^m {d{\lambda _i}K({\lambda _1},{\lambda _2})K({\lambda _2},{\lambda _3}) \ldots K({\lambda _{m - 1}},{\lambda _m})K({\lambda _m},{\lambda _1}){e^{i\sum \limits _{i = 1}^m {{k_i}{\lambda _i}} }}} } \\
&= \frac{L}{\pi }... | {
"cite_spans": []
} | 10.1103/PhysRevD.98.086026 | 1806.05316 | Spectral form factors and late time quantum chaos | [
"Junyu Liu"
] | [
"hep-th",
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d6b3387c0b3dfd5be5849c4b4b3094474c9512d6 | subsection | 15 | 57 | Higher point form factor: theorem | So we finally get the useful formulaTheorem 2.2 (Convolution formula for finite large L)&\int {\prod \limits _{i = 1}^m {d{\lambda _i}K({\lambda _1},{\lambda _2})K({\lambda _2},{\lambda _3}) \ldots K({\lambda _{m - 1}},{\lambda _m})K({\lambda _m},{\lambda _1}){e^{i\sum \limits _{i = 1}^m {{k_i}{\lambda _i}} }}} } \\
&=... | {
"cite_spans": []
} | 10.1103/PhysRevD.98.086026 | 1806.05316 | Spectral form factors and late time quantum chaos | [
"Junyu Liu"
] | [
"hep-th",
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] | 2,018 | en | Physics | [
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6021828f0fcb261afa814633575c7755c9e72370 | subsection | 16 | 57 | Four point form factor | Now let us consider the four point form factor as an example{\mathcal {R}_{4}}=\sum \limits _{a,b,c,d=1}^{L}{\int {D\lambda }{{e}^{i({{\lambda }_{a}}+{{\lambda }_{b}}-{{\lambda }_{c}}-{{\lambda }_{d}})t}}}Before our computation, we will define the following building block functions&{r_1}(t) = \frac{{{J_1}(2t)}}{t}\\
&{... | {
"cite_spans": []
} | 10.1103/PhysRevD.98.086026 | 1806.05316 | Spectral form factors and late time quantum chaos | [
"Junyu Liu"
] | [
"hep-th",
"cond-mat.str-el",
"quant-ph"
] | 2,018 | en | Physics | [
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cf1a005e00f208ed69b87226a87c5f9ba461095c | subsection | 17 | 57 | Four point form factor | Add them together we get& {\mathcal {R}_{4}}=L(L-1)(L-2)(L-3)\int {D\lambda }{{e}^{i({{\lambda }_{1}}+{{\lambda }_{2}}-{{\lambda }_{3}}-{{\lambda }_{4}})t}} \\
& +2L(L-1)(L-2)\operatorname{Re}\int {D\lambda }{{e}^{i(2{{\lambda }_{1}}-{{\lambda }_{2}}-{{\lambda }_{3}})t}} \\
& +L(L-1)\int {D\lambda }{{e}^{i(2{{\lambda }... | {
"cite_spans": []
} | 10.1103/PhysRevD.98.086026 | 1806.05316 | Spectral form factors and late time quantum chaos | [
"Junyu Liu"
] | [
"hep-th",
"cond-mat.str-el",
"quant-ph"
] | 2,018 | en | Physics | [
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