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+FLAME: A small language model for spreadsheet formulas
+Harshit Joshi1 , Abishai Ebenezer1 , Jos´e Cambronero2∗ , Sumit Gulwani2∗ ,
+Aditya Kanade3∗ , Vu Le2∗ , Ivan Radiˇcek4∗ , Gust Verbruggen5∗
+1Microsoft, India
+2Microsoft, USA
+3Microsoft Research, India
+4Microsoft, Croatia
+5Microsoft, Belgium
+{t-hjoshi, t-aebenezer, jcambronero, sumitg, kanadeaditya, levu, ivradice, gverbruggen}@microsoft.com
+Abstract
+The widespread use of spreadsheet environments
+by billions of users presents a unique opportunity
+for formula-authoring assistance. Although large
+language models, such as Codex, can assist in
+general-purpose languages, they are expensive to
+train and challenging to deploy due to their large
+model sizes (up to billions of parameters). More-
+over, they require hundreds of gigabytes of train-
+ing data. We present FLAME, a T5-based model
+trained on Excel formulas that leverages domain
+insights to achieve competitive performance with a
+substantially smaller model (60M parameters) and
+two orders of magnitude less training data. We cu-
+rate a training dataset using sketch deduplication,
+introduce an Excel-specific formula tokenizer for
+our model, and use domain-specific versions of
+masked span prediction and noisy auto-encoding
+as pretraining objectives. We evaluate FLAME on
+formula repair, formula auto-completion, and a
+novel task called syntax reconstruction.
+FLAME
+(60M) can outperform much larger models, such
+as Codex-Davinci (175B), Codex-Cushman (12B),
+and CodeT5 (220M), in 6 out of 10 settings.
+1
+Introduction
+Despite a much larger user base, spreadsheet environments
+do not have access to nearly the same range of productivity
+tools as available for general programming environments. The
+latter typically have code completion, refactoring, linting, and
+a wide range of extensions for additional functionality, like
+generating tests, inserting code snippets, and summarizing
+code. Many of these advanced programming assistance tools
+are driven by advances in large language models trained on
+code (LLMCs). Codex [Chen et al., 2021a] is used for code
+completion [GitHub, 2021] and repair [Joshi et al., 2022],
+AlphaCode [Li et al., 2022a] solves competitive programming
+problems, [Li et al., 2022b] built a code review system, and
+many other models show great performance in code related
+tasks [Xu et al., 2022; Fried et al., 2022; Nijkamp et al., 2022].
+∗Listed in alphabetical order
+Formula Autocompletion
+Last Mile Repair
+Syntax Reconstruction
+0.90
+0.85
+0.80
+0.75
+FLAME 
+(16M) 
+FLAME 
+(60M) 
+CodeT5 
+(220M) 
+Codex Cushman 
+(12B)
+Codex Davinci 
+(175B) 
+Model Parameters (log-scale)
+Performance
+0.40 Z
+Figure 1: A summary of model comparisons in fine-tuned setting for
+different formula assistance tasks. We show the results under a top-5
+cutoff on a public excel benchmark. Note that all Codex-Davinci
+results are few-shot, and Autocompletion is zeroshot for all systems
+except CodeT5. For Autocompletion, results represent the fraction of
+benchmarks successfully (based on a sketch match metric) completed
+given 90% of the prefix.
+To capture the complexity and variety of code and com-
+ments in different languages, these models need billions of
+parameters—the smallest variant of Codex, used by GitHub
+Copilot, has 12 billion parameters. As a result, these models
+are trained for long periods on corpora containing millions of
+programs. For example, Incoder 6.7B used 159GB of code
+over a period of 24 days on 248 V100 GPUs. In addition to
+training costs, inference on large models is expensive due to
+extensive hardware requirements. For example, using Codex-
+Davinci to process 1000 tokens, including the prompt, costs
+$0.02 USD [OpenAI, 2023]. In a spreadsheet environment
+used by billions, these costs quickly add up.
+In this paper, we present FLAME, a Formula LAnguage Model
+for Excel trained exclusively on Excel formulas. FLAME is
+based on T5-small [Raffel et al., 2020] and has only 60 mil-
+lions parameters, yet it can compete with much larger models
+(up to 175B parameters) on three formula authoring tasks:
+last-mile repair, formula auto-completion and syntax recon-
+struction. Syntax reconstruction is a novel task where all de-
+limiters are removed from a formula, resulting in a flat stream
+arXiv:2301.13779v1  [cs.PL]  31 Jan 2023
+
+=SUMIF(B1:B5, A1:A5, "Yes")
+=SUMIF(B1:B5, "Yes", A1:A5)
+Last Mile Repair
+=AVERAGEIFS(B4:M4
+=AVERAGEIFS(B4:M4, B4:M4, ">0")
+Formula Autocompletion
+Syntax Reconstruction
+IFERROR VLOOKUP A2 Sheet2 $A$1:$E$22 5 0 "Not available"
+=IFERROR(VLOOKUP(A2, Sheet2!$A$1:$E$22, 5, 0), "Not available")
+Figure 2: We consider three downstream tasks: Last Mile Repair,
+Formula Autocompletion, and Syntax Reconstruction. Red and green
+colors denote the input and the expected output, respectively. Yellow
+text denotes the buggy part of the formula in the repair task, where
+the user has swapped the correct order of arguments resulting in a
+type error. Each task shows a case that FLAME successfully solves.
+of tokens, and the model must recover the original formula.
+Figure 1 shows a high-level summary of results as a function
+of model size on a public dataset, where FLAME can outper-
+form larger models in all three tasks. Figure 2 provides real
+examples, solved by FLAME, for each of these tasks.
+There are three main challenges involved in training a model
+for Excel formulas: obtaining diverse training data, tokenizing
+their unique structure, and pretraining with objectives that
+teach the model about this distinctive structure. Spreadsheets
+contain many duplicate formulas due to copying down for-
+mula cells. We reduced our corpus from 927M formulas down
+to 6.1M by comparing formulas based on syntax, creating
+540MB of training data. We combine formulas insights with
+byte pair encoding (BPE) to train an Excel-specific tokenizer.
+In addition to two generic objectives (tail-masking and de-
+noising auto-encoding), we introduce two new pretraining
+objectives designed for formulas: language-aware masked
+span prediction and user-inspired denoising.
+We extensively evaluate FLAME on three downstream tasks,
+showing that our proposed solutions to the modeling chal-
+lenges significantly improve the performance of FLAME over
+T5-based models and can compete with much larger models.
+Specifically, we find that FLAME can outperform other models
+in 6 out of 10 settings in our evaluation.
+We make the following contributions:
+• We present FLAME, the first language model designed
+exclusively for Excel formulas (§3). To this end, we
+introduce domain-specific dataset curation (§3.2), tok-
+enization (§3.3), and pretraining objectives (§3.4).
+• We extensively evaluate FLAME on three formula assis-
+tance tasks: last-mile repair, formula autocompletion,
+and syntax reconstruction (§4.3).
+• We compare our performance to two variants of Codex
+(latest version of Cushman and Davinci) and CodeT5,
+and finetune Cushman for downstream tasks (§4.1). We
+show that FLAME can outperform larger models in 6 out
+of 10 settings (§5.1).
+• We analyze the contribution of different design choices
+for FLAME (§5.2,§5.3)
+2
+Related Work
+Language models for code
+Multiple popular language
+model architectures have been successfully adapted to code.
+CodeBERT [Feng et al., 2020] trained BERT (encoder) on nat-
+ural language and code. CodeT5 [Wang et al., 2021] trained
+T5 (encoder-decoder) on a similar corpus. Codex [Chen et
+al., 2021a], PolyCoder [Xu et al., 2022], or CodeGen [Ni-
+jkamp et al., 2022] are all trained variants of GPT (decoder).
+These models are trained on multiple programming languages
+and use pretraining objectives to understand or generate code
+and natural language, but do not adapt them for specific lan-
+guages. In contrast, FLAME exploits a single domain to use
+domain-specific objectives, such as span masking that respects
+programming language tokens, to learn a better representation.
+Evaluating code models
+Many tasks have been presented
+to evaluate code models, and CodeXGLUE [Lu et al., 2021]
+bundles most of these. These tasks are categorized by the
+modality (text/code) of their input and output. FLAME is trained
+on formulas exclusively and is focused on formula tasks. We
+now describe related work for these tasks.
+Formula repair
+A popular code authoring task is repairing
+small mistakes. DeepFix [Gupta et al., 2017], BIFI [Yasunaga
+and Liang, 2021], Dr.Repair [Yasunaga and Liang, 2020], and
+TFix [Berabi et al., 2021] use deep learning to perform syntax,
+compilation, or diagnostics repair in general-purpose program-
+ming languages. LaMirage [Bavishi et al., 2022] generates
+repair engines for low-code languages and coins the term last-
+mile repair for these types of fixes. RING [Joshi et al., 2022]
+uses Codex to fix last-mile errors across multiple languages,
+but it requires additional information, such as examples of
+repairs and compiler messages.
+Formula autocompletion
+The generative nature of LLMCs
+makes them serve as code-completion engines. This feature
+has been shipped in commercial products, such as GitHub
+Copilot in Visual studio Code [GitHub, 2021] and IntelliCode
+in Visual Studio [Svyatkovskiy et al., 2020]. Spreadsheet-
+Coder [Chen et al., 2021b] is a model designed for predicting
+simple formulas from context in the spreadsheet.
+Syntax reconstruction
+Syntax reconstruction, where all de-
+limiters in a formula are removed, resembles component-based
+program synthesis, where partial programs are combined into
+a program that satisfies a specification. Components are pro-
+vided by a user [Jha et al., 2010], generated by a model [Rah-
+mani et al., 2021], or defined by an API [Feng et al., 2017].
+3
+FLAME: Approach
+We now describe the FLAME architecture and how it overcomes
+the three key challenges (data, tokenization, and training) in
+pretraining a general language model for formulas.
+3.1
+Architecture
+To facilitate both formula understanding and generation,
+FLAME follows an encoder-decoder architecture based on T5
+[Raffel et al., 2020]. Encoder models like CodeBERT [Feng
+et al., 2020] show remarkable code understanding capabilities.
+Decoder models like CodeGen [Nijkamp et al., 2022] and
+
+User-inspired Denoising
+INDEX(summary!N:N; MATCH(A350;
+summary!$D:$D, 0, 0))
+INDEX(summary!N:N; MATCH(A350;
+summary!$D:$D; 0))
+Change Function Arity
+Comma to Semi colon
+17 user-inspired noise operators 
+INDEX(summary!N:N, MATCH(A350,
+summary!$D:$D, 0))
+INDEX(summary!N:N, MAT<mask>
+Tail Masking
+INDEX(summary!N:N, <mask>(A350,
+summary!$D:$D, 0<mask>)
+low mask rate, low average span length 
+INDEX2(summary!N:N, yeMATCH(A350,
+summary!$[D:$D, 0))
+Random Noising
+INDEX(<mask>!N:N, MATCH<mask>A350,
+summary!<mask>:$D, 0<mask>)
+high mask rate, low average span length 
+Language-Aware Span Masking
+different combinations of high and low
+masking rate and average span lengths
+Figure 3: Four pretraining objectives used by FLAME. For each batch, we randomly (with weighted probability) choose one of the four objectives.
+Generic objectives (tail masking and random noise) are shown with a yellow header, while formula-specific variants (language-aware span
+masking and user-inspired noise) are shown with a green header. We depict inserted tokens with red and deleted tokens with blue.
+Codex [Chen et al., 2021a] perform well on code generation.
+Encoder-decoder models seek to blend these strengths.
+3.2
+Training Data
+We start from a dataset of 927M formulas drawn from a corpus
+of 1.8M publicly available Excel workbooks.1 Each workbook
+contains one or more worksheets, and each worksheet contains
+zero or more formulas. Formulas in spreadsheets are often
+repeated with minor cell reference changes across rows or
+columns. For example, a user can drag a formula to another
+cell to repeat a computation on neighboring cell values.
+We compute formula sketches to preserve a single instance
+of each unique formula per workbook. In a formula sketch,
+numeric constants, string constants and cell references are
+replaced by their token type. For example, the sketch of
+=SUM(A1:A10) is =SUM(cell:cell). After applying sketch
+deduplication, we are left with 6.1M formulas. Note that ap-
+plying this globally to the corpus, rather than per workbook,
+results in only 591K formulas. We found this globally dedu-
+plicated corpus to be insufficient for training as it skews the
+distribution of formulas —see evaluation (§5.2) for details.
+3.3
+Tokenizing Formulas
+Tokenization is an essential part of language models [Domingo
+et al., 2018]. A popular method for tokenization is byte pair
+encoding (BPE) [Sennrich et al., 2016]. BPE iteratively joins
+consecutive tokens that appear together most frequently until
+a target vocabulary size is reached. However, this procedure
+can have adverse effects on formulas. For example, SUM and (
+are combined to get SUM(, which can reduce expressiveness
+and hurt performance for tasks like repair.
+Our tokenizer considers punctuation, whitespace, built-in
+function names, and digits as individual tokens [Chowdhery
+et al., 2022] and applies BPE [Radford et al., 2019] to the re-
+maining parts of formulas, like string constants. Excel is case
+insensitive (with the exception of string contents) so we con-
+vert all input tokens to lowercase to map differently capitalized
+tokens to a single token. For example, without lowercasing,
+the same function SUM and sum will map to different tokens.
+Example 1. A formula
+=SUMIF(B1:B5, "Not available", A1:A5)
+1These workbooks were collected as part of a large Excel corpus
+planned for public release by a separate group of authors.
+is tokenized as
+= sumif ( b 1 :
+b 5 , ␣ " not ␣ available "
+, ␣ a 1 :
+a 5 )
+with space tokens denoted by ␣.
+3.4
+Pretraining Objectives for Training
+In this section, we describe the combination of generic and
+Excel-specific pretraining objectives, as summarized in Fig-
+ure 3, that we use to train FLAME.
+Masking objectives
+We use two forms of masking to pre-train FLAME, an Excel-
+specific variant of masked span prediction (MSP), and a
+generic tail masking objective.
+Language-aware masked span prediction
+In contrast to
+traditional MSP, spans must respect Excel lexer bounds. For
+example, when an Excel cell reference BC18 is divided into
+four tokens B C 1 8, we ensure that either all or none of its
+constituent tokens is masked. Consecutive masked tokens are
+represented with a single <mask> token. Inspired by Mixture-
+of-Denoisers [Tay et al., 2022], we mask spans of tokens using
+combinations of high (35%) and low (15%) masking rates, and
+big (6 tokens) and small (2 tokens) average span lengths.
+Generic tail masking
+We perform tail masking at the char-
+acter level and allow partial masks of complete tokens. We
+keep the leading {30%,40%,··· ,70%} tokens of the input
+sequence and append a <mask> token.
+Noisy Auto-encoding
+Previous work in natural language processing has used denois-
+ing auto-encoding during pretraining [Lewis et al., 2020]. We
+incorporate two such objectives in FLAME.
+Random Noise
+We introduce generic noise by randomly
+inserting, deleting, or updating tokens in the input sequence.
+The insertion and update operators randomly sample a token
+from the vocabulary.
+Excel-specific user-inspired noise
+We introduce noise op-
+erators that mirror mistakes that real users might make when
+writing Excel formulas. For example, users often write formu-
+las with the incorrect function arity for in-built functions such
+as SUMIF. We implement 17 noise operators (Appendix A)
+
+based on a combination of help forum and code analysis. We
+randomly choose one of these noise operators when introduc-
+ing noise into an input sequence.
+Note that for all pretraining objectives, FLAME needs to
+generate a complete formula (rather than just mask values).
+Combining pretraining objectives
+Rather than applying all pretraining objectives on every batch
+and then combining losses, we pick a single objective for
+each batch. We use the following probabilities {MSP: 50%,
+tail masking: 20%, user-inspired denoising: 20%, random
+denoising: 5%} for choosing the objective to be applied, and
+with a 5% probability, we leave the sequence intact.
+4
+Experimental Setup
+We now describe our experimental setup. We start with the
+baseline models we compare against (§4.1), the training setup
+(§4.2), and then detail each downstream task in our evaluation,
+along with their corresponding datasets (§4.3).
+4.1
+Baselines and Configurations
+We compare FLAME to the following much larger language
+models, summarized in Table 1:
+• CodeT5: a 220 million parameter T5-based encoder-
+decoder model trained on both natural language and code.
+We present fine-tuned results.
+• Codex-Cushman: a 12 billion parameter autoregressive,
+decoder-only, GPT-3-based model trained on both natural
+language and code. We present both zeroshot and fine-
+tuned results.
+• Codex-Davinci: a 175 billion parameter autoregressive,
+decoder-only, GPT-3-based model trained on both natural
+language and code. We present zeroshot and few-shot
+results. We do not have resources to fine-tune Davinci.
+For Codex-based baselines, we use nucleus sampling [Holtz-
+man et al., 2019] (temperature=0.7) and sample 50 sequences
+per task. We sort these sequences based on their average token
+log probabilities following [Joshi et al., 2022]. We detail the
+prompts in Appendix B. For CodeT5, we use beam search with
+a beam width of 50, and we consider the top 50 sequences.
+4.2
+Training Details
+We pretrain FLAME for 10 epochs and finetune CodeT5 and
+FLAME on a cluster with 16 AMD MI200s, 96 cores and 900
+GB RAM. We finetune FLAME for 2 epochs for each down-
+stream task and finetune CodeT5 for 25 epochs with a patience
+of 5 epochs. We carry out all Codex experiments on a cluster
+with 8 V100s, 40 cores, and 672 GB RAM. For Codex fine-
+tuning we use low-rank adaptation (LoRA) [Hu et al., 2021].
+Refer to Appendix C for more details.
+4.3
+Downstream Tasks
+We consider three different downstream tasks.
+System
+Architecture
+Number of parameters
+Codex-Cushman
+Decoder
+12 billion
+Codex-Davinci
+Decoder
+175 billion
+CodeT5 (base)
+Encoder-Decoder
+220 million
+FLAME (ours)
+Encoder-Decoder
+60 million
+Table 1: Architecture and size comparison of baselines and FLAME
+Last-mile Repair
+Last-mile repair refers to repairs that require few edits and fix
+syntax and simple semantic errors, such as wrong function call
+arity. In this setting, FLAME is given the buggy formula as the
+input sequence, and the task is to generate the user’s intended
+(and syntactically correct) formula without any last-mile error.
+Example 2. The user has used the wrong call arity for
+ISERROR. Red highlights the error in the buggy formula, and
+green denotes the required edit to match the groundtruth.
+Buggy Formula: =IF(ISERROR(G6 *1.2, "" ) )
+Groundtruth Formula: =IF(ISERROR(G6 *1.2 ) , "")
+Fine Tuning
+We create a finetuning dataset for all systems
+by taking 200K well-formed formulas from Excel help forums.
+We then randomly apply our user-inspired noise operators to
+generate broken versions.
+Evaluation Metric
+We compute an exact match with re-
+spect to the ground truth repair. We consider the top 1 and top
+5 candidates produced by each system per formula and report
+the exact match fraction.
+Benchmarks
+We evaluate all systems on two benchmarks.
+We use the collection of 273 labeled Excel formulas used
+in recent last-mile repair literature [Joshi et al., 2022]. The
+authors sourced these formulas from Excel help forums. We
+refer to this benchmark set as Forum.
+We also reserve a split of randomly sampled 500 formulas
+derived using the same procedure as our finetuning dataset to
+create a Test benchmark set.
+Autocompletion
+Code completion is a popular task for language models trained
+on code, both due to its autoregressive nature and the practical
+value of code completion as a feature in developers’ workflows.
+In this setting, FLAME is given a formula prefix, and the task is
+to generate the complete formula.
+Example 3. Formula Autocompletion
+Formula Prefix: =B2<=EDATE(
+Formula Completion: =B2<=EDATE(TODAY(),-33)
+Fine Tuning
+We curated a finetuning dataset for autocom-
+pletion by splitting 189k formulas and sampling a prefix length
+of {0.2,··· ,0.7,0.8} fraction of tokens.
+Evaluation Metric
+When completing formulas, some parts
+can be hard to predict due to lack of context [Guo et al.,
+2021], such as cell references, sheet names, string literals, and
+numerics. Therefore, in addition to exatch match, we also
+consider sketch match for autocompletion with respect to the
+ground truth. Precisely, for sketch match, we use the same
+sketch procedure described in §3. This uses the Excel lexer
+
+Model
+Last Mile Repair
+Syntax Reconstuction
+Forum
+Test
+Forum
+Test
+T@1
+T@5
+T@1
+T@5
+T@1
+T@5
+T@1
+T@5
+Cushman
+0.79
+0.88
+0.87
+0.93
+0.70
+0.80
+0.84
+0.91
+Davinci (FS)
+0.76
+0.89
+0.54
+0.77
+0.62
+0.77
+0.61
+0.73
+CodeT5 (220M)
+0.70
+0.84
+0.84
+0.90
+0.70
+0.84
+0.82
+0.89
+CodeT5 (60M)
+0.72
+0.83
+0.82
+0.89
+0.65
+0.81
+0.83
+0.89
+FLAME
+0.76
+0.89
+0.83
+0.91
+0.75
+0.89
+0.84
+0.89
+Table 2: Fine-tuned performance for Last Mile Repair and Syntax reconstruction tasks. Codex-Davinci uses few-shots and is denoted by an
+FS suffix). FLAME outperforms larger models at last-mile repair in the Forum benchmark at top-5, and comes in second at top-1. In syntax
+reconstruction, FLAME outperforms all models at both cutoffs in the Forum benchmark. Bold denotes best performing model and Underline
+represents second best.
+Models
+Exact Match
+Sketch Match
+0.25
+0.50
+0.75
+0.90
+0.99
+0.25
+0.50
+0.75
+0.90
+0.99
+Cushman
+0.0
+0.04
+0.27
+0.61
+0.86
+0.12
+0.26
+0.47
+0.71
+0.86
+Davinci
+0.0
+0.03
+0.31
+0.64
+0.85
+0.10
+0.25
+0.53
+0.76
+0.85
+CodeT5
+0.0
+0.02
+0.10
+0.27
+0.21
+0.03
+0.09
+0.20
+0.39
+0.22
+FLAME
+0.01
+0.06
+0.34
+0.70
+0.93
+0.10
+0.24
+0.55
+0.84
+0.94
+Table 3: Zeroshot autcompletion performance of FLAME, Codex-Cushman and Codex-Davinci, and fine-tuned CodeT5 (as denoted by FT
+suffix). Given {0.25,0.50,0.75,0.90,0.99} fraction of formula prefix, we report the proportion of formulas completed in the top 5. We observe
+that FLAME outperforms all the large language models in the exact match setting and most (3/5) of the sketch match settings. Bold denotes best
+performing model and Underline represents second best.
+to tokenize a formula and preserves built-in function names
+but replaces all other tokens with their token type. We then
+compare the sketches of the formulas for a match. For instance,
+in Example 3, predicting the numeric −33 is highly contextual,
+so in a sketch we match with its token type, Numeric.
+Benchmarks
+We evaluate autocompletion on a single bench-
+mark, consisting of the 273 ground truth formulas from the
+Forum last-mile repair benchmark. For each formula, given
+exact match or sketch match metric, we predict completions
+at 0.25, 0.5, 0.75, 0.90 and 0.99 fractions of formula prefix.
+Syntax Reconstruction
+We introduce a new task that we term syntax reconstruction.
+The input to this task consists of Excel formulas which we
+have processed to remove any delimiters, resulting in a flat
+stream of lexer tokens. Excel delimiters are defined to be the
+following set of tokens: {( ) !
+, ; { } [ ] .}. The
+model is then tasked with generating the original formula with
+appropriate delimiters.
+Example 4. Syntax Reconstruction given the excel tokens.
+Tokens: MAX 0 MOD C10 - B10 1 - D10
+Reconstruction: MAX(0,MOD(C10-B10,1)-D10)
+Since, by definition, syntax reconstruction cannot introduce
+tokens into the output that are not delimiters or not in the orig-
+inal input token stream, FLAME employs constrained decoding
+to greedily remove invalid candidates from the search space.
+Our tokenizer design, particularly splitting on punctuation,
+makes this decoding strategy easier to implement.
+Fine Tuning
+We curate a finetuning dataset by sampling
+200k formulas from the publicly available Excel corpus that
+we used for FLAME’s pretraining. We keep the subset that con-
+tains at least one delimiter (139k) and remove all delimiters.
+Evaluation Metric
+We compute an exact match with re-
+spect to the ground truth and consider the top 1 and top 5
+candidates produced by each system per formula.
+Benchmarks
+We derive a benchmark set from the last-
+mile repair benchmarks by removing the delimiters for every
+groundtruth formula. We refer to this benchmark as Forum.
+Finally, we also consider a Test split that reflects the same
+preparation as the fine tuning dataset.
+5
+Evaluation
+We explore the following research questions in our evaluation:
+• RQ1: How does FLAME perform on formula intelligence
+tasks compared to substantially larger language models?
+• RQ2: How do pretraining design decisions such as data
+curation, model size, pretraining objectives, and tokenizer
+affect FLAME’s downstream performance?
+• RQ3: How do various decoding strategies affect different
+downstream-task performances for FLAME?
+5.1
+RQ1: Larger Language Models
+We now compare FLAME to substantially larger language mod-
+els on our three formula intelligence tasks.
+Last Mile Repair and Syntax Reconstruction
+We finetune FLAME, CodeT5, and Codex-Cushman for last-
+mile repair and syntax reconstruction, and use few-shot
+prompts with three shots for Codex Davinci. Although one of
+
+Model
+Last Mile Repair
+Syntax Reconstuction
+Forum
+Test
+Forum
+Test
+T@1
+T@5
+T@1
+T@5
+T@1
+T@5
+T@1
+T@5
+Cushman
+0.55
+0.85
+0.41
+0.63
+0.27
+0.53
+0.23
+0.46
+Davinci
+0.60
+0.82
+0.51
+0.75
+0.51
+0.65
+0.31
+0.45
+FLAME
+0.71
+0.88
+0.74
+0.85
+0.41
+0.53
+0.50
+0.58
+Table 4: Zeroshot last-mile repair and syntax reconstruction performance of FLAME and Codex models. FLAME outperforms all the larger
+models in Last Mile Repair task and solves more benchmarks than Codex-Cushman for the Syntax Reconstruction task. Bold denotes best
+performing model and Underline represents second best.
+Model
+Zeroshot
+Finetuned
+LMR
+SR
+AC (EM)
+AC (SM)
+LMR
+SR
+Forum
+Test
+Forum
+Test
+0.75
+0.90
+0.75
+0.90
+Forum
+Test
+Forum
+Test
+FLAME (60M)
+0.71
+0.74
+0.41
+0.50
+0.34
+0.70
+0.55
+0.84
+0.76
+0.83
+0.75
+0.84
+FLAME (16M)
+0.68
+0.64
+0.23
+0.42
+0.24
+0.59
+0.54
+0.76
+0.73
+0.78
+0.73
+0.78
+Global Deduplication
+0.57
+0.56
+0.16
+0.2
+0.15
+0.45
+0.41
+0.59
+0.68
+0.76
+0.73
+0.81
+T5 (Generic objectives and tokenizer)
+0.11
+0.12
+0.02
+0.05
+0.07
+0.22
+0.25
+0.37
+0.62
+0.82
+0.49
+0.74
+Table 5: We compare multiple pretraining design decisions: model size, pretraining data curation, domain-specific pretraining objectives and
+tokenizer. We consider at top-1 for Last-Mile Repair (LMR) and Syntax Reconstruction (SR) and top-5 for Autocompletion (AC) with Exact
+Match (EM) and Sketch Match (SM). For details refer to Appendix D. Smaller model performs worse across the board. Curating data with
+global deduplication reduces performance by up to 30 points. Removing domain-specific objectives and tokenizer impacts performance most.
+our pretraining objectives closely resembles last-mile repair
+(noisy auto-encoding) we find that finetuning FLAME helps
+direct it towards a particular task.
+We summarize the results in Table 2 and observe that on
+the Forum last-mile repair benchmark FLAME outperforms all
+models at top-5, and is second best to Codex-Cushman at top-
+1. In the Test benchmark, we find that FLAME is second-best
+to Codex-Cushman at top-5 and is close to CodeT5’s second-
+best performance at top-1. In the Test benchmark, Davinci’s
+performance is substantially worse than the fine-tuned models.
+On further analysis, we found that all models solve 73% of
+the Forum benchmark. FLAME solves 4% of the benchmarks
+that no other model solves and fails on 1% of the benchmarks
+that all other models fix. FLAME also generates syntactically
+correct formulas for 98% of the benchmarks in top 5. In
+Figure 4, we show examples where FLAME gets the correct
+fix, and other models do not, and vice versa. We note that in
+some cases, FLAME’s fixes appear to be more natural, but fail
+to match the user’s ground truth repair.
+For syntax reconstruction Forum, we find that FLAME outper-
+forms other models across the top-1 and top-5. Interestingly,
+CodeT5 also solves more syntax reconstruction tasks than
+both Codex models. We hypothesize that since syntax recon-
+struction is a new task, as compared to the more traditional
+repair problem, after fine-tuning, encoder-decoder models per-
+form better than decoder-only models, as shown by [Tay et
+al., 2022]. In Test, we find that FLAME performs similar to
+Codex-Cushman (same at top-1 and -2 points lower at top-5).
+We find that 54% of the Forum syntax reconstruction bench-
+marks are solved by all the models, 1% is solved only by
+FLAME, and there are no benchmarks that all other models
+solve but FLAME doesn’t. We attribute this performance to our
+=IF('Jan 13'!B2="", 'Feb 13'!B2="", 'Mar 13'!B2="", 'Apr 13'!B2="", yes, no)   
+=IF(AND('Jan 13'!B2="", 'Feb 13'!B2="", 'Mar 13'!B2="", 'Apr 13'!B2=""), "yes", "no") 
+Buggy Formula
+Ground Truth Fix
+FLAME
+Codex-Cushman
+Codex-Davinci
+CodeT5
+X
+X
+X
+=VLOOKUP($Z25,$X$25:$Y:31,2,FALSE) 
+=VLOOKUP($Z25,$X$25:$Y31,2,FALSE) 
+Buggy Formula
+Ground Truth Fix
+FLAME
+Codex-Cushman
+Codex-Davinci
+CodeT5
+X
+=VLOOKUP($Z25,$X$25:$Y$31,2,FALSE)
+FLAME
+Example 1
+Example 2
+Figure 4: Repair tasks with diverging performance. In Example 1, the
+user did not use the AND function and missed double quotes around
+string literals yes and no. FLAME fixes this (in top-5), while other
+models fail. In Example 2 FLAME’s top candidate is syntactically valid
+but does not match the user’s fix, while other models’ predictions do.
+pretraining design choices. First, FLAME learns to generate
+syntactically correct code as a result of its noisy auto-encoding
+pretraining objective. Second, FLAME learns the natural distri-
+bution of formulas by generating complete sequences during
+pretraining, rather than just mask values and sentinel tokens.
+Zeroshot Performance
+FLAME’s pretraining objectives al-
+low us to consider zeroshot performance for both last-mile
+repair and syntax reconstruction. In Table 4, we observe that
+FLAME outperforms Codex models for last-mile repair across
+all benchmarks. We attribute this to the closeness of our noisy
+auto-encoding pretraining objectives and the last-mile repair
+task. We find that in the syntax reconstruction task, FLAME out-
+performs Codex-Cushman. We believe this is because syntax
+reconstruction can be considered an extreme case of repair.
+
+Formula Autocompletion
+The autoregressive nature of Codex models and FLAME’s pre-
+training objectives allows us to evaluate their zeroshot per-
+formance2 for formula auto-completion. Note that we fine-
+tune CodeT5 for this task as it is pretrained on smaller span
+lengths (1 to 5 tokens) and generates special mask tokens (e.g.,
+<MASK1>) in a zeroshot setting. We compute exact match and
+sketch match metrics with top-5 results.
+In Table 3, we observe that FLAME performs better than all
+the larger models on the exact match metric and 3 out of 5 pre-
+fix lengths for sketch match. We note that Codex-Cushman and
+Codex-Davinci fail to complete 14% and 15% of the bench-
+marks with 0.99 fraction of the prefix, respectively, whereas
+FLAME fails to complete 6% of the benchmarks. We observe
+significantly lower performance by CodeT5, likely due to
+the lack of longer masks spans during pretraining. Surpris-
+ingly, Codex-Davinci performs slightly worse than the smaller
+Codex-Cushman for 3 out of 5 prefix lengths. Inspection of
+completions shows that Codex-Davinci tends to generate more
+tokens than required when completing these benchmark tasks.
+We also observe cases where models succeed with a shorter
+prefix but fail given a longer prefix.
+5.2
+RQ2: Pretraining design decisions
+We investigate FLAME’s data curation, model size, the use of
+domain-specific pretraining objectives, and domain-specific
+tokenizer, and present results in Table 5.
+Training data curation
+Previous work [Lee et al., 2021; Kandpal et al., 2022] have
+shown that deduplication can improve the performance of
+language models and reduce the memorization of training
+data. Therefore, we curate a pretraining dataset by performing
+workbook-level sketch-based formula deduplication. Alterna-
+tively, one might consider performing global (pooled across
+all workbooks) sketch-based deduplication. This alternative
+results in a pretraining set of 591K formulas. Table 5 shows
+that training on this smaller corpus results in a lower perfor-
+mance model . We find that FLAME’s zeroshot performance
+falls by 14 points and finetuned performance falls by 18 points
+for last-mile repair in Forum benchmarks.
+Model size
+We trained two variants of FLAME with 16M and 60M parame-
+ters. Table 5 compares FLAME-16M and FLAME-60M. We find
+that performance declines slightly across tasks/benchmarks
+when we reduce the model size to 16M. However, note that
+FLAME-16M can still outperform larger models such as Codex
+in 5 out of 10 zeroshot and finetuned settings, highlighting the
+efficacy of our design choices for FLAME.
+Pretraining objectives and Tokenizer
+To evaluate the effectiveness of our domain-specific pretrain-
+ing objectives and tokenizer, we pretrained a 60M parameters
+T5 model with generic pertaining objectives and tokenizer.
+Specifically, this model uses tail-masking, masked span pre-
+diction without accounting for lexer token boundaries, and
+2We finetuned Codex-Cushman and FLAME but observed worse
+performance, possibly from over-fitting.
+MAX C2 Sum C3:C4 SUM C5:C7 1
+MAX(C2, Sum(C3:C4),SUM(C5:C7),1)
+MAX(C2,Sum!C3:C4,SUM(C5:C7),1)
+Tokens
+Formula
+T5 (Generic Pretraining and Tokenizer)
+Figure 5: Failing case of syntax reconstruction. Due to the different
+capitalization of Sum and SUM, the model treats them as different
+tokens, converting them to an identifier and a function, respectively.
+random denoising objectives. Additionally, it uses the CodeT5
+tokenizer trained on our pretraining data. Table 5 shows that
+this variant performs worse across all tasks and benchmarks,
+both in a zeroshot and finetuned setting. We attribute the huge
+drop, up to 62 points, in last-mile repair tasks in zeroshot to
+our user-inspired denoising pretraining objective. Moreover,
+we hypothesize that FLAME’s good syntax reconstruction per-
+formance can be attributed to the domain-specific tokenizer.
+Figure 5 illustrates how the generic tokenizer treats tokens
+with different capitalizations, resulting in incorrect generation.
+5.3
+RQ3: Decoding strategy
+In Table 6, we evaluate FLAME using four different decoding
+strategies, Beam Search, Group Beam Search [Vijayakumar et
+al., 2016], Nucleus Sampling [Holtzman et al., 2019] and Top
+K Sampling [Fan et al., 2018]. We find FLAME to perform bet-
+ter with group beam search decoding (group size of 2) for all
+the formula intelligence tasks. However, for autocompletion
+with sketch match, nucleus sampling showed superior perfor-
+mance. We believe this is because autocompletion requires
+more diverse results, particularly at shorter prefixes. Refer to
+Appendix E for autocompletion table.
+Decoding Method
+LMR (Forum)
+SR (Forum)
+T@1
+T@5
+T@1
+T@5
+Beam Search
+0.76
+0.88
+0.75
+0.89
+Group Beam
+0.76
+0.89
+0.75
+0.89
+Nucleus Sampling
+0.72
+0.85
+0.7
+0.84
+Top K
+0.67
+0.86
+0.67
+0.84
+Table 6: Performance by decoder strategy for last mile repair (LMR)
+and syntax reconstruction (SR). Beam and Grouped Beam Search
+have similar performance, and outperform Nucleus, Top K Sampling.
+6
+Conclusions and Future Work
+We present FLAME, a small (60M parameter) language model
+for spreadsheet formulas, which captures domain-specific
+properties in its data curation, tokenization, and pretraining
+objectives. We implemented FLAME for Excel formulas and
+evaluate on three downstream tasks: last-mile repair, autocom-
+pletion, and a novel task that we term syntax reconstruction.
+We compare with the much larger models CodeT5, Codex-
+Cushman, and Codex-Davinci. When fine-tuned, FLAME can
+achieve top performance in 6 of our 10 experimental settings,
+despite having two orders of magnitude fewer parameters.
+Future work will explore downstream tasks that require
+additional spreadsheet context (e.g. tables). To tackle such
+tasks we will explore extending our pretraining objectives
+to incorporate context and the extent to which FLAME can
+integrate with existing table encoder models.
+
+Acknowledgments
+We thank Microsoft Research Cambridge for sharing the Ex-
+cel corpus used for pretraining FLAME. We thank OCTO at
+Microsoft (in particular Gopi Kumar and the AMD vTeam)
+for providing us with compute resources. We also thank the
+Excel team for their feedback and encouragement in pursuing
+this work.
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+
+A
+User noise operators
+We implement the following noise operators:
+1. Wrong Range: we replace the range operator :, with
+one of the following symbols: {; , space "}, or we
+delete the range operator.
+2. Malformed Range:
+A range consists of 4 el-
+ements:
+col1,
+row1,
+col2,
+row2
+written
+as
+col1row1:col2row2. We randomly delete one of these
+elements. For eg: col1:col2row2
+3. Space between Function and Arguments in a Call:
+We introduce a space between the function name and
+the opening parentheses for built-in functions. For exam-
+ple: SUM(A1:A10) converts to SUM (A1:A10)
+4. Change number of arguments: We change the num-
+ber of arguments for functions with fixed function arity.
+For example, IF has a minimum arity of 2 and maxi-
+mum arity of 3. Specifically, if a function contains ar-
+guments equal to its minimum function arity, then we
+randomly delete one argument. Whereas, if the func-
+tion’s max arity is equal to the number of arguments,
+then we randomly copy one of the existing arguments
+and pass it as an additional argument to the function.
+For example, IF(A2>10, True, False) can become
+IF(A2>10, True, False, False)
+5. Swap arguments: If a function takes different types of
+arguments, then we swap these arguments. For example:
+IF(A1>10, 1, 2) can become IF(1, A1>10, 2).
+6. Space between relational operators:
+We add space
+between relational operators, such as < =.
+7. Swap relational operators: We swap relational opera-
+tors, such as <= turns to =<
+8. Inequality noise operator: In Excel <> is the inequality
+operator. We replace this with the incorrect != or =!.
+9. Invalid Equality: We also corrupt the equality operator.
+The equality operator in Excel is =, we replace it with ==
+or ===.
+10. Malformed Sheet Name: Multi-word sheet names in
+Excel need to be enclosed within single quotes (’<sheet
+name>’). We randomly choose to either delete the single
+quotes or replace them with double quotes. For example,
+’Sheet 1’!A10 can become "Sheet 1"!A10.
+11. Remove exclamation Mark:
+In Excel, sheet names
+are followed by an exclamation mark to denote sheet
+reference. We delete this exclamation mark.
+12. Malformed Strings: We corrupt strings by either delet-
+ing the double quotes or replacing them with single
+quotes.
+13. Add Comma and Remove Parentheses: We randomly
+choose to either insert a comma before a closing parenthe-
+sis or insert a comma and delete the closing parentheses.
+14. Add random operators: We define a set of operators
+that we randomly insert into the formula at a random
+position. These operators are: {+ - * / ^& < > = .
+)
+#}
+15. Add operator at the end:
+We randomly add one of
+the operators mentioned previously at the end of the se-
+quence.
+16. Add Parentheses: We add opening and closing paren-
+thesis at random places.
+17. Corrupting Unreliable tokens: Following [Bavishi et
+al., 2022], we randomly add, delete or replace unreliable
+tokens. Unreliable tokens are tokens where users often
+make mistakes, defined to be delimiters.
+B
+Codex Prompts
+For all our codex experiments, we use the following prompts
+for zeroshot and finetuning and use a temperature of 0.7
+B.1
+Repair - Zeroshot and Finetuning
+##### Fix bugs in the below code
+### Buggy Excel
+<Buggy Formula>
+### Fixed Excel
+<Fixed Ground Truth Formula>
+##### Fix bugs in the below code
+### Buggy Excel
+=SUMIFS(
+Master!$P:$P,
+Master!$F:$F,$A7,
+Master856!$E:212Systems$B7
+)
+### Fixed Excel
+B.2
+Syntax Reconstruction - Zeroshot and
+Finetuning
+### Excel Tokens
+<Flat Stream of Tokens>
+### Complete Excel Formula
+<Formula>
+### Excel Tokens
+INDEX Table1 SMALL IF
+Table1 COMPANY_NAME = $E$1
+ROW Table1 COMPANY_NAME - 1
+ROW 2:2 3
+### Complete Excel Formula
+B.3
+Autocomplete - Zeroshot
+### Excel Formula
+<Partial Excel Formula>
+<Formula>
+### Excel Formula
+IF(FALSE,NA(
+
+Models
+Exact Match
+Sketch Match
+0.25
+0.50
+0.75
+0.90
+0.99
+0.25
+0.50
+0.75
+0.90
+0.99
+FLAME (60M)
+0.01
+0.06
+0.34
+0.70
+0.93
+0.10
+0.24
+0.55
+0.84
+0.94
+FLAME (16M)
+0
+0.03
+0.24
+0.59
+0.89
+0.11
+0.25
+0.54
+0.76
+0.90
+Global Deduplication
+0
+0.03
+0.15
+0.45
+0.64
+0.10
+0.25
+0.41
+0.59
+0.70
+T5 (Generic objectives and tokenizer)
+0
+0.07
+0.07
+0.22
+0.21
+0.01
+0.09
+0.25
+0.37
+0.29
+Table 7: Design choice experiments for autocompletion task. We compare multiple pretraining design decisions: model size, pretraining data
+curation, domain-specific pretraining objectives and tokenizer. We consider top-5 for Autocompletion (AC) with Exact Match (EM) and Sketch
+Match (SM). We note that FLAME outperforms all the models.
+Models
+Exact Match
+SketchMatch
+0.25
+0.5
+0.75
+0.9
+Total
+0.25
+0.5
+0.75
+0.9
+Total
+Beam Search
+0.00
+0.06
+0.33
+0.71
+0.92
+0.10
+0.25
+0.54
+0.82
+0.94
+Group Beam Search (groups = 2)
+0.01
+0.06
+0.34
+0.70
+0.93
+0.10
+0.24
+0.55
+0.84
+0.94
+Nucleus Sampling
+0.00
+0.04
+0.26
+0.59
+0.92
+0.14
+0.30
+0.53
+0.74
+0.92
+TopK Sampling
+0.00
+0.04
+0.25
+0.62
+0.92
+0.15
+0.30
+0.55
+0.76
+0.92
+Table 8: Performance by decoder strategy for Autocompletion (top 5) with Exact Match and Sketch Match. Beam Search outperforms all the
+strategies – Group Beam Search with a group size of 2, Nucleus Sampling, and Top K Sampling.
+C
+Training details
+We use the following HuggingFace configuration to train
+FLAME:
+{
+"architectures": [
+"T5ForConditionalGeneration"
+],
+"d_ff": 1024,
+"d_kv": 64,
+"d_model": 512,
+"decoder_start_token_id": 0,
+"dropout_rate": 0.1,
+"bos_token_id": 1,
+"eos_token_id": 2,
+"feed_forward_proj": "gated-gelu",
+"initializer_factor": 1.0,
+"is_encoder_decoder": true,
+"layer_norm_epsilon": 1e-06,
+"model_type": "t5",
+"num_decoder_layers": 8,
+"num_heads": 6,
+"num_layers": 8,
+"output_past": true,
+"pad_token_id": 0,
+"relative_attention_num_buckets": 32,
+"tie_word_embeddings": false,
+"vocab_size": 16479
+}
+We use an AdaFactor optimizer, with 1e-4 learning rate,
+clip factor of 1.0, with scale parameters and relative steps set
+to false. For fine-tuning, we use a weight decay of 0.1 We use
+linear learning rate schedule with 100 warm-up steps.
+D
+Design Decision (Autocompletion)
+We detail our autocompletion evaluation where we evaluate
+FLAME against different variations in Table 7. We observe that
+FLAME beats all the different model variants.
+E
+Decoder Autocompletion
+In Table 8, we detail autocompletion results for different de-
+coding strategies. We find that Beam Search beats other decod-
+ing methods in 7 out of 10 prefix lengths, and Top K Sampling
+beats others in Sketch Match for smaller fractions of prefixes.
+

1tAzT4oBgHgl3EQft_25/content/tmp_files/2301.01685v1.pdf.txt

@@ -0,0 +1,2226 @@
+arXiv:2301.01685v1  [math.AP]  4 Jan 2023
+Global existence and decay of small solutions for
+quasi-linear second-order uniformly dissipative
+hyperbolic-hyperbolic systems
+Matthias Sroczinski∗
+January 5, 2023
+Abstract
+This paper is concerned with quasilinear systems of partial differential equations
+consisting of two hyperbolic operators interacting dissipatively. Its main theorem es-
+tablishes global-in-time existence and asymptotic stability of strong solutions to the
+Cauchy problem close to homogeneous reference states. Notably, the operators are not
+required to be symmetric hyperbolic, instead merely the existence of symbolic sym-
+metrizers is assumed. The dissipation is characterized by conditions equivalent to the
+uniform decay of all Fourier modes at the reference state. On a technical level, the
+theory developed herein uses para-differential operators as its main tool. Apparently
+being the first to apply such operators in the context of global-in-time existence for
+quasi-linear hyperbolic systems, the present work contains new results in the field of
+para-differential calculus. In the context of theoretical physics, the theorem applies
+to recent formulations for the relativistic dynamics of viscous, heat-conductive fluids
+notably such as that of Bemfica, Disconzi and Noronha [1] (Phys. Rev. D, 98:104064,
+2018.).
+Keywords. hyperbolic systems, initial value problem, global existence, asymptotic
+stability, para-differential operators, fluid mechanics
+AMS subject classifications.
+Primary 35A01, 35B35, 35L72, 35L15, 35S50,
+35Q35, 35Q75
+∗Department
+of
+Mathematics,
+University
+of
+Konstanz,
+78457
+Konstanz,
+Germany.
+matthias.sroczinski@uni-konstanz.de, https://orcid.org/0000-0002-5472-2741
+1
+
+1
+Introduction and main result
+In this paper, we study systems of partial differential equations that are given by the su-
+perposition of two hyperbolic operators and show that homogeneous states are nonlinearly
+stable in the sense that small perturbations thereof lead to global-in-time decaying solutions.
+Concretely, we consider the Cauchy problem for quasi-linear systems of the form
+d
+�
+j=0
+Aj(u(t, x))uxj(t, x) =
+d
+�
+j,k=0
+(Bjk(u(t, x))uxj(t, x))xk,
+x0 = t ≥ 0, x = (x1, . . . , xd) ∈ Rd,
+(1.1)
+u(0, x) = u0(x),
+ut(0, x) = u1(x),
+x ∈ Rd,
+(1.2)
+where both the operator on the right hand side and the operator on the left hand side are
+hyperbolic and each of them acts dissipatively on the trajectories generated by the other one.
+Such systems occur in theroretical physics as recent formulations for the (special-)relativistic
+dynamics of viscous, heat conductive fluids [15, 16, 17, 1, 12, 2]. Our results apply to these
+formulations. The main theorem is the following.
+1.1 Theorem. Consider d ≥ 3, s > d/2 + 1, ¯u ∈ Rn and let (1.1) satisfy conditions
+(HA), (HB) and (D) from Section 3. Then there exist constants δ > 0 and C = C(δ) > 0
+such that the following holds: For all u0, u1 with u0 − ¯u ∈ Hs+1(Rd, Rn) ∩ L1(Rd, Rn), u1 ∈
+Hs(Rd, Rn)∩L1(Rd, Rn) as well as ∥u0 − ¯u∥Hs+1, ∥u1∥Hs, ∥u− ¯u∥L1, ∥u1∥L1 < δ there exists a
+unique global solution u of (1.1), (1.2) satisfying u − ¯u ∈ C([0, ∞), Hs+1) ∩ C1([0, ∞), Hs),
+l = 0, . . . , s + 1 and, for all t ∈ [0, ∞),
+∥u(t) − ¯u∥Hs + ∥ut(t)∥Hs−1 ≤ C(1 + t)− d
+4(∥u0 − ¯u∥Hs + ∥u1∥Hs−1 + ∥u0 − ¯u∥L1 + ∥u1∥L1),
+(1.3)
+∥u(t) − ¯u∥2
+Hs+1 + ∥ut(t)∥2
+Hs +
+� t
+0
+∥u(τ) − ¯u∥2
+Hs+1 + ∥ut(τ)∥2
+Hs dτ
+(1.4)
+≤ C(∥u0 − ¯u∥2
+Hs+1 + ∥u1∥2
+Hs + ∥u0 − ¯u∥2
+L1 + ∥u1∥2
+L1)
+While conditions (HA) and (HB) specify the assumed hyperbolicity, condition (D), essentially
+obtained in [14], characterizes the needed decay behaviour for the Fourier modes of the
+associated linearized system.
+Based on the famous Kawashima-Shizuta condition [24, 34], analogous results are well-known
+for symmetric hyperbolic-parabolic systems and first-order hyperbolic systems with relax-
+ation, cf. [9, 33, 41, 19, 26, 42, 5] among others.1 Regarding dissipative second-order hyper-
+bolic systems there are fewer results available, cf. notably [11, 27, 32] and references therein,
+all of those treat systems whose structure is different form the one we consider here. The
+1Note that the often available reformulations of (1.1) as first-order hyperbolic systems do typically not
+satisfy the assumptions of these works.
+2
+
+most prominent example for equations satisfying condition (D) are probably damped wave
+equations with a non-linear convection term, which alternatively can be viewed as conserva-
+tion laws with hyperbolic artificial viscosity. In this case, (D) reduces to Whitham’s famous
+sub-characteristic condition [40] and various in-depth results on the asymptotic behaviour
+of solutions have been achieved in this context, cf. [31, 25, 39, 20, 10, 23]. Closest related
+to the present work are [35, 36, 37], there a predecessor of Theorem 1.1 was shown for the
+systems proposed in [16, 17].
+The theory developed in the present work requires novel techniques in the use of para-
+differential operators. Developed by Bony [6] and Meyer [30, 29], such operators have been
+used in the context of hyperbolic equations by G´erard and Rauch [18], Taylor [38] and
+M´etivier [28].
+However, quite different from these works, we will in particular need to
+precisely understand how the norms of para-differential operators depending on the functions
+inducing their symbols.
+The paper is organized as follows. In the crucial Section 2 general results on para-differential
+operators needed for the argumentation in Section 3 and 4 will be established. The present
+work apparently being the first that uses such operators to treat global-in-time solutions to
+quasi-linear hyperbolic systems, we prove corresponding new results on that dependence. The
+technical highlight in this regard will be a modified version of the strong G˚arding inequality.
+In Section 3 we construct a para-differential operator which is specifically associated with the
+system’s dissipativity. Section 4 is dedicated to the proof of Theorem 1.1. The challenging
+part is the treatment of the highest derivatives.
+Here we have to use the sophisticated
+estimates of Section 2 and the construction of Section 3. Finally, Section 5 shows that models
+of equations of dissipative relativistic fluid dynamics satisfy the assumptions of Theorem 1.1.
+2
+Results on para-differential operators
+A tour through the theory of para-differential operators from scratch to fine properties, this
+section relies on Appendix C of Benzoni-Gavage and Serre [4] and Section 9 of H¨ormander
+[22], however with strong attention to symbols induced by what later will be the solution to
+the PDE system considered. In its initial part interpolating between brevity and legibility,
+the section culminates in the aforementioned novel version of the strong G˚arding inequality.
+2.1
+Notation, definitions and basics
+For topological vector-spaces V, W we write L(V, W) for the space of continuous linear oper-
+ators form V to W (or L(V ) if W = V ). Throughout this section consider fixed dimensions
+n, d ∈ N and let m denote some real number. For x, ξ ∈ Rd we just write xξ for their
+Euclidian scalar product.
+3
+
+Let E be a finite-dimensional C-Banach space. We denote the E-valued Schwartz space by
+S(Rd, E), and by S′(Rd, E) := L(S(Rd), E) the space of continuous linear mappings from
+S(Rd) to E, i.e. the space of E-valued temperate distributions, both equipped with the
+standard locally convex topologies. For f ∈ S(Rd, E) the Fourier transform is
+(Ff)(ξ) = ˆf(ξ) = (2π)−d/2
+�
+Rd f(x)e−ixξdx
+with inverse
+(F −1 ˆf)(x) = (2π)−d/2
+�
+Rd
+ˆf(ξ)eixξdξ.
+We write F1 and F2 for the Fourier transform with respect to the first and the second variable
+for functions f ∈ S(Rd × Rd, E), i.e.
+(F1f)(η, y) = F(f(·, y))(η) = (2π)−d/2
+�
+Rd f(x, y)e−ixηdx,
+(F2f)(x, ξ) = F(f(x, ·))(ξ) = (2π)−d/2
+�
+Rd f(x, y)e−iyξdy.
+As usual we extend F, F1, F2 to continuous operators on S′(Rd, E), S′(Rd × Rd, E) and
+unitary operators on L2(Rd, E), L2(Rd × Rd, E) also denoted by F, F1, F2.
+We will use ⟨ξ⟩ := (1 + |ξ|2)
+1
+2, ξ ∈ Rd, Λm := F −1⟨·⟩mF. As usual
+Hm(Rd, E) := {u ∈ L2(Rd, E) : Λmu ∈ L2(Rd, E)},
+are the L2-based E-valued Sobolev spaces with norm
+∥u∥Hm(Rd,E) := ∥Λmu∥L2(Rd,E).
+If E is a Hilbert space we consider the scalar product on Hm(Rd, E)
+⟨u, v⟩Hm(Rd,E) := ⟨Λmu, Λmv⟩L2(Rd,E).
+We also use L∞-based Sobolev spaces
+W k,∞(Rd, E) := {u ∈ L∞(Rd, E) : ∂α
+x u ∈ L∞(Rd, E), |α| ≤ k}
+with norm
+∥u∥W k,∞(Rd,E) = max
+|α|≤k ∥∂α
+x u∥L∞(Rd,E).
+We often just write Hm, ∥u∥m, ⟨u, v⟩m, W k,∞ instead of Hm(Rd, E), ∥u∥Hm(Rd,E), ⟨u, v⟩Hm(Rd,E),
+W k,∞(Rd, E) if there is no concern for confusion, and ∥u∥ for ∥u∥0.
+For A ∈ Cn×n we denote the adjoint matrix by A∗ = ¯At and for T ∈ L(S(Rd, Cn)) we write
+T ∗ for the adjoint operator with respect to the L2(Rd, Cn) inner product. As usual we call T
+4
+
+self-adjoint if T = T ∗ and positive (strictly positive) if ⟨Tf, f⟩0 ≥ 0 (⟨Tf, f⟩ > 0), in which
+case we also write T ≥ 0 (T > 0).
+Next, we turn to the basic definitions concerning pseudo-differential operators which will be
+used in the present paper. We consider the following symbol classes.
+2.1 Definition.
+(i) Sm := Sm(Rd, Cn×n) is the set of all functions a ∈ C∞(Rd×Rd, Cn×n)
+for which for any α, β ∈ N0 there exists Cαβ > 0 such that
+|∂β
+x∂α
+ξ a(x, ξ)| ≤ Cαβ⟨ξ⟩m−|α|.
+(2.1)
+With semi-norms being the optimal constants in (2.1), Sm is a Fr´echet space.
+(ii) Sm
+1,1 := Sm
+1,1(Rd, Cn×n) is the set of functions a ∈ C∞(Rd × Rd) for which for any
+α, β ∈ Nd
+0 there exist Cαβ > 0 such that
+|∂β
+x∂α
+ξ a(x, ξ)| ≤ Cαβ⟨ξ⟩m−|α|+|β
+(2.2)
+for all (x, ξ) ∈ Rd × Rd. With semi-norms being the optimal constants in (2.2), Sm
+1,1 is
+a Fr´echet space.
+(iii) For a ∈ Sm
+1,1 the mapping op[a] ∈ L(S(Rd, Cn)) defined by
+(op[a]f)(x) := (2π)− d
+2
+�
+eixξa(x, ξ)Ff(ξ) dξ.
+(2.3)
+is called the pseudo-differential operator with symbol a. We also write a := Sym[op[a]].
+As first shown in [7, 8] for a ∈ Sm
+1,1 the operator op[a] extends to a bounded operator from
+Hl+m to Hl only if op[a]∗ also has a symbol in S1,1
+m . But the operator norm of op[a] can
+in general not be controlled by semi-norms of a uniformly over this subspace. As for our
+applications to dissipative hyperbolic systems it is essential that the norm of op[a] is small
+if the semi-norms of a are small we have to make sure that the symbols occurring in the
+present work belong to the following smaller subspaces.
+2.2 Definition. For L ∈ (0, 1], Sm,L
+1,1
+is the subspace of all a ∈ Sm
+1,1 such that F1a vanishes
+on NL := {(η, ξ) ∈ Rd × Rd : |η + ξ| + 1 < L|ξ|} in the sense of distributions, i.e.
+a(F1φ) = 0 for all φ ∈ S(Rd × Rd) with supp φ ⊂ NL.
+(2.4)
+2.3 Proposition. Let L ∈ (0, 1]. For all l ∈ R and a ∈ Sm,L
+1,1
+op[a] extends to a continuous
+operator form Hl+m to Hl and op is itself continuous from Sm,L
+1,1
+to L(Hl+m, Hl).
+Proof. Cf. [22], Proposition 9.3.1.
+The symbols in Sections 2 and 3 will be induced by functions (x, ξ) �→ F(u(x), ξ) where
+F ∈ C∞(Rn × Rd), u ∈ W k,∞(Rd, Rn) for some k ∈ R, i.e. they belong to the following
+symbol class.
+5
+
+2.4 Definition. For any k ∈ N0 the set Γm
+k of symbols of order m with regularity k is the
+set of functions A : Rd × Rd �→ Cn×n such that,
+(i) for almost all x ∈ Rd the mapping ξ �→ A(x, ξ) is in C∞(Rd, Cn×n)
+(ii) for any α ∈ Nd
+0 and ξ ∈ Rd the mapping x �→ ∂α
+ξ A(x, ξ) belongs to W k,∞(Rd, Cn×n)
+and there exists Cα > 0 not depending on ξ such that
+∥∂α
+ξ A(·, ξ)∥W k,∞ ≤ Cα⟨ξ⟩m−|α|.
+(2.5)
+With the semi-norms being the optimal constants in (2.5), Γm
+k is a Fr´echet space.
+Para-differential operators associated with symbols in Γm
+k are defined as follows.
+2.5 Definition. For ǫ = (ǫ1, ǫ2) with 0 < ǫ1 < ǫ2 < 1 we call a function χ ∈ C∞(Rd × Rd)
+an admissible ǫ-cut-off if χ is even with respect to each variable, χ(Rd × Rd) ⊂ [0, 1],
+χ(η, ξ) =
+�
+1,
+|η| ≤ ǫ1|ξ| and |ξ| ≥ 1
+0,
+|η| ≥ ǫ2⟨ξ⟩ or |ξ| ≤ ǫ2
+(2.6)
+for all η, ξ ∈ Rd and for all α, β ∈ Nd there exists Cα,β > 0 such that for all ξ, η ∈ Rd
+|∂β
+η ∂α
+ξ χ(η, ξ)| ≤ Cα,β⟨ξ⟩−|α|−|β|.
+2.6 Proposition. Let χ be an admissible ǫ-cut-off. Set Kχ := F −1
+1 (χ) and consider the
+function Rχ : Γm
+k → C∞(Rd × Rd) given by
+Rχ(A) := Kχ ∗1 A,
+A ∈ Γm
+k .
+Then Rχ defines a bounded linear operator from Γm
+k to Sm,1−ǫ2
+1,1
+∩ Γm
+k . Here
+(Kχ ∗1 A)(x, ξ) =
+�
+Rd Kχ(x − y, ξ)A(y, ξ) dy.
+Proof. Apart from the aspect that a is not only in Sm
+1,1 but even in Sm,1−ǫ2
+1,1
+the proof can
+be found in [4], Proposition C.16. But that aspect follows in a straightforward manner as
+|η + ξ| + 1 ≤ (1 − ǫ2)|ξ| implies |ξ| − |η| + 1 ≤ (1 − ǫ2)|ξ| and thus |η| ≥ ǫ2⟨ξ⟩ and χ vanishes
+for such η, ξ.
+2.7 Definition. Let χ be an admissible ǫ-cut-off.
+For A ∈ Γm
+k the (χ-)para-differential
+operator with symbol A is defined by
+Opχ[A] := op[Rχ(A)].
+As Rχ ∈ L(Γm
+k , Sm,1−ǫ2
+1,1
+), Opχ = op ◦Rχ defines a continuous linear operator from Γm
+k to
+L(Hl+m, Hl) (l ∈ R). In particular the L(Hl+m, Hl)-norm of Opχ[A] can be estimated by a
+constant depending on l, χ and a finite sum of Γk
+m-semi-norms of A.
+6
+
+The following shows that, regarding its dependence on χ, opχ[A] is determined by A up to
+a lower order operator, if k ≥ 1.
+2.8 Lemma. Let χ be an admissible ǫ-cut-off and k ≥ 1. Then the following holds:
+(i) The mapping Rχ − Id is a continuous operator from Γm
+k to Γm−1
+k−1 .
+(ii) If ˜χ is an admissible ˜ǫ-cut-off, then Rχ − R˜χ is a continuous operator from Γm
+k to
+Sm−1,1−τ
+1,1
+∩ Γm−1
+k−1 with τ = max{ǫ2, ˜ǫ2}.
+Proof. Cf. [4], Proposition C.13, Corollary C.5.
+We end this subsection by stating two additional results on para-differential operators for
+later usage. The proofs are contained in [4], Appendix C. To simplify notation we fix an
+admissible ǫ-cut-off χ and suppress the dependence of Rχ and Opχ on χ in the following.
+We call an operator K infinitely smoothing if K ∈ L(Hs, Hl) for all s, l ∈ R.
+2.9 Lemma. Let b ∈ Sm be constant with respect to the first variable. Then the following
+holds:
+(i) op(b) − Op[b] is infinitely smoothing.
+(ii) Op[b] = Op[b∗]
+(iii) Op[Ab] = Op[A]F −1bF for any A ∈ Γµ
+k.
+2.10 Lemma. For each k > 0 there exists C > 0 such that for all f ∈ L∞ ∩ Hk, A ∈
+W 1,∞ ∩ Hk
+∥A − Op[A]f∥k ≤ C(∥A∥Hk∥f∥L∞ + ∥A∥W 1,∞∥f∥Hk−1).
+2.2
+Adjoints and products
+For the argumentation in Section 3 it will be essential to control the norms of operators
+Op[A∗] − Op[A]∗, Op[BA] − Op[B] Op[A], A ∈ Γm
+1 , B ∈ Γµ
+1, µ ∈ R, in terms of the semi-
+norms of A and B. While for a ∈ Sm,L
+1,1 , b ∈ Sm,L
+1,1
+there exist symbols g ∈ Sm
+1,1, h ∈ Sm+µ
+1,1
+such that op[a]∗ = op[g], op[b] op[a] = op[h] and that, provided ∂xja ∈ Sm
+1,1(j = 1, . . . , d),
+op[a∗] − op[a]∗ ∈ L(Hl+m−1, Hl), op[b] op[a] − op[ba] ∈ L(Hl+m+µ−1, Hl), l ∈ R, it is not
+true in general that g, h are again in some class Sm,L
+1,1 , Sm+µ,L
+1,1
+which would allow to control
+their operator norms. However, for our purposes it is sufficient to consider symbols of the
+particular form a = R(A), b = R(B) for A ∈ Γm
+1 , B ∈ Γµ
+1 and we will show that in this case
+the symbols of op[a]∗ = Op[A]∗, op[b] op[a] = Op[B] Op[A] are in fact again in Sm,L
+1,1 , Sm+µ,L
+1,1
+for some L ∈ (0, 1].
+As a first step note that for symbols in S(Rd × Rd, Cn×n) there exist neat formulas for the
+symbols of adjoint and product operators, which also can be found in [22].
+7
+
+2.11 Lemma. If a ∈ S(Rd × Rd), then op[a]∗ = op[g] with F1g(η, ξ) = (F1a(−η, η + ξ))∗.
+2.12 Lemma. If a, b ∈ S(Rd × Rd, Cn×n), then op[b] op[a] = op[h] with
+F1h(η, ξ) =
+�
+Rd F1b(η − θ + ξ, θ)F1a(θ − ξ, ξ)dθ.
+The significance of this result lies in the following observation.
+2.13 Lemma. Let A ∈ Γm
+0 .
+Then there exists a sequence (aν)ν≥1 ⊂ S(Rd × Rd, Cn×n)
+such that op[aν]u → Op[A]u, ν → ∞ in S(Rd, Cn) for all u ∈ S(Rd, Cn). Furthermore
+for all δ ∈ (ǫ2, 1), ǫ2 being the constant of the ǫ-cut-off, there exists ν0 > 0 such that
+supp F1aν ⊂ {(η, ξ) ∈ Rd × Rd : |η| ≤ δ⟨ξ⟩} for all ν ≥ ν0.
+Proof. The first part of the statement is shown as in [21], proof of Theorem 18.1.8. However,
+we have to slightly modify the construction to also obtain the second part. Set a := R(A).
+Choose ˆφ ∈ S(Rd) with supp ˆφ ⊂ B0(1), F −1 ˆφ(0) = 1 and define φ := F −1 ˆφ,
+aν(x, ξ) := φ(x/ν)φ(ξ/ν)a(x, ξ),
+x, ξ ∈ Rd.
+The asserted convergence then follows as ibid.
+It remains to show the statement concerning the supports. Set ψν(x, ξ) := φ(x/ν)φ(ξ/ν)
+(ξ, η ∈ Rd, ν ≥ 1). As F1(aν) = (2π)−d/2(F1ψν) ∗1 (F1a) and F1a = χF1A, it is sufficient
+to show that for given δ ∈ (ǫ2, 1) and ν sufficiently large χ(η − θ, ξ)F1ψν(θ, ξ) = 0 for all
+θ, η, ξ ∈ Rd |η| ≥ δ⟨ξ⟩. Clearly
+F1ψν(θ, ξ) = νd ˆφ(θν)φ(ξ/ν).
+As by construction ˆφ(θν) = 0 for |θ| ≥ ν−1 we can assume |θ| ≤ ν−1. Then |η| ≥ δ⟨ξ⟩ yields
+|η − θ| ≥ |η| − |θ| ≥ δ⟨ξ⟩ − ν−1.
+Hence choosing ν so large that ν−1 ≤ δ − ǫ2 gives (note ⟨ξ⟩ ≥ 1)
+|η − θ| ≥ δ⟨ξ⟩ − (δ − ǫ2)⟨ξ⟩ = ǫ2⟨ξ⟩.
+But this implies χ(η − θ, ξ) = 0, which finishes the proof.
+2.14 Proposition. Let A ∈ Γm
+0 . Then there exists b = b(A) ∈ Sm,1−ǫ2
+1,1
+such that Op[A]∗ =
+op[b(A)]. Furthermore if A ∈ Γ1 the operator
+T : Γm
+1 → Sm−1,1−ǫ2
+1,1
+, a �→ b(A) − R(A)∗
+is continuous. In particular the mapping
+A �→ Op[A∗] − Op[A]∗ = op[R(A)∗ − b(A)]
+is continuous from Γm
+1 to L(Hl+m−1, Hl) for any l ∈ R.
+8
+
+Proof. Set a := R(A). As A ∈ Sm,1−ǫ2
+1,1
+, the existence of b := b(A) ∈ Sm
+1,1 with op[b] = Op[A]∗
+follows by [22], Lemma 9.4.1.
+Next we prove that F1b vanishes on N1−ǫ2. If A ∈ S(Rd×Rd, Cn×n) also a ∈ S(Rd×Rd, Cn×n)
+and Lemma 2.11 gives
+F1b(η, ξ) = (F1a(−η, η + ξ))∗.
+If now |η + ξ| + 1 ≤ (1 − ǫ2)|ξ| then ǫ2|ξ| ≤ |η| and thus
+ǫ2⟨η + ξ⟩ ≤ ǫ2(1 + |η + ξ|) ≤ (1 − ǫ2)ǫ2|ξ| ≤ (1 − ǫ2)|η| ≤ |η|,
+which implies F1a(−η, η + ξ) = (χF1A)(−η, η + ξ) = 0.
+For general A choose a sequence (aν)ν≥1 ⊂ S(Rd × Rd, Cn×n) with op[aν]u → Op[A]u in
+S(Rd, Cn) for all u ∈ S(Rd, Cn). This implies op[aν]∗ → Op[a]∗ = op[b] in S′(Rd × Rd, Cn×n)
+and it is straightforward to show that this yields bν → b ∈ S′(Rd × Rd, Cn×n), where
+F1bν(η, ξ) = F1aν(−η, η+ξ). By Lemma 2.13 F1aν(η, ξ) vanishes for |η| ≥ δ⟨ξ⟩, if δ ∈ (ǫ2, 1).
+As seen above this yields bν ∈ Sm,1−δ
+1,1
+. In conclusion b = limν→∞ bν ∈ Sm,1−δ
+1,1
+for all δ > ǫ2,
+i.e. b ∈ Sm,1−ǫ2
+1,1
+.
+Lastly A ∈ Γm
+1 directly gives ∂δ
+xA ∈ Γm
+0 and hence ∂δ
+xR(A) = R(∂δ
+xA) ∈ Sm
+1,1 (|δ| = 1) . By
+[22], Lemma 9.6.1 (applied to N = 1, mN = m − 1) we now obtain b − R(A) ∈ Sm−1
+1,1
+and its
+Sm
+1,1-semi-norms are bounded by a constant times a sum of finitely many Sm
+1,1-semi-norms of
+∂δ
+xR(A) (|δ| = 1). As also b − R(A) ∈ Sm−1,1−ǫ2, the assertion follows by the continuity of R
+and op.
+Concerning the analysis of product operators we first consider the difference Rχ(AB) −
+Rχ(A)Rχ(B).
+2.15 Lemma. For A ∈ Γm
+1 , B ∈ Γµ
+1 and an ǫ-cut-off χ with ǫ2 < 1/2 we have R��(B)Rχ(A) ∈
+Sm+µ,1−2ǫ2
+1,1
+. Furthermore the bilinear operator
+T : Γm
+1 × Γm
+1 → Sm+µ−1,1−2ǫ2
+1,1
+,
+(A, B) �→ Rχ(AB) − Rχ(A)Rχ(B)
+is continuous.
+Proof. We suppress the superscript χ in the following. As R(A) ∈ Sm
+1,1, R(B) ∈ Sµ
+1,1, it
+is clear that R(B)R(A) ∈ Sm+µ
+1,1 . Thus regarding the first assertion we need to show that
+R(B)R(A) vanishes on N1−2ǫ2.
+Since F1(R(B)R(A)) = (2π)−d/2F1R(B) ∗1 F1R(B) and
+F1R(A) = χF1A, F1R(B) = χF1B, it is sufficient to prove that χ(η − θ, ξ)χ(θ, ξ) vanish for
+all θ, η, ξ ∈ Rd with |η +ξ|+1 ≤ (1−2ǫ2)|ξ|. Take such θ, η, ξ. If χ(θ, ξ) ̸= 0 then |θ| ≤ ǫ2⟨ξ⟩
+and |η + ξ| + 1 ≤ (1 − 2ǫ2)|ξ| implies |η| ≥ 2ǫ2|ξ| + 1. Together this yields
+|η − θ| ≥ |η| − |θ| ≥ 2ǫ2|ξ| + 1 − ǫ2ξ ≥ ǫ2⟨ξ⟩.
+Now χ(η − θ, ξ) vanishes for such η, θ, ξ, wich completes the argument.
+9
+
+In regard to the continuity we write
+R(BA) − R(B)R(A) = R(BA) − BA + B(A − R(A)) − (R(B) − B)R(A).
+Hence it follows from Lemma 2.8 (i) and the continuity of R that T is continuous as an
+operator to Γm+µ−1
+0
+. Thus the proof is finished if we show that each Sm+µ−1
+1,1
+-semi-norm can
+be bounded by a constant times a finite sum of Γm+µ−1
+0
+-semi-norms of T(A, B). We show
+that even the following holds: For all α, β ∈ N0 there exists Cβ > 0 such that
+|∂β
+x∂α
+ξ T(A, B)(x, ξ)| ≤ Cαβ|∂α
+ξ T(A, B)(x, ξ)|⟨ξ⟩|β|,
+x, ξ ∈ Rd.
+(2.7)
+By Bernstein’s Lemma applied to ∂α
+ξ T(a, b)(·, ξ) (cf.
+e.g.
+[4], Lemma C.3) this can be
+deduced from the fact that for all ξ ∈ Rd
+supp
+�
+(FT(A, B))(·, ξ)
+�
+⊂ B(0, 2ǫ2⟨ξ⟩).
+In fact,
+supp(F(R(ba)(·, ξ)) ⊂ supp(χ(·, ξ)) ⊂ B(0, 2ǫ2⟨ξ⟩)
+holds by definition of χ and that F1(R(b)R(a)) vanishes for all η, ξ with |η| > 2ǫ2⟨ξ⟩ follows
+by the same argumentation as in the first part of the proof.
+We can now prove our main proposition concerning products of para-differential operators.
+2.16 Proposition. Let A ∈ Γm
+0 , B ∈ Γµ
+0. Then for L := (1 − ǫ2)2 there exists h(B, A) ∈
+Sµ+m,L
+1,1
+such that Op[B] Op[A] = op[h(B, A)]. Furthermore the operator
+Γm
+1 × Γm
+1 → L(Hl+µ+m−1, Hl),
+(B, A) �→ Opχ[B] Op[A] − Op[BA]
+is continuous for all l ∈ R.
+Proof. The existence of a h = h(B, A) ∈ Sm+µ
+1,1
+such that
+op[h(B, A)] = op[R(B)] op[R(A)] = Op[B] Op[A]
+follows direclty from [22], Lemma 9.5.1 as R(A) ∈ Sm,1−ǫ2
+1,1
+. We now prove that h satisfies
+(2.4) for L = (1 − ǫ2)2. First assume a := R(A), b := R(B) ∈ S(Rd × Rd). By Lemma 2.12
+F1h(η, ξ) =
+�
+Rd F1b(η − θ + ξ, θ)F1a(θ − ξ, ξ)dθ.
+(2.8)
+Let η, ξ ∈ Rd with |η + ξ| + 1 ≤ (1 − ǫ2)2|ξ|. If F1a(θ − ξ, ξ) = F1R(A)(θ − ξ, ξ) ̸= 0 we have
+|θ − ξ| ≤ ǫ2⟨ξ⟩ ≤ ǫ2 + ǫ2|ξ|, which gives (1 − ǫ2)|ξ| ≤ |θ| + ǫ2. We arrive at
+|η + ξ − θ| ≥ |θ| − |η + ξ| ≥ |θ| − (1 − ǫ2)2|ξ| + 1 ≥ |θ| − (1 − ǫ2)|θ| − (1 − ǫ2)ǫ2 + 1
+= ǫ2θ + ǫ2 + (1 − ǫ2)2 ≥ ǫ2⟨θ⟩.
+10
+
+But this implies F1b(η + ξ − θ, θ) = F1R(B)(η + ξ − θ, θ) = 0, which finishes the argument.
+For general A, B choose sequences (aν)ν≥1, (bν)ν≥1, ⊂ S(Rd × Rd) with op[aν]u → Op[A]u,
+op[bν]u → Op[B]u in S(Rd × Rd, Cn) for all u ∈ S(Rd × Rd, Cn) as constructed im Lemma
+2.13. Then clearly
+op[hν]u = op[bν] op[aν]u → Op[B] Op[A]u = op[h]u
+in S(Rd, Cn), where hν is defined by (2.8) with a, b replaced by aν, bν. This implies hν → h
+in S′(Rd × Rd, Cn×n). As for all 1 > δ > ǫ2 supp F1aν, supp F1bν ⊂ {(η, ξ) ∈ Rd × Rd : |η| ≤
+δ⟨ξ⟩} for ν sufficiently large we get by the same reasoning as above that for all 1 > δ > ǫ2
+hν vanishes on N(1−δ)2 for ν sufficiently large, which proves that h vanishes on N(1−ǫ2)2.
+To prove the second assertion note that by Lemma 2.8 (ii), the mapping G �→ Opχ[G] −
+Op˜χ[G] is continuous from Γk
+1 to L(Hl+k−1, Hl), k, l ∈ R, for any admissible cut-offs χ, ˜χ.
+Hence we can assume w.l.o.g ǫ2 < 1
+2. By Lemma 2.15 and the continuity of op
+(B, A) �→ Op[BA] − op[R(B)R(A)] = op[R(BA) − R(B)R(A)]
+is also continuous as mapping from Γm
+1 × Γm
+1 to L(Hl+µ+m−1, Hl). What is left to show ist
+the continuity of
+(B, A) �→ Op[B] Op[A] − op[R(B)R(A)] = op[h(B, A) − R(B)R(A)].
+As R(A) ∈ Sm,1−ǫ2
+1,1
+and
+∂xjR˜χ(A) = R(∂xjA) ∈ Sm
+1,1,
+∂xjR˜χ(B) = R(∂xjB) ∈ Sm
+1,1,
+j = 1, . . . , d.
+all semi-norms of h(B, A) − R(B)R(A) can be estimated by a constant times a finite sum
+of products of semi norms of ∂xjR(A), ∂xkR(B). Thus as h(B, A) − R(B)R(A) ∈ Sm−1,L
+1,1
+for
+l = min{1 − 2ǫ2, (1 − ǫ2)2} the assertion follows from the continuity of op and R.
+2.3
+Estimates for operators with symbols induced by Sobolev func-
+tions
+In Section 3 the results of Sections 2.1, 2.2 are applied to symbols of the form (x, ξ) �→
+F(u(x), ξ), where F ∈ C∞(U × Rd, Cn×n) (U ⊂ Rn some 0-neighbourhood) and u ∈
+Hs(Rd, Rn) for s sufficiently large. For this purpose we prove the results below.
+In the following let U ⊂ RN be a 0-neighbourhood.
+2.17 Definition. We denote by Sm(U) := Sm(U, Cn×n) the set of all functions F ∈ C∞(U ×
+Rd, Cn×n) for which for any α, β ∈ Nd
+0 there exists Cαβ > 0 such that for all (u, ξ) ∈ U × Rd
+|∂β
+x∂α
+ξ F(u, ξ)| ≤ Cαβ⟨ξ⟩m−|α|.
+(2.9)
+11
+
+For functions F : U × Rd → Cn×n and u : Rd → U we consider the composition
+Fu : Rd × Rd → Cn×n, (x, ξ) �→ F(u(x), ξ).
+2.18 Lemma. Let F ∈ Sm(U) and u ∈ Hs with s > d/2. Then Fu ∈ Γm
+k for k = [s − d/2]
+and for all α ∈ Nd
+0 and each Γm
+k -semi-norm pα(Fu) it holds
+pα(Fu) ≤ Cα(∥u∥s, F),
+and if additionally F(0, ξ) = 0, then
+pα(Fu) ≤ ˜Cα(∥u∥s, F)∥u∥s,
+where Cα, ˜Cα depend on α, F and continuously on ∥u∥s.
+Proof. By Sobolev embedding Hs ֒→ W k,∞. Thus we have Fu(·, ξ) ∈ W k,∞ and
+∥∂α
+ξ Fu(·, ξ)∥W k,∞ ≤ C(∥u∥W k,∞)∥∂α
+ξ F(·, ξ)∥W k,∞(U) ≤ C(∥u∥s)Cα(F)⟨ξ⟩m−|α.
+all ξ ∈ Rd. If F(0, ξ) = 0, we even get, for all ξ ∈ Rd,
+∥∂α
+ξ Fu(·, ξ)∥W k,∞ ≤ C(∥u∥W k,∞)∥u∥W k,∞∥∂α
+ξ Fu(·, ξ)∥W k,∞(U) ≤ C(∥u∥s, F)∥u∥s⟨ξ⟩m−|α|.
+The following proposition will be central for the energy estimates in Section 3. It follows
+directly by the continuity of Op : Γm
+k → L(Hl+m, Hl) and Lemma 2.18 as well as Propositions
+2.14, 2.16 and the facts that Op[F0]∗ = op[F ∗
+0 ] and op[G0F0] − op[G0] op[F0] is infinitely
+smoothing by Lemma 2.9.
+2.19 Proposition. Let F ∈ Sm(U), l ∈ R. Then for all u ∈ Hs with s > d/2 there exists
+Cl = Cl(F, ∥u∥) > 0 depending on l, F and monotonically increasingly on ∥u∥s such that:
+(i) ∥ Op[Fu]∥L(Hl+m,Hl) ≤ Cl(∥u∥s) and for F(0, ·) = 0 ∥ Op[Fu]∥L(Hl+m,Hl) ≤ Cl∥u∥s,
+(ii) for s > d/2 + 1, Op[Fu]∗ − Op[F ∗
+u] ∈ L(Hl−1+m, Hm) and
+∥ Op[Fu]∗ − Op[F ∗
+u]∥L(Hl−1+m,Hm) ≤ Cl∥u∥s
+(iii) for G ∈ Sµ(U) and s > d/2 + 1 there exist Cl,2 = Cl,2(G, ∥u∥s) depending on G and
+monotonically increasingly on ∥u∥s such that
+∥ Op[Gu] Op[Fu] − Op[GuFu]∥L(Hl+µ−1+m,Hm) ≤ Cl,2Cl∥u∥s
+up to an infinitely smoothing operator, which is determined by F(0, ·), G(0, ·).
+12
+
+2.20 Proposition. Let F ∈ Sm(U) and u ∈ C1([0, T], Hs) (T > 0) for s > d/2. Then for
+each l ∈ R the mapping
+[0, T] → L(Hl+m, Hl),
+t �→ Op[Fu(t)]
+is continuously differentiable and there exists Cl depending on l and F but not on u such
+that for all t ∈ [0, T]
+∥ d
+dt Op[Fu(t)]∥L(Hl+m,Hl) ≤ Cl∥∂tu(t)∥s0
+(2.10)
+Proof. If
+d
+dtFu(t) ∈ Γm
+0 we get by continuity and linearity of Op
+d
+dt Op[Fu(t)] = Op[∂tFu(t)]
+To prove this and (2.10) it is sufficient to show that for any α ∈ Nd
+0 there exists Cα = Cα(F)
+auch that for all ξ ∈ Rd
+∥∂α
+ξ ∂tFu(t)(·, ξ)∥L∞ ≤ Cα∥∂tu(t)∥s⟨ξ⟩m−|α|.
+Let α ∈ Nd
+0 and set F α
+u(t) := ∂α
+ξ Fu(t). We have for all x, ξ ∈ Rd
+∂tF α
+u(t)(x, ξ) =
+n
+�
+j=1
+∂tuj∂ujF α(u(t, x), ξ)
+Due to F ∈ Sm(U) this yields
+∥∂tF α
+u(t)(x, ξ)∥L∞ ≤ ∥∂tu(t)∥L∞
+�
+|β|=1
+∥∂β
+uF α(·, ξ)∥L∞ ≤ Cα(F)∥∂tu∥s⟨ξ⟩m−|α|.
+Lastly we prove a version of the strict G˚arding inequality for F ∈ Sm(U). First consider the
+following lemma which is a modification of a construction in [21], proof of Thm. 18.1.6.
+2.21 Lemma. There exists an even function ψ ∈ S(Rd × Rd) with unit integral, Op[ψ] =
+Op[ψ]∗, ⟨op[ψ]v, v⟩ ≥ 0 (v ∈ S(Rd)) and F1ψ compactly supported.
+Proof. Choose an even function ˆφ ∈ C∞
+0 (Rd × Rd) with L2-norm one and set φ = F −1
+1
+ˆφ. By
+definition F1φ is compactly supported and clearly φ is even and has L2-norm one. Next, let
+ψ ∈ S(Rd) be the symbol of op[ψ]∗ op[ψ]. As ibid. it follows that ψ is even and has unit
+integral. op[ψ] = op[ψ]∗, ⟨op[ψ]v, v⟩L2 ≥ 0 (u ∈ S(Rd)) holds by definition. Now, let ρ be
+the symbol of op[φ]∗. By Lemma 2.11 we get
+F1ρ(η, ξ) = (F1φ)∗(−η, η + ξ),
+η, ξ ∈ Rd
+13
+
+and thus by Lemma 2.12
+F1ψ(η, ξ) =
+�
+Rd F1ρ(η − θ, θ + ξ)F1φ(θ, ξ)dθ =
+�
+Rd F1φ(θ − η, η + ξ)F1φ(θ, ξ)dθ.
+As F1φ is compactly supported, we can choose C > 0 such that F1φ(θ, ξ) = 0 if |θ| ≥ C
+or |ξ| ≥ C. Then by definition F1ψ(η, ξ) = 0 if |ξ| ≥ C. Given |η| ≥ 2C and |θ| ≤ C we
+conclude |θ − η| ≥ |η| − |θ| ≥ C, i.e. F1φ(θ − η, η + ξ) = 0. In conclusion we have proven
+that F1ψ is in fact compactly supported. In particular ψ ∈ S(Rd × Rd).
+Next we introduce a method to decompose symbols in Sm
+1,1 into an infinite sum of infinitely
+smoothing symbols; cf. [22].
+First, choose a function ρ ∈ D(Rd) even and monotonically decaying along rays such that
+ρ(Rd) ⊂ [0, 1] and
+ρ(ξ) =
+�
+1,
+|ξ| ≤ 1
+2
+0,
+|ξ| ≥ 1 .
+For ν ∈ N0 define ρν, ζν ∈ D(Rd) by
+ρν(ξ) := ρ(ξ/2ν),
+ζν(ξ) = ρν+1(ξ) − ρν(ξ), ξ ∈ Rd
+Additionally set ζ−1 := ρ.
+2.22 Definition. For a function a : Rd × Rd → Cn×n and ν ≥ −1 define
+aν(x, ξ) := a(x, ξ)ζν(ξ).
+Note that a = �
+ν≥−1 aν.
+It is straightforward to show the following.
+2.23 Lemma. Let a ∈ Sm
+1,1. Then aν ∈ S−r for all r ∈ R and for any α, β ∈ N0 x, ξ ∈ Rd
+|∂β
+x∂α
+ξ aν(x, ξ)|⟨ξ⟩r ≤ C2ν(r+m−|α|+|β|) �
+γ≤α
+Cγβ(a),
+where Cγβ(a) are semi-norms of a.
+2.24 Proposition. Let s > d/2, u ∈ Hs+2 and F ∈ Sm(U) such that there exists an R > 0
+with F(y, ξ) + F(y, ξ)∗ ≥ 0 for all y ∈ U and ξ ∈ Rd with |ξ| > R. Then there exists
+C = C(∥u∥s+2, F) > 0 and for all q ∈ R there exists c = c(∥u∥s+2, F, q) > 0, both increasing
+functions of ∥u∥s+2, such that for all v ∈ S(Rd, Cn)
+⟨(Op[Fu] + Op[Fu]∗)v, v⟩L2 ≥ −C∥u∥
+1
+2
+s+2∥v∥2
+(m−1)/2 − c∥v∥2
+−q.
+14
+
+Proof. In the following it is straightforward to see that all constants can be chosen to be
+increasing functions of ∥u∥s+2. First note that by Proposition 2.19 for all l ∈ R
+∥ Op[Fu] + Op[Fu]∗ − Op[Fu + F ∗
+u]∥L(Hl+m−1,Hl) ≤ Cl∥u∥s+1.
+Thus
+⟨(Op[Fu] + Op[Fu]∗)v, v⟩L2 ≥ ⟨Op[Fu + F ∗
+u]v, v⟩L2 − C∥u∥s+1∥v∥2
+(m−1)/2,
+v ∈ S(Rd).
+Hence it is sufficent to prove the result for Op[Fu] + Op[Fu]∗ replaced by Op[Fu + F ∗
+u], i.e.
+we can assume w.l.o.g F(u, ξ) = F(u, ξ)∗ ≥ 0.
+It holds R(Fu) = R(F ∗
+u) = R(Fu)∗. By assumption this gives pointwise in Rd × {|ξ| ≥ R}
+for all v ∈ Cn
+⟨(R(Fu))v, v⟩Cn ≥ ⟨(R(Fu) − Fu)v, v⟩Cn ≥ −|R(Fu) − Fu||v|2
+≥ −(|R(Fu − F0) − (Fu − F0)| + |R(F0) − F0|)|v|2.
+By Lemma 2.18 Fu−F0 ∈ Γm
+2 with all semi-norms bounded by a positive constant depending
+on F times ∥u∥s+2. By Lemma 2.8 (i) this yields R(Fu − F0) − (Fu − F0) ∈ Γm−1
+1
+with semi-
+norms bounded in the same way. Thus
+|R(Fu − F0) − (Fu − F0)| ≤ C0∥u∥s+2⟨ξ⟩m−1.
+Using also that R(F0) − F0 has compact support we conclude that for all q ∈ R
+|R(Fu − F0) − (Fu − F0)| + |R(F0) − F0| ≤ C0∥u∥s+2⟨ξ⟩m−1 + c0
+q⟨ξ⟩−q.
+Therefore on Rd × {|ξ| ≥ R}
+a := R(Fu) + C0∥u∥s+2⟨ξ⟩m−1 + c0⟨ξ⟩−r ≥ 0
+and a = a∗, a ∈ Sm,1−ǫ2
+1,1
+. As
+Op[Fu] = op[R(Fu)] = op[a] − C0∥u∥s+2 op[⟨ξ⟩m−1] − c0 op[⟨ξ⟩−r]
+it is now sufficient to show
+⟨op[a]v, v⟩L2 ≥ −C∥u∥1/2
+s+2∥v∥2
+(m−1)/2 − c∥v∥−q
+for all q ∈ R.
+To this end we proceed similarly as in the proof of Theorem 9.7.1 in [22] but with a crucial
+modification. First, decompose a = �
+ν≥−1 aν according to Definition 2.22. As for all ν0
+¯aν0 := �ν0
+ν=−1 aν ∈ S−q for any q ∈ R with norm depending on µ, ν0 according to Lemma 2.23,
+i.e. ∥ op[¯aν0]v∥ ≤ cν0,µ∥v∥−r, we only need to consider �
+ν≥ν0 aν for some ν0 ∈ N. Naturally,
+in a first step we choose ν0 large enough to obtain 2ν0−2 > R and thus by assumption
+15
+
+aν(x, ξ) ≥ 0 for all x, ξ ∈ Rd, ν ≥ ν0. But we will later see that we may have to choose ν0
+even larger.
+W.l.o.g. assume u ̸= 0. Otherwise the result readily follows as F0 ≥ 0 is constant with
+respect to x and Op[F0] − op[F0] is infinitely smoothing.
+Choose an even function ψ ∈ S(Rd × Rd) with unit integral such that op[ψ] = op[ψ]∗,
+⟨op[ψ]v, v⟩ ≥ 0 (v ∈ S(Rd)) and F1ψ compactly supported as constructed in Lemma 2.21.
+For ν ∈ N0 set qν := 2ν/2 and write aν = bν + hν with
+bν(x, ξ) :=
+�
+Rd
+�
+Rd ψ((x − y)qνµ, (ξ − θ)/(qνµ))aν(y, θ) dy dθ
+(2.11)
+=
+�
+Rd
+�
+Rd ψ(y, θ)aν(x − y/(qνµ), ξ − θqνµ) dy dθ,
+(2.12)
+where µ := ∥u∥s+2. As aν ≥ 0 and op[ψ] is a positive operator it is straightforward to obtain
+the positivity of bν. Hence the theorem is proven provided
+⟨op[h]v, v⟩L2 ≥ −Cµ∥u∥
+1
+2
+k+1∥v∥(m−1)/2,
+v ∈ S(Rd).
+(2.13)
+To this end we show h ∈ Sm−1,L
+1,1
+for some L ∈ (0, 1) and that all semi-norms of h are bounded
+by a constant times ∥u∥
+1
+2
+s+2. Then (2.13) follows from Proposition 2.3.
+First we verify h ∈ Sm−1
+1,1
+and the estimate on the semi-norms, i.e.
+|
+�
+ν≥ν0
+∂β
+x∂α
+ξ hν(x, ξ)| ≤ Cαβ∥u∥
+1
+2
+s+2⟨ξ⟩m−1−|α|+|β|.
+(2.14)
+Let α = β = 0. Fix ξ ∈ Rd and consider ν ∈ N0 with |ξ| < 2ν−2 or |ξ| > 2ν+2. As aν(y, θ) = 0
+for 2ν−1 ≤ |θ| ≤ 2ν+1 we then have hν(x, ξ) = −bν(x, ξ) and it follows by basic estimates (cf.
+[22]) that in the support of the first integrand in (2.11)
+|ξ − θ| ≥ 1
+5(2ν + |ξ|)
+and thus
+|ξ − θ|/qν = 2−ν/2|ξ − θ| ≥ 1
+5(2ν + |ξ|)
+1
+2.
+(2.15)
+As a ∈ Sm
+1,1 and supp aν ⊂ {(x, θ) ∈ Rd × Rd : 2ν−1 ≤ |θ| ≤ 2ν}
+|aν(y, θ)| ≤ C⟨θ⟩m ≤ C(1 + 2ν)m.
+Hence ψ ∈ S(Rd × Rd) and (2.15) yield
+|hν| ≤ Cm(1 + 2ν)m
+� �
+(|ξ − θ|/(qνµ))−2(|m|+1)(1 + |ξ − θ|/(qνµ))−n−1(1 + |(x − y)|qνµ)−n−1dy dθ
+≤ Cm,nµ2(|m|+1)(1 + 2ν)m(2ν + |ξ|)−2|m|−2
+≤ Cm,nµ2(|m|+1)(1 + |ξ|)m−12−ν.
+(2.16)
+16
+
+Thus
+�
+{ν:|ξ|<2ν−2 or |ξ|>2ν+2}
+|hν| ≤ C∥u∥
+1
+2
+s+2⟨ξ⟩m−1.
+(2.17)
+Now consider ν ∈ N0 with 2ν−2 ≤ |ξ| ≤ 2ν+2. As ψ is an even function with unit integral we
+get from (2.12)
+hν = aν − bν =
+� �
+ψ(y, θ)
+�
+aν(x, ξ) − aν(x − y/(qνµ), ξ − θqνµ)
+�
+dy dθ
+=
+� �
+ψ(y, θ)
+�
+�
+|α+β|<2
+∂β
+x∂α
+ξ aν(x, ξ)(−y)β(−θ)α − aν(x − y/(qνµ), ξ − θqνµ)
+�
+dy dθ.
+(2.18)
+By Taylor’s fomula we can estimate (w.l.o.g. assume |θ| ≤ |ξ|)
+��
+�
+|α+|β|<2
+∂β
+x∂α
+ξ aν(x, ξ)(−y)β(−θ)α − aν(x − y/(qνµ), ξ − θqνµ)
+��
+≤ C
+�
+|α|+|β|=2
+sup
+x,ξ∈Rd |∂β
+x∂α
+ξ aν(x, ξ)||yβθα|(qνµ)|α|−|β|.
+(2.19)
+Note that a = R(Fu). By Lemma 2.18 Fu ∈ Γm
+2 and thus ∂β
+xFu ∈ Γm
+2−|β| for |β| ≤ 2. Hence
+for each γ ∈ Nd
+0, ξ ∈ Rd
+∥∂γ
+ξ ∂β
+xFu(·, ξ)∥W 2−|β|,∞ ≤ Cγ⟨ξ⟩m−|γ|
+and for |β| ≥ 1 we also have ∂β
+xFu|u=0 = 0. Thus again by Lemma 2.18
+∥∂γ
+ξ ∂β
+xFu(·, ξ)∥W 2−|β|,∞ ≤ Cγ∥u∥s+2⟨ξ⟩m−|γ|.
+Clearly
+∂β
+xa = ∂β
+xR(Fu) = R
+�
+∂β
+xFu).
+and we conclude from Proposition 2.6 that ∂β
+xa ∈ Sm
+1,1 and for all x, ξ ∈ Rd
+|∂β
+x∂γ
+ξ a(x, ξ)| ≤ Cγ⟨ξ⟩m−|γ|
+�
+1,
+|β| = 0,
+∥u∥s+2,
+1 ≤ |β| ≤ 2 .
+Then by Lemma 2.23
+sup
+x,ξ∈Rd |∂β
+x∂γ
+ξ aν| ≤ Cγ2ν(m−|α|) ≤ Cγ
+�
+1,
+|β| = 0,
+∥u∥s+2,
+1 ≤ |β| ≤ 2
+(2.20)
+From (2.18), (2.19), (2.20) and µ = ∥u∥
+1
+4
+s+2, qν = 2ν/2 we now get for 2ν−2 ≤ |ξ| ≤ 2ν+2
+|hν| ≤ C
+�
+ψ(y, θ)(|θ|2 + |θ||y| + |y|2) dy dθ
+�
+2ν(m−2)2ν∥u∥
+1
+2
+s+2 + 2ν(m−1)∥u∥s+2 + 2νm∥u∥s+22−ν∥u∥
+− 1
+2
+s+2
+�
+≤ Cµ2ν(m−1)∥u∥
+1
+2
+s+2 ≤ C∥u∥
+1
+2
+s+2⟨ξ⟩m−1,
+17
+
+where we used ψ ∈ S(Rd × Rd) and 2ν−2 ≤ |ξ| ≤ 2ν+2 in the last line. Together with (2.17)
+this shows (2.14) for α = β = 0.
+Now note that ∂β
+x∂α
+ξ bν is given by (2.12) with aν replaced by ∂β
+x∂α
+ξ aν. Hence we obtain
+(2.14) for α, β ̸= 0 by applying the argumentation above with aν replaced by ∂β
+x∂α
+ξ aν and m
+replaced by m − |α| + |β|.
+To finish the proof we show that F1h vanishes on NL = {(η, ξ) ∈ Rd × Rd : |η + ξ| < L|ξ|}
+with L := min{1 − ǫ2, 1
+2}. Then the estimate on the operator norm follows by the continuity
+of op. As a = R(Fu) ∈ Sm,1−ǫ2
+1,1
+, it suffices to prove that F1b vanishes on N 1
+2.
+By standard arguments on convolution and Fourier transform we have for all g ∈ S(Rd ×Rd)
+bν(F1g) = (µqν)−d/2
+�
+Rd
+�
+Rd aν(y, θ)F1f(y, θ) dθ dy,
+where
+f(η, θ) =
+�
+Rd F1ψ(η/(qνµ), (ξ − θ)/qνµ)g(η, ξ)dη.
+(2.21)
+Let supp g ⊂ N1/2. By construction we have supp F1ψ ⊂ {(ξ, η) ∈ Rd × Rd : |η|, |ξ| ≤ D, }
+for some D > 0. Next choose ν0 ∈ N so large that 3Dµ ≤ qν0/2. Then for ν ≥ ν0 on the
+support of the integrand of (2.21) we have |η|, |ξ − θ| ≤ Dqνµ and |ξ + η| + 1 < 1
+2|ξ|. The
+first and third inequality yield
+|ξ| < 2|η| ≤ 2Dqνµ
+and thus the second one gives
+|θ| ≤ Dqνµ + |ξ| < 3Dqνµ ≤ qνqν0/2 ≤ 2ν/22ν0/2−1 ≤ 2ν−1.
+But this implies bν(y, θ) = 0 for all y ∈ Rd. Therefore we have proven bν(F1g) = 0 for
+all ν ≥ ν0 and supp g ⊂ {(ξ, η) ∈ Rd × Rd : |ξ + η| <
+1
+2|ξ|}. Hence this also holds for
+b = �
+ν≥ν0 bν.
+3
+Dissipativity
+Throughout this section we consider (1.1), (1.2) with smooth matrix families Aj, Bjk : U →
+Rn×n, u0, u1 : Rd → U and u : [0, T] × Rd → U for some domain U ⊂ Rn. Carrying out the
+differentiation with respect to xk on the right-hand side and distinguishing between space
+and time derivatives we write (1.1) as
+−B00(u)utt =
+d
+�
+j,k=1
+Bjk(u)uxjxk+
+d
+�
+j=1
+(B0j(u)+Bj0(u)ut)xj−A0(u)ut−
+d
+�
+j=1
+Aj(u)uxj+Q(u, Dt,xu),
+18
+
+where Q is of the form
+Q(u, Dt,xu) =
+n
+�
+l=1
+d
+�
+j,k=0
+Qljk(u)ul
+xkuxj.
+We will see in the proofs that the specific form of the matrices Qljk(u) does not play any role.
+Hence multiplying (1.1) by (−B00)−1, we can assume −B00 = In without loss of generality,
+which we will always do in the following.
+Next, denote by
+B(u, ξ) :=
+d
+�
+j,k=1
+Bjk(u)ξjξk,
+C(u, ξ) :=
+d
+�
+j=1
+(B0j(u) + Bj0(u))ξj,
+A(u, ξ) :=
+d
+�
+j=1
+Aj(u)ξj,
+ξ = (ξ1, . . . , ξn) ∈ Rd.
+the symbols of the second and first order parts, respectively. Then the hyperbolicity of both
+sides of (1.1) is expressed by the following conditions:
+(HA) (a) there exists a smooth bounded family of hermitian uniformly positive definite ma-
+trices Σ : U → Rn×n such that Σ(u)A0(u) is symmetric and uniformly positive on U,
+(b) the matrix family A0(u)−1A(u, ξ) permits a symbolic symmetrizer H(u, ξ),
+(HB) with
+B(u, ξ) =
+�
+0
+|ξ|In
+−|ξ|−1B(u, ξ)
+iC(u, ξ)
+�
+,
+ξ = (ξ1, ..., ξd) ∈ Rd,
+the matrix family iB(u, ξ) permits a symbolic symmetrizer H(u, ξ).
+Above we use the following notion of a symbolic symmetrizer (cf. e.g. [38]).
+3.1 Definition. Let K ∈ C∞(U × Rd \ {0}, Cn×n). A symbolic symmetrizer for K is a
+smooth mapping S ∈ C∞(U × Rd \ {0}, Cn×n) positive homogeneous of degree 0 with respect
+to the second argument, bounded as well as all its derivatives on U ×Sd−1 such that for some
+c > 0 and all (u, ξ) ∈ U × Rd \ {0}
+S(u, ξ) = S(u, ξ)∗ ≥ cIn,
+and S(u, ξ)K(u, ξ) = (S(u, ξ)K(u, ξ))∗.
+3.2 Remark. K admits a symbolic symmetrizer if K is positive homogeneous of degree 1,
+for all (u, ω) ∈ U ×Sd−1 all eigenvalues of K(u, ξ) are real, semi-simple (i.e. their geometric
+and algebraic multiplicities coincide) and their multiplicities do not depend on (u, ω) (cf.
+[38], Proposition 5.2 C). If this holds for A0(u)−1A(u, ξ) or B(u, ξ) the respective operator
+is often called constantly hyperbolic.
+19
+
+We now fix a homogeneous state ¯u ∈ U and assume the following dissipativity conditions on
+the coefficient matrices.
+Condition (D). Matrices Aj(¯u), Bjk(¯u) have three properties:
+(D1) For every ω ∈ Sd−1, all restrictions, as a quadratic form, of
+W1 = H(¯u, ω)(A0(¯u))−1�
+− B(¯u, ω) + (A0(¯u))−1(A(¯u, ω))(A0(¯u))−1A(¯u, ω)
++ C(¯u, ω)(A0(¯u))−1A(¯u, ω)
+�
+,
+on the eigenspaces E = J−1
+E (Cn) of
+W0 = (A0(¯u))−1A(¯u, ω)
+are uniformly negative in the sense that
+J∗
+E (W1 + W ∗
+1 ) JE ≤ −¯c J∗
+EJE
+with one ¯c > 0.
+(D2) For every ω ∈ Sd−1, all restrictions, as a quadratic form, of
+W1 = H(¯u, ω)A(¯u, ω),
+A(¯u, ω) =
+�
+0
+0
+−iA(¯u, ω)
+−A0(¯u)
+�
+(3.1)
+on the eigenspaces E = J −1
+E (C2n) of
+W0 = B(¯u, ω)
+(3.2)
+are uniformly negative in the sense that
+J ∗
+E (W1 + W∗
+1) JE ≤ −¯c IE
+with one ¯c > 0..
+(D3) All solutions (λ, ξ) ∈ C × (Rd \ {0}) of the dispersion relation of (1.1) at ¯u = 0 have
+Re(λ) < 0.
+3.3 Remark. Note that as (D) is an open condition there exists a neighbourhood of ¯u such
+that Bjk(u), Aj(u) satisfy (D) with ¯u replaced by u for all u ∈ U0 with ¯c independent of u.
+The following remark is useful in the proofs below.
+3.4 Remark. It is straightforward to show that (D1) and (D2) are equivalent to the same
+conditions with W0, W1 replaced by
+¯W0 := H(¯u, ω)
+1
+2A(¯u)−1A(0, ω)H(¯u, ω)− 1
+2,
+¯W1 := H(¯u, ω)− 1
+2W1H(¯u, ω)− 1
+2
+and W0, W1 replaced by
+¯
+W0 := H(¯u, ω)
+1
+2B(¯u, ω)H(¯u, ω)− 1
+2,
+¯
+W1 := H(¯u, ω)
+1
+2A(¯u, ω)H(¯u, ω)− 1
+2.
+20
+
+From now on we always assume (HA), (HB) and (D). As we could also consider (1.1), (1.2)
+in the variable u − ¯u, we can w.l.o.g. restrict our argumentation to the case ¯u = 0.
+We write (1.1) as the first-order in time system
+ut = v
+vt =
+d
+�
+j=1
+(Bj0 + B0j)(u)vxj +
+d
+�
+j,k=1
+Bjk(u)uxjxk − A0(u)v −
+d
+�
+j=1
+Aj(u)uxj + Q(u, Dt,xu) (3.3)
+and denote by
+¯
+M(u, ξ) :=
+�
+0
+In
+M(u, ξ)
+N(u, ξ)
+�
+,
+(3.4)
+with
+M(u, ξ) = −iA(u, ξ) − B(u, ξ),
+N(u, ξ) = iC(u, ξ) − A0(u),
+the Fourier symbol of (3.3). We also define
+M(u, ξ) := Z(ξ) ˜
+M(u, ξ)Z(ξ)−1
+Z(ξ) =
+�⟨ξ⟩In
+0
+0
+In
+�
+.
+First we treat the linearization of (1.1) at the reference state u = 0, i.e.
+d
+�
+j=0
+Aj(0)uxj =
+d
+�
+j,k=0
+Bij(0)uxixj.
+(3.5)
+Such linear systems were studied in [14], however under the stronger assumptions, that the
+coefficient matrices are symmetric and A0 is positive definite. Then (HA) is clearly satisfied
+with FA = In and H = A0. Also, condition (HB) (b) ibid.
+requires the existence of a
+matrix family S : Sd−1 → Cn×n such that iS(ω)B(0, ω)S(ω)−1 is real symmetric. But one
+can easily check that this can be relaxed to the assumption that iS(ω)B(0, ω)S(ω)−1 is
+hermitian, which is satisfied in the present context for S(ω) := H(0, ω)
+1
+2. Lastly, we want
+to point out that (D1), (D2) ibid. were stated in the equivalent form mentioned in Remark
+3.4.
+We will make plausible below that the weaker conditions in the present work are still sufficient
+to retrieve the main result of [14], namely:
+3.5 Proposition. There exist a c > 0 and a family ξ �→ T (ξ), Rd → C2n×2n of linear
+transformations of C2n which, together with their inverses T (ξ)−1, are uniformly bounded,
+such that
+T (ξ)M(0, ξ)T −1(ξ) + (T (ξ)M(0, ξ)T −1(ξ))∗ ≤ −cρ(ξ)I2n,
+ξ ∈ Rd,
+(3.6)
+where ρ(ξ) = |ξ|2/(1 + |ξ|2).
+21
+
+As outlined in [14] this brings about the pointwise decay of solutions in Fourier space and
+thus the following decay estimate for the inhomogeneous linear Cauchy problem.
+3.6 Corollary. For any s ∈ N0 there exists C > 0 such that the following holds: For all
+u0 ∈ Hs+1 ∩ L1, u1 ∈ Hs ∩ L1 and f ∈ C([0, T], Hs ∩ L1) the solution u of
+f +
+d
+�
+j=0
+Aj(0)uxj =
+d
+�
+j,k=0
+Bij(0)uxixj
+with u(0) = u0, ut(0) = u1 satisfies
+∥u(t)∥s+1 + ∥ut(t)∥s ≤ C(1 + t)− d
+4(∥u0∥s+1 + ∥u0∥L1 + ∥u1∥s + ∥u1∥L1)
+C
+� t
+0
+(1 + t − τ)− d
+4(∥f(τ)∥s + ∥f(τ)∥L1) dτ
+for all t ∈ [0, T].
+Proof of Proposition 3.5. As stated above the proof can be found essentially in [14]. We just
+illustrate at which points it has to be slightly modified.
+The existence of a bounded family T (ξ) ⊂ Gl2n2 satisfying (3.6) is proven separately for the
+three different regimes |ξ| ≤ r0, r0 ≤ |ξ| ≤ r∞ and |ξ| ≥ r∞ for suitable r0, r∞ > 0. In
+the latter two cases only ((H)B) and conditions (D2), (D3) are used. The symmetry of the
+matrices plays no role whatsoever.
+For small values of |ξ| writting ξ = ξω for ξ > 0, ω ∈ Sd−1 one finds a bounded family of
+invertible R(ξ, ω) with R(ξ, ω)−1 also bounded and (supressing the argument u = 0)
+R(ξ, ω) ¯
+M(ξω)R(ξ, ω)−1 =
+�
+X(ξ, ω)
+0
+0
+Y (ξ, ω)
+�
+,
+where
+X(ξ, ω) = iξ(A0)−1A(ω)
++ ξ2(A0)−1�
+− B(ω) + (A0)−1(A(ω))(A0)−1A(ω) + C(ω)(A0)−1A(ω)
+�
++ O(ξ3)
+Y (ξ, ω) = −A0 + O(ξ3).
+This is due to the fact that A0(0) is invertible and again makes no use of the symmetry.
+Hence for
+ˇR(ξ, ω) =
+�
+H(ω)
+1
+2
+0
+0
+F
+1
+2
+A
+�
+ˇR(ξ, ω)
+we get
+ˇR(ξ, ω)M(ξω) ˇR(ξ, ω)−1 =
+�
+iξ ¯W0 + ξ2 ¯W1 + O(ξ3)
+0
+−F
+1
+2
+A A0F
+− 1
+2
+A .
+�
+.
+2For m ∈ N Glm denotes the space of invertible m × m-matrices.
+22
+
+with ¯W0, ¯W1 as in Remark 3.4. Since F
+1
+2
+A A0F
+− 1
+2
+A
+is positive definite the existence of the
+family T (ξ) now follows for sufficiently small ξ by condition (D1) and [14], Lemma 5.3
+In Section 4 we will see that, given d ≥ 3, s > d/2 + 1, Corollary 3.6 directly implies the
+decay of a solution to the quasi-linear problem (1.1) in Hs−1 but only provided that its
+Hs-norm is a-priori known to be small. To close this gap we need to show that the Hs-norm
+of a small solution can be bounded by the initial conditions and L2-norms of lower order
+derivatives. The rest of this section is devoted to a construction preparing such a result.
+In the following for ξ ∈ Rd we write ξ = ξω with ξ = |ξ| ∈ [0, ∞), ω = ξ/|ξ| ∈ Sd−1.
+For r > 0, u ∈ Rn, ξ ∈ Rd and ω ∈ Sd−1 by Bn(u, r), Bd(ξ, r), BS(ω, r) we denote the balls
+with radius r and center u, ξ, ω with respect to the metrices on Rn, Rd, Sd−1.
+For some
+ω∗ ∈ Sd−1 and δ > 0 we use
+P(ω∗, δ) = Bn(0, δ) × [0, δ) × BS(ω∗, δ)
+.
+3.7 Proposition. There exist r > 0, c∞ > 0 and a mapping D∞ ∈ C∞(Ω∞, C2n×2n), Ω∞ :=
+¯U0 × {ξ ∈ Rd : |ξ| ≥ r−1}, ¯U0 := Bn(0, r) ⊂ U, such that:
+(i) For all (u, ξ) ∈ Ω∞
+D∞(u, ξ) = D∞(u, ξ)∗ ≥ c∞In,
+and
+D∞(u, ξ)M(u, ξ) + (D∞(u, ξ)M(u, ξ))∗ ≤ −c∞I2n.
+(ii) For any α, β ∈ Nd
+0 there exist Cαβ > 0 with
+|∂β
+u∂α
+ξ D∞(u, ξ)| ≤ Cαβ⟨ξ⟩−|α|,
+(u, ξ) ∈ Ω∞.
+(3.7)
+Proof. Consider the mapping K : U × (0, ∞) × Sd−1 → C2n×2n defined by
+K(u, η, ω) =
+�
+0
+In
+−iηA(u, ω) − B(u, ω)
+−iC(u, ω) − ηA0(u)
+�
+,
+ω ∈ Sd−1.
+(3.8)
+and H(u, ω) denote the symmetrizer of B(u, ω) as in condition (HB) (b). Set
+W(u, η, ω) := H(u, ω)
+1
+2K(u, η, ω)H(u, ω)− 1
+2.
+Since
+K(0, 0, ω) =
+�
+0
+In
+−B(0, ω)
+iC(0, ω)
+�
+= B(0, ω)
+3Note that in said Lemma it is sufficient to assume that iM(0, ω) is selfadjoint instead of requiring
+iM(0, ω) to be real symmetric.
+23
+
+and
+∂K
+∂η (0, 0, ω) =
+�
+0
+0
+−iA(0, ω)
+−A0(0)
+�
+= A(0, ω)
+W satisfies
+W(0, 0, ω) = ¯
+W0,
+∂W(0, 0, ω)
+∂η
+= ¯
+W1,
+with ¯
+W0, ¯
+W1 as in Remark 3.4. Now fix ω0 ∈ Sd−1. By virtue of condition (D2) it follows
+from Lemma 5 in [14] that there exists δ0 > 0, c0 > 0 and T0 ∈ C∞(P(ω∗, δ0), Gl2n) with
+T −1
+0
+also bounded such that pointwise on P(ω0, δ0)
+T0WT −1
+0
++ (T0WT −1
+0
+)∗ ≤ −˜cηI2n
+for some ˜c > 0. Hence ˜D0 := H
+1
+2T ∗
+0 T0H
+1
+2 ∈ C∞(P(δ0, ω0), C2n×2n) satisfies
+˜D0(u, ξ, ω) = ˜D0(u, ξ, ω)∗ ≥ cI2n,
+(u, ξ, ω) ∈ P(δ0, ω0)
+for some c > 0 and thus
+˜D0K + ( ˜D0K)∗ ≤ −c˜cηI.
+In conclusion we have shown the following: For each ω ∈ Sd−1 there exist δω > 0, cω > 0
+and Dω ∈ C∞(P(ω, δω), C2n×2n) such that for all (u, ξ, ¯ω) ∈ P(ω, δω)
+Dω(u, η, ¯ω) = Dω(u, η, ¯ω)∗ ≥ cωI
+Dω(u, η, ¯ω)K(u, η, ¯ω) + (Dω(u, η, ¯ω)K(u, η, ¯ω))∗ ≤ −cωξ2I.
+(3.9)
+As Sd−1 is compact we may choose ω1, . . . , ωr such that
+¯l�
+l=1
+BS(ωl, δl/2) = Sd−1 (δl := δωl).
+Set r0 = min{δ1, . . . , δr}, c0 = min{cω1, . . . , cωr}. Then for l = 1, . . . , ¯l and Pl := Bn(0, r0) ×
+[0, r0) × BS(ωl, δl) choose functions φl ∈ C∞(Sd−1, [0, 1]) with supp φl ⊂ BS(ωj, δl), φl = 1
+on BS(ωj, δj/2) and extend Dl := Dωl trivially by 0 to a function defined on Bn(0, r0) ×
+[0, r0) × Sd−1 =: Ω0. Define
+D0 : Ω0 → C2n×2n : (u, η, ω) �→
+¯l
+�
+l=1
+φl(ω)Dl(u, η, ω).
+Then D0 ∈ C∞(Ω0, C2n×2n), and D0(u, η, ω) is hermitian for all (u, η, ω) ∈ Ω0. Furthermore
+for (u, η, ω) ∈ Ω0 we have ω ∈ BS(ωk, δ/2) for some k ∈ {1, . . . , ¯l} and thus as Dl(u, η, ω) ≥ 0
+D0(u, η, ω) =
+¯l
+�
+l=1
+φl(ω)Dl(u, η, ω) ≥ Dk(u, η, ω) ≥ c0I.
+with the same reasoning we see
+D0(u, η, ω)K(u, η, ω) + (D0(u, η, ω)K(u, η, ω))∗ ≤ −c0ηI2n,
+(u, η, ω) ∈ Ω0.
+24
+
+Now note that for all u, ξ, ω
+ξK(u, 1/ξ, ω) =
+�
+0
+ξIn
+−iA(u, ω) − ξB(u, ω)
+−iξC(u, ω) − A0(U)
+�
+= ˜Z(ξ)M(u, ξω) ˜Z(ξ)−1,
+where
+˜Z(ξ) =
+� ⟨ξ⟩
+ξ In
+0
+0
+In
+�
+.
+As clearly ˜Z, ˜Z−1 ∈ C∞((r−1
+0 , ∞), C2n×2n) are symmetric and positive definite on (r−1
+0 , ∞),
+for r := r0/2, Ω∞ := Bn(0, r) × {ξ ∈ Rd : |ξ| ≥ r−1} the mapping
+D∞ : Ω∞ → C2n×2n, (u, ξ) �→ ˜Z(|ξ|)D0(u, 1/|ξ|, ξ/|ξ|) ˜Z(|ξ|)
+is in C∞(Ω∞, C2n×2n) and for all (u, ξ) ∈ Ω∞
+D∞(u, ξ) = D∞(u, ξ)∗ ≥ c∞I2n
+for some c∞ > 0. Since for ξ = ξω ∈ U0
+D∞(u, ξ)M(u, ξ) = ξ ˜Z(ξ)D0(u, 1/ξ, ω) ˜Z(ξ)K(u, 1ξ, ω) = ξ ˜Z(ξ) ˜D0(u, 1/ξ, ω)K(u, 1/ξ) ˜Z(ξ),
+we also have
+D∞(u, ξ)M(u, ξ) + (D∞(u, ξ)M(u, ξ))∗ ≤ −c∞I2n
+for some c∞ > 0.
+It remains to verify (3.7). First note that the functions ξ �→ ⟨|ξ|⟩/|ξ|, ξ �→ ξk/|ξ|, k = 1, . . . , d
+and ξ �→ 1/|ξ| are positive homogeneous of degree 0 and −1, respectively. Thus for any
+α ∈ Nd
+0 there exists Cα > 0 such that for all ξ ∈ Rd with |ξ| > 2r−1
+0
+|DαZ(ξ)| + |Dα(ξk/|ξ|)| + |Dα(1/|ξ|)| ≤ Cα⟨ξ⟩−|α|.
+Since D0 as well as all of its derivatives are bounded on Bn(0, r0/2) × [0, r0/2] × Sd−1 the
+estimate (3.7) follows by product and chain rule.
+25
+
+4
+Proof of Theorem 1.1
+To begin with, we remark that local well-posednes sof (1.1), (1.2) follows from the existing
+theory for hyperbolic systems of any order [38].4 Our task thus consists in showing that
+under an a priori smallness assumption the solution satisfies the decay and energy estimates
+(1.3) and (1.4), for, w.l.o.g., ¯u = 0. Then we can extend them globally by standard methods
+(cf. e.g. [24], proof of Theorem 3.6). We show the following.
+4.1 Proposition. Consider d ≥ 3, s > d/2 + 1 and assume (HB), (HA) and (D). Then
+there exist constants µ > 0, δ = δ(µ) > 0, and C = C(µ, δ) > 0 (all independent of T)
+such that the following holds: For all u0 ∈ Hs+1, u1 ∈ Hs with ∥u0∥s+1 + ∥u1∥s < δ and all
+u ∈ C0([0, T], Hs+1) ∩ C1([0, T], Hs) satisfying (1.1), (1.2) and
+sup
+t∈[0,T]
+∥u(t)∥2
+s+1 + ∥ut(t)∥2
+s +
+� T
+0
+∥u(τ)∥2
+s+1 + ∥ut(τ)∥2
+s dτ ≤ µ
+we have for all t ∈ [0, T]
+∥u(t)∥s + ∥ut(t)∥s−1 ≤ C(1 + t)− d
+4(∥u0∥s + ∥u0∥L1 + ∥u1∥s−1 + ∥u1∥L1),
+(4.1)
+∥u(t)∥2
+s+1 + ∥ut(t)∥2
+s +
+� t
+0
+∥u(τ)∥2
+s+1 + ∥ut(τ)∥2
+s ≤ C(∥u0∥2
+s+1 + ∥u0∥2
+L1 + ∥u1∥2
+s1 + ∥u1∥2
+L1)
+(4.2)
+We split the proof into two parts corresponding to the following two assertions.
+4.2 Proposition. In the situation of Proposition 4.1 there exist µ > 0, δ > 0, and C >
+0 such that the following holds: For all u0 ∈ Hs+1 ∩ L1, u1 ∈ Hs ∩ L1 with ∥u0∥s+1 +
+∥u1∥s, ∥u0∥L1 + ∥u1∥L1 < δ and all u ∈ C0([0, T], Hs+1) ∩ C1([0, T], Hs) satisfying (1.1),
+(1.2) and
+sup
+t∈[0,T]
+∥u(t)∥2
+s+1 + ∥ut(t)∥2
+s+1 +
+� T
+0
+∥u(τ)∥2
+s+1 + ∥ut(τ)∥2
+s dτ ≤ µ
+(1.3) holds for all t ∈ [0, T].
+4.3 Proposition. In the situation of Proposition 4.1 there exist µ > 0, and C > 0 such that
+the following holds: For all u0 ∈ Hs+1, u1 ∈ Hs and all u ∈ C0([0, T], Hs+1) ∩ C1([0, T], Hs)
+satisfying (1.1), (1.2) and
+sup
+t∈[0,T]
+∥u(t)∥2
+s+1 + ∥ut(t)∥2
+s+1 +
+� T
+0
+∥u(τ)∥2
+s+1 + ∥ut(τ)∥2
+s dτ ≤ µ
+we have for all t ∈ [0, T]
+∥u(t)∥2
+s+1 + ∥ut(t)∥2
+s +
+� t
+0
+∥u(τ)∥2
+s + ∥ut(τ)∥2
+s−1dτ
+≤ C(∥u0∥2
+s+1 + ∥u1∥2
+s) + C
+� t
+0
+∥u(τ)∥2
+s + ∥ut(τ)∥2
+s−1 dτ.
+(4.3)
+4For example, the recent result in [3], which applies to the class we study in Section 5, is of this type.
+26
+
+From there Proposition 4.1 clearly follows by multiplying (4.3) with a sufficiently small factor
+integrating, (1.3) with respect to t, and adding the resulting inequalities.
+For notational reasons we write the first order representation (3.3) of (1.1) in the compact
+form
+Ut = L(u)U + (0, Q(u, Dx,tu))t
+(4.4)
+with U = (u, ut),
+L(u) =
+�
+0
+In
+�d
+j,k=1 Bjk(u)∂xj∂xk − �d
+j=1 Aj(u)∂xj
+�d
+j=1( ¯Bj0 + ¯B0j)(u)∂xj − A0.
+�
+Proof of Proposition 4.2. As s > d/2+1 we find by Moser type inequalities (cf. [4] Appendix
+C and the references therein)
+∥(L2(u) − L2(0))U∥s−1 + ∥(L2(u) − L2(0))U∥L1 �� Cµ∥u∥s−1(∥u∥s+1 + ∥ut∥s),
+where L(u)U = (U2, L2(u)U). Furthermore
+∥Q(u, Dx,tu))∥s−1 + ∥Q(u, Dx,tu)∥L1 ≤ Cµ∥u∥s∥ut∥s−1.
+Now writing system (4.4) as L(0)U = (0, L2(0)−L2(u)+Q(u, Dx,tu)) and applying Corollary
+3.6 to f = (L2(0) − L2(u)) + Q(u, Dx,tu) with s replaced by s − 1 yields
+∥u(t)∥s + ∥ut(t)∥s−1 ≤ C(1 + t)− d
+4(∥u0∥s + ∥u0∥L1 + ∥u1∥s−1 + ∥u1∥L1)
++ Cµ sup
+τ∈[0,t]
+(∥u(τ)∥s+1 + ∥ut(τ)∥s)
+� t
+0
+(1 + t − τ)− d
+4(∥u(τ)∥s + ∥ut∥s−1)dτ.
+(4.5)
+As t → (1 + t)− d
+4 is square-integrable over [0, ∞) for d ≥ 3 this gives (1.3) as in e.g. [24],
+proof of Proposition 3.3.
+Proof of Proposition 4.3. From now Cµ always denotes some constant depending monoton-
+ically increasing on µ, whose concrete value may change at every instance.
+For 0 < ǫ < 1 let Jǫ be the Friedrichs mollifier and set V = (Λu, ut), W := Wǫ := ΛsJǫ(Λu, ut)
+and
+Mu(x, ξ) = Mu(t)(x, ξ) = M(u(t, x), ξ)
+=
+�
+0
+⟨ξ⟩In
+�
+− B(u, ξ) − A(u, ξ)
+�
+⟨ξ⟩−1
+C(u, ξ) − A0(u)
+�
+.
+We start with the following observation.
+27
+
+4.4 Lemma. W satisfies the differential equation
+Wt = Op[Mu]W + R1,
+(4.6)
+for some R1 ∈ L2 satisfying
+∥R1∥ ≤ Cµ∥V ∥2
+s + C∥V ∥s−1
+(4.7)
+Proof. Set
+˜L(u) :=
+�ΛIn
+0
+0
+In
+�
+L(u)
+�Λ−1In
+0
+0
+In
+�
+.
+Then
+Vt = Op[Mu]V + ˜R1
+(4.8)
+where
+˜R1 = (˜L(u) − Op[Mu])V + (0, Q(u, Dx,tu)).
+As we have already seen in the proof of Proposition 4.2 (now with s − 1 replaced by s)
+∥Q(u, Dx,tu)∥s ≤ Cµ∥V ∥2
+s.
+By Lemma 2.9
+∥(˜L(0) − Op[M0])V ∥s ≤ C∥V ∥s−1
+and due to Lemma 2.9 (iii) all terms appearing in
+(˜L(u) − ˜L(0) − Op[Mu − M0])V
+are of the form (a(u) − Op[au])f, where a is a smooth function with a(0) = 0 and f ∈
+{∂l
+t∂β
+xu| l ≤ 1, l + |β| ≤ 2} ⊂ Hs−1 ֒→ L∞. Hence Lemma 2.10 yields
+∥(˜L(u) − ˜L(0) − Op[Mu − M0])V ∥s ≤ Cµ(∥u∥s∥V ∥s + ∥V ∥s−1).
+In conclusion we have shown
+∥ ˜R1∥s ≤ Cµ(∥V ∥2
+s + ∥V ∥s−1).
+(4.9)
+Now apply ΛsJǫ to (4.8) and obtain
+Wt = Op[Mu]W + R1,
+(4.10)
+where
+R1 = [ΛsJǫ, Op[Mu]]V + ΛsJǫ ˜R1
+Note that (Jǫ)ǫ∈(0,1) is a family of pseudo-differential operators, constant with respect to x,
+with symbols uniformly bounded in S0. Thus we get from (4.9)
+∥ΛsJǫR1∥ ≤ Cµ∥V ∥2
+s + C∥V ∥s−1
+and from Proposition 2.19 (iii)
+∥[ΛsJǫ, Op[Mu]]V ∥ ≤ Cµ∥u∥s∥V ∥s + C∥V ∥s−1,
+which proves the assertion.
+28
+
+Next, let D∞ ∈ C∞(Bnr (0) × {ξ ∈ Rd : |ξ| ≥ r}, C2n×2n) be the mapping constructed in
+Proposition 3.7 and extend it trivially by zero to a function defined on Bn
+r (0)×Rd := U0×Rd.
+Choose a function φ ∈ C∞(Rd), with 0 ≤ φ ≤ 1, φ(ξ) = 0 for |ξ| ≤ 2r and φ(ξ) = 1 for
+|ξ| ≥ 3r. Set
+D(v, ξ) := φ(ξ)D∞(v, ξ),
+(v, ξ) ∈ U0 × Rd
+Let µ be sufficiently small such that u(t, x) ∈ Bnr (0) for all (t, x) ∈ [0, T] × Rd and define
+Du(x, ξ) := Du(t)(x, ξ) = D(u(t, x), ξ),
+(t, x, ξ) ∈ [0, T] × Rd × Rd.
+Choose another function ψ ∈ C∞(Rd), with 0 ≤ ψ ≤ 1, ψ(ξ) = 0 for |ξ| ≥ 5r, ψ(ξ) = 1 for
+|ξ| ≤ 4r and define
+˜Du(x, ξ) = Du(x, ξ) + ψ(ξ)I2n.
+4.5 Lemma. The family of operators (Gu(t))t∈[0,T] defined by
+Gu(t) := 1
+2(Op[ ˜Du(t)] + Op[ ˜Du(t)]∗) + op[ ˜D0] − Op[ ˜D0]
+is self-adjoint and uniformly positive definite in L(L2) for µ sufficiently small. Furthermore
+1
+2
+d
+dt
+�
+GuW, W⟩ = Re⟨Gu Op[Mu]W, W⟩ + R2,
+for some R2 ∈ R with
+|R2| ≤ Cµ∥W∥(∥V ∥2
+s + ∥V ∥s∥W∥ + ∥V ∥s−1).
+Proof. By Proposition 3.7 ˜D, D ∈ S0(U) and ˜Du = ˜D∗
+u is uniformly positive definite. In
+particular, op[ ˜D0] = op[ ˜D0]∗ is a self-adjoint and uniformly positive definite operator on
+L(L2) (cf. Lemma 2.9). Due to ibid. also Op[ ˜D0]∗ = Op[ ˜D0], i.e.
+Gu = op[ ˜D0] + 1
+2(Op[ ˜Du − ˜D0] + Op[ ˜Du − ˜D0]∗).
+Proposition 2.19 (i) gives
+∥ Op[ ˜Du − ˜D0]∥L(L2) ≤ Cµ∥u∥s,
+which yields the first assertion.
+Now apply Gu to (4.6), take the L2 scalar product with W and consider the real part to find
+Re⟨GuWt, W⟩ = Re⟨Gu Op[Mu]W, W⟩ + Re⟨GuR1, W⟩ := Re⟨Gu Op[Mu]W, W⟩ + R21.
+(4.11)
+Due to (4.7) and ∥Gu∥L(L2) ≤ Cµ,
+∥R21∥ ≤ Cµ∥W∥(∥V ∥2
+s + ∥V ∥s−1).
+(4.12)
+29
+
+As Gu is self-adjoint we get
+Re⟨GuWt, W⟩ = 1
+2
+d
+dt
+�
+GuW, W⟩ − Re
+�� d
+dtGu
+�
+W, W⟩
+(4.13)
+and 2.20 (iv) yields
+2∥ d
+dtGu∥L(L2) ≤
+�� d
+dt Op[ ˜Du]
+��
+L(L2) ≤ Cµ∥ut∥s.
+(4.14)
+The second statement then clearly follows from (4.11)-(4.14).
+The last step consists in showing the following.
+4.6 Lemma. It holds
+Re⟨Gu Op[Mu]W, W⟩ ≤ −c∥W∥2 + Cµ∥W∥2(∥u∥
+1
+2
+s+1 + ∥u∥s) + Cµ∥W∥2
+−1).
+From Lemmas 4.5, 4.6 we obtain
+1
+2
+d
+dt⟨GuW, W⟩+c∥W∥2 ≤ Cµ∥W∥(∥V ∥2
+s+∥V ∥s∥W∥+∥V ∥
+1
+2)+Cµ(∥V ∥2
+s−1+∥W∥2
+−1). (4.15)
+As Λ−kW = Λ−kWǫ → V as ǫ → 0 uniformly with respect to t for 0 ≤ k ≤ s and Gu is
+uniformly postive definite, we find by integrating (4.15)
+∥V (t)∥2
+s +
+� t
+0
+∥V ∥2
+s dτ ≤ Cµ(∥V (0)∥ +
+� t
+0
+∥V (τ)∥3
+s + ∥V (τ)∥
+5
+2s + ∥V (τ)∥s−1)dτ,
+t ∈ [0, T],
+which yields the assertion since ∥V ∥2
+s = ∥u∥2
+s+1 + ∥ut∥2.
+Proof of Lemma 4.6. Set κ := op[ ˜D0] − Op[ ˜D0], which is infinitely smoothing. Then
+Gu = 1
+2(Op[ ˜Du] + Op[ ˜Du]∗) + κ.
+As ∥Mu∥L(Hl,Hl−1) ≤ Cµ, l ∈ R, due to Proposition 2.19 (i) we find
+Re⟨κ Op[Mu]W, W⟩ ≤ Cµ∥W∥2
+−1
+By construction ˜Du = ˜D∗
+u and thus 2.19 (ii) yields
+Re
+�
+(1
+2(Op[ ˜Du]∗ − Op[Du]) Op[Mu]W, W
+�
+≤ Cµ∥u∥s∥W∥2.
+Next note that ˜Du(x, ·) − Du(x, ·) is compactly supported with support not depending on t.
+Therefore Op[ ˜Du − Du] is infinitely smoothing and
+Re⟨Op[ ˜Du − Du] Op[Mu]W, W⟩ ≤ Cµ∥W∥2
+−1.
+30
+
+In conclusion
+Re⟨Gu Op[Mu]W, W⟩ = Re⟨Op[Du] Op[Mu]W, W⟩ + Re⟨κ Op[Mu]W, W⟩
++ 1
+2 Re⟨(Op[ ˜Du]∗ − Op[ ˜Du]) Op[Mu])W, W⟩ + Re⟨Op[ ˜Du − Du]MuW, W⟩
+≤ Re⟨Op[Du] Op[Mu]W, W⟩ + Cµ(∥u∥s∥W∥2 + ∥W∥2
+−1)
+(4.16)
+By Proposition 2.19 (iii)
+∥(Op[Du] Op[Mu] − Op[DuMu])W∥ ≤ Cµ∥u∥s∥W∥ + C∥W∥−1.
+Hence
+Re⟨Op[Du] Op[Mu]W, W⟩ ≤ Re⟨Op[DuMu]W, W⟩ + Cµ∥u∥s∥W∥2 + C∥W∥2
+−1.
+(4.17)
+Set Xu := DuMu + c∞/2I2n with c∞ as in Lemma 3.7. Note that c∞ does not depend on µ.
+Since Op[I2n] − IdL2 is infinitely smoothing we conclude
+Re⟨Op[DuMu]W, W⟩ ≤ Re⟨Op[Xu]W, W⟩ − c∞
+2 ∥W∥2 + C∥W∥2
+−1.
+(4.18)
+By Proposition 3.7
+Xu(x, ξ) + X ∗
+u(x, ξ) = DuMu(x, ξ) + (DuMu)(x, ξ)∗ + c∞ ≤ 0,
+for x ∈ Rd und ξ ∈ Rd with |ξ| ≥ 3r. Since u ∈ Hs+1 and s + 1 ≥ d/2 + 2, Proposition 2.24
+applied to −Xu gives
+Re⟨Op[Xu]W, W⟩ ≤ Cµ(∥u∥
+1
+2
+s+1∥W∥2 + ∥W∥−1).
+(4.19)
+(4.18) and (4.19) lead to
+Re⟨Op[DuMu]W, W⟩ ≤ −c∥W∥2 + Cµ(∥u∥
+1
+2
+s+1∥W∥2 + ∥W∥2
+−1)
+(4.20)
+for c independent of u. Clearly the assertion follows from (4.16), (4.17), (4.18) and (4.20).
+5
+A class of examples from dissipative relativistic fluid
+dynamics
+We consider the Euler-augmented Navier-Stokes formulation of dissipative relativistic fluid
+dynamics on flat Minkowski space-time derived in [12] as a generalization of a model proposed
+in [1]. For barotropic fluids it consists of a system of four equations which, using Einstein’s
+summation convention, read
+Aαβγ(ψǫ)∂ψγ
+∂xδ =
+∂
+∂xβ
+�
+Bαβγδ(ψǫ)∂ψγ
+∂xδ
+�
+,
+α = 0, 1, 2, 3,
+(5.1)
+31
+
+where all Greek indices run from 0 to 3, Aαβγ, Bαβγδ are contravariant tensors and the
+unknown function ψǫ = (ψ0, ψ1, ψ2, ψ3)t determining the state of the fluid is a 4-vector with
+respect to the Minkowski-metric of flat space-time. More specifically ψǫ = uǫ/θ with uǫ being
+the 4-velocity, θ the temperature of the fluid. We show that the results of the present work
+imply non-linear stability of the homogeneous reference state ¯ψ = ¯uǫ/¯θ, where ¯uǫ = (1, 0, 0, 0)
+represents the fluid’s rest frame and ¯θ > 0 is a constant temperature.
+For a fluid with equation of state p = ρ/r, 1 ≤ r < ∞, p being the pressure, ρ the specific
+internal energy, the coefficent matrices evaluated at ¯ψ are given by [12] (w.lo.g. assume
+¯θ = 1)5
+A0( ¯ψ) =
+�
+r
+0
+0
+I3
+�
+,
+Aj( ¯ψ) =
+�
+0
+(ej)t
+ej
+0
+�
+,
+B00( ¯ψ) =
+�
+−r2µ
+0
+0
+−νI3
+�
+,
+B0j( ¯ψ) = Bj0( ¯ψ) = 1
+2
+�
+0
+−(µr + ν)(ej)t
+−(µr + ν)ej
+0
+�
+,
+Bij( ¯ψ) =
+�
+−νδij
+0
+0
+ηδij + 1
+2(−µ + 1
+3η + ζ)(ei ⊗ ej + ej ⊗ ei)
+�
+,
+i, j = 1, 2, 3,
+where η, ζ > 0 quantify the fluid’s viscosity, ν, µ > 0 with µ > ˜η := 4
+3η + ζ reflect a frame
+change and Aβ(ψǫ) := (Aαβγ(ψǫ))0≤α,γ≤3, Bβγ(ψǫ) := (Bαβγδ(ψǫ))0≤α,γ≤3, β, δ = 0, . . . , 3.
+We do not give the detailed non-linear formulation at this point and just refer to [12]. The
+only information we need for the argumentation below is the fact that for all β, δ = 0, . . . , 3
+and all states ψǫ the coefficient matrices Aβ(ψǫ), Bβδ(ψǫ), β, δ = 0, . . . , 3 are symmetric (cf.
+ibid.).
+We show (HA), (HB), (D) for the matrices (−B00)−1Bβδ, (−B00)−1Aβ.
+(HA) is straightforward: As −B00(ψǫ), A0(ψǫ) are positive definite at ψǫ = ¯ψ and symmetric
+for all states they are symmetric positive definite also in a neighbourhood of ¯ψ. Thus (HA)
+(a) is satisfied with FA(u) = −B00(u) and (HA) (b) with H(u) = A0(u).
+Regarding (HB) Freist¨uhler proved ibid. that at the reference state ψǫ = ¯ψ for each ω ∈ S2
+the matrix
+˜B(ψǫ, ω) =
+�
+0
+I4
+−(−B00)− 1
+2B(ψǫ, ω)(−B00)− 1
+2
+i(−B00)− 1
+2C(ψǫ, ω)(−B00)− 1
+2
+�
+,
+where
+B(ψǫ, ω) =
+d
+�
+ij=0
+Bij(��ǫ)ωiωj,
+C(ψǫ, ω) = 2
+d
+�
+j=0
+B0j(ψǫ)ωj,
+ω = (ω1, . . . , ω2) ∈ S2,
+5Here e1, e2, e3 and δij denote the conanical basis of R3 and the Kronecker symbol, respectively.
+32
+
+has four simple and two semi-simple purely imaginary eigenvalues. This is then also true for
+B(ψǫ, ω) :=
+�
+0
+I4
+(−B00)−1B(ψǫ, ω)
+i(−B00)−1C(ψǫ, ω),
+�
+= T −1 ˜B(ψǫ, ω)T
+with T = diag((−B00)
+1
+2, (−B00)
+1
+2). Now in the present context the geometric multiplicities
+of purely imaginary eigenvalues of B(ψǫ, ω) are state invariant properties. Therefore there
+exists a symbolic symmetrizer of B due to Remark 3.2.
+To see this invariance note that (even in the general setting in Section 3) the eigenvectors
+v = v(u, ω) ∈ C2n \ {0} to an eigenvalue λ = λ(u, ω) ∈ C of B(u, ω) are exactly of the form
+v = (v1, λv1) with v1 ∈ Cn such that eλt+iξv1 is a plane wave solution to the linearization
+of (1.1) at u. As (5.1) is a covariant expression, eλt+iξv being a plane wave solution with
+λ ∈ iR is also a covariant property (cf. e.g. [13]).
+It remains to show (D1), (D2), (D3). In the following we only consider matrices evaluated
+at ¯ψ. The Fourier-symbols correpsonding to the differential operators in (5.1) are given by
+A(ω) =
+d
+�
+j=1
+Ajωj =
+�0
+ωt
+ω
+0
+�
+,
+B(ω) =
+d
+�
+j,k=1
+Bjkωjωk =
+�−r2µ
+0
+0
+η + (−µ + 1
+3η + ζ)ω ⊗ ω
+�
+,
+C(ω) = 2
+d
+�
+j=1
+B0jξj
+�
+0
+−(µr + ν)ωt
+−(µr + ν)ω
+0
+�
+,
+ω = (ω1, ω2, ω3) ∈ Sd−1.
+It is straightforward to see that for any ω ∈ Sd−1 the matrices A0, Aj(ω), B00, Bjk(ω), C(ω)
+all decompose in sense of linear operators as A0 = A0
+l ⊕ A0
+t, A(ω) = Al ⊕ At, B00 =
+B00
+l
+⊕ B00
+t , B(ω) = Bl ⊗ Bt, C(ω) = Cl ⊕ Ct with respect to the orthogonal decomposition
+C4 = (C×ωC)⊕({0}×{ω}⊥). Thus we can verify the conditions for A0
+l , Al, B00
+l , Bl, Cl and
+A0
+t, At, B00
+t , Bt, Ct separately. We have
+A0
+t = I2,
+At = 0,
+B00 = −νI2,
+Bt = ηI2,
+Ct = 0.
+As η > 0, these matrices correspond to coefficients of damped wave equations and it is
+well-known that such equations satisfy (D). One can also check this easily by virtue of [14],
+Theorem 4 and Lemma 5.
+Next
+A0
+l =
+�r
+0
+0
+1
+�
+,
+Al =
+�0
+1
+1
+0
+�
+,
+B00
+l
+=
+�−r2µ
+0
+0
+−ν
+�
+,
+Bl =
+�−ν
+0
+0
+˜η − µ
+�
+,
+Ct =
+�
+0
+−(µr + ν)
+−(µr + ν)
+0
+�
+.
+It was shown in [14] that
+˜Aj = (−B00
+l )− 1
+2Aj
+l (−B00
+l )− 1
+2,
+˜Bjk
+l
+= (−B00
+l )− 1
+2Bjk
+l (−B00
+l )− 1
+2
+satisfy (D). But then also ˇAj := (−B00
+l )−1Aj
+l , ˇBjk := (−B00
+l )−1Bjk
+l
+satisfy (D).
+33
+
+To see this note that ˇAj = S ˜AjS−1, ˇBjk = S ˜BjkS−1 with S = (−B00)
+1
+2. Hence we have
+ˇW0 = S ˜W0S−1 for ˇW0 and ˜W0 as in (D1) for the matrices ˇAj, ˇBjk and ˜Aj, ˜Bjk, respectively.
+Further the symbolic symmetrizer ˜H = ˜A0 of ( ˜A0)−1 ˜A(ω) the matrix ˇH = S−1 ˜A0S−1 is a
+symbolic symmetrizer for ( ˇA0)−1 ˇA(ω). This yields ˇW1 = S−1 ˜W1S−1 with ˇW1, ˜W1 as in (D1)
+for the respective matrices. If now v is an eigenvector of ˇW0, S−1v is an eigenvector of ˜W0
+and as ˜Aj, ˜Bjk satisfy (D1) we get
+⟨( ˜W1 + ˜W ∗
+1 )S−1v, S−1v⟩ ≤ −c|S−1v|2 ≤ −ˇc|v|2,
+i.e. ⟨( ˇW1 + ˇW ∗
+1 )v, v⟩ ≤ −ˇc|v|2, which proves (D1) for ˇAj, ˇBjk.
+(D2) follows analogously since with S = diag((−B00)
+1
+2, (−B00)
+1
+2) the matrix S−1 ˜H(ω)S−1
+is a symbolic symmetrizer for ˇB(ω) if ˜H(ω) is a symbolic symmetrizer for ˜B(ω).
+Lastly, (D3) is satisfied trivially, as the matrices introduce equivalent systems of PDEs and
+thus solutions to the dispersion relation are identical for the two systems.
+Statements and declarations
+Funding. This work was supported by DFG Grants No. FR 822/10-1, 10-1/2)
+Competing interests. The author has no competing interests to declare that are relevant
+to the content of this article.
+Acknowledgement. The author would like to sincerely thank Heinrich Freist¨uhler for his
+highly helpful suggestions and comments as well as many fruitful discussions.
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+

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@@ -0,0 +1,708 @@
+arXiv:2301.01916v1  [math.CV]  5 Jan 2023
+THE SHARP BOUND OF THE THIRD HANKEL
+DETERMINANT FOR INVERSE OF CONVEX FUNCTIONS
+BISWAJIT RATH1, K. SANJAY KUMAR 2, D. VAMSHEE KRISHNA3
+Abstract. The objective of this paper is to find the best possible upper bound
+of the third Hankel determinant for inverse of convex functions.
+1. Introduction
+Denote H the family all analytic functions in unit disk D = {z ∈ C : |z| < 1}
+and A be the subfamily of functions f normilized by the conditions
+f(0) = f ′(0) − 1 = 0, i.e, of the type
+(1.1)
+f(z) =
+∞
+�
+n=1
+anzn, a1 := 1,
+and S be the subfamily of A, possessing univalent (schlicht) mappings. For f ∈ S,
+has an inverse f −1 given by
+(1.2)
+f −1(w) = w +
+∞
+�
+n=2
+tnwn, |w| < ro(f);
+�
+ro(f) ≥ 1
+4
+�
+.
+A typical problem in geometric function theory is to study a functional made up of
+combination of the coefficients of the original functions. For the positive integers
+r, n, Pommerenke [16] characterized the rth- Hankel determinant of nth-order for
+f given in (1.1), defined as follows:
+(1.3)
+Hr,n(f) =
+an
+an+1
+· · ·
+an+r−1
+an+1
+an+2
+· · ·
+an+r
+...
+...
+...
+...
+an+r−1
+an+r
+· · ·
+an+2r−2
+.
+The problem of finding sharp estimates of the third Hankel determinant obtained
+for r = 3 and n = 1 in (1.2), given by
+(1.4)
+H3,1(f) :=
+a1 = 1
+a2
+a3
+a2
+a3
+a4
+a3
+a4
+a5
+= 2a2a3a4 − a3
+3 − a2
+4 + a3a5 − a2
+2a5,
+is technically much tough than r = n = 2.
+In recent years, many authors are working on obtaining upper bounds (see [2, 8,
+17, 18, 20]) and a few papers were devoted to the estimation of sharp upper bound
+to H3,1(f), for certain subclasses of analytic functions (see [3, 6, 7, 9, 10, 19]).
+2020 Mathematics Subject Classification. 30C45, 30C50.
+Key words and phrases. Holomorphic function, univalent function, Hankel determinant, Inverse
+of Convex, Carath´eodory function.
+1
+
+2
+B. RATH, K. S. KUMAR, D. V. KRISHNA
+Recently Lecko et al. [6] obtained the sharp bound for the class of convex function
+denoted as Sc, defined by
+(1.5)
+Re
+�
+1 + zf ′′(z)
+f ′(z)
+�
+> 0.
+Motivated by these results, in this paper we obtain sharp estimate for H3,1(f −1)
+when f ∈ Sc as 1/36.
+The collection P, of all functions p, each one called as Carath´eodory function [5]
+of the form,
+(1.6)
+p(z) = 1 +
+∞
+�
+t=1
+ctzt,
+having a positive real part in D. In view of (1.4) and (1.5), the coefficients of the
+functions in Sc can be expressed in terms of coefficients of functions in P. We then
+obtain the upper bound of H3,1(f −1), buliding our analysis on the familiar formulas
+of coefficients c2 (see, [15, p. 166]), c3 (see [11, 12]) and c4 can be found in [10].
+The foundation for proofs of our main results is the following lemma and we
+adopt the procedure framed through Libera and Zlotkiewicz [12].
+Lemma 1.1. If p ∈ P, is of the form (1.5) with c1 ≥ 0, such that c1 ∈ [0, 2] then
+2c2 = c2
+1 + νµ,
+4c3 = c3
+1 + 2c1νµ − c1νµ2 + 2ν
+�
+1 − |µ|2�
+ρ,
+and
+8c4 = c4
+1 + 3c2
+1νµ +
+�
+4 − 3c2
+1
+�
+νµ2 + c2
+1νµ3 + 4ν
+�
+1 − |µ|2� �
+1 − |ρ|2�
+ψ
++ 4ν
+�
+1 − |µ|2� �
+c1ρ − cµρ − ¯µρ2�
+,
+where ν := 4 − c2
+1, for some µ, ρ and ψ such that |µ| ≤ 1, |ρ| ≤ 1 and |ψ| ≤ 1.
+2. Main result
+Theorem 2.1. If f ∈ Sc, then
+��H3,1(f −1)
+�� ≤ 1
+36
+and the inequality is sharp for p0(z) = (1 + z3)/(1 − z3).
+Proof. For f ∈ Sc, there exists a holomorphic function p ∈ P such that
+(2.1)
+�
+1 + zf ′′(z)
+f ′(z)
+�
+= p(z) ⇔ {f ′(z) + zf ′′(z)} = p(z)f ′(z)
+Using the series representation for f and p in (2.1), a simple calculation gives
+a2 = c1
+2 , a3 = c2
+1 + c2
+6
+, a4 = 1
+12
+�1
+2c3
+1 + 3
+2c1c2 + c3
+�
+and a5 = 1
+20
+�1
+6c4
+1 + c2
+1c2 + 1
+2c2
+2 + 4
+3c1c3 + c4
+�
+(2.2)
+Now from the defination (1.2), we have
+(2.3)
+w = f(f −1) = f −1(w) +
+∞
+�
+n=2
+an(f −1(w))n.
+
+THE SHARP BOUND OF THE HANKEL DETERMINANT
+3
+Further, we have
+(2.4)
+w = f(f −1) = w +
+∞
+�
+n=2
+tnwn +
+∞
+�
+n=2
+an(w +
+∞
+�
+n=2
+tnwn)n.
+Upon simplification, we obtain
+(t2 + a2)w2 + (t3 + 2a2t2 + a3)w3 + (t4 + 2a2t3 + a2t2
+2 + 3a3t2 + a4)w4
++(t5 + 2a2t4 + 2a2t2t3 + 3a3t3 + 3a3t2
+2 + 4a4t2 + a5)w5 + ...... = 0.
+(2.5)
+Equating the coefficients of like power in (2.5), upon simplification, we obtain
+t2 = −a2; t3 = {−a3 + 2a2
+2}; t4 = {−a4 + 5a2a3 − 5a3
+2};
+t5 = {−a5 + 6a2a4 − 21a2
+2a3 + 3a2
+3 + 14a4
+2}.
+(2.6)
+Using the values of an(n = 2, 3, 4, 5) from (2.2) in (2.6), upon simplification, we
+obtain
+t2 = −c1
+2 , t3 = 1
+6
+�
+2c2
+1 − c2
+�
+, t4 = 1
+24
+�
+−6c3
+1 + 7c1c2 − 2c3
+�
+and t5 =
+1
+120
+�
+−6c4 + 22c1c3 − 46c2
+1c2 + 7c2
+2 + 24c4
+1
+�
+.
+(2.7)
+Now,
+H3,1(f −1) =
+t1 = 1
+t2
+t3
+t2
+t3
+t4
+t3
+t4
+t5
+,
+(2.8)
+Using the values of tj, (j = 2, 3, 4, 5) from (2.7) in (2.8), it simplifies to give
+H3,1(f −1) =
+1
+8640
+�
+4c6
+1 − 24c4
+1c2 + 12c3
+1c3 + 39c2
+1c2
+2 − 44c3
+2 + 36c1c2c3
+−36c2
+1c4 − 60c2
+3 + 72c2c4
+�
+.
+(2.9)
+In view of (2.9), using the values of c2, c3 and c4 from lemma 1.1, gives
+24c4
+1c2 =12
+�
+c6
+1 + c4
+1νµ
+�
+;
+12c3
+1c3 =3
+�
+c6
+1 + 2c4
+1νµ − c4
+1νµ2 + 2c3
+1ν(1 − |µ|2)ρ
+�
+44c3
+2 =11
+2
+�
+c6
+1 + 3c4
+1νµ + 3c2
+1ν2µ2 + ν3µ3�
+;
+39c2
+1c2
+2 =39
+4
+�
+c6
+1 + 2c4
+1νµ + c2
+1ν2µ2�
+;
+36c1c2c3 =9
+2
+�
+c6
+1 + 3c4
+1νµ + 2c2
+1ν2µ2 − c4
+1νµ2 − c2
+1ν2µ3
++2ν
+�
+c3
+1 + c1νµ
+� �
+1 − |µ|2�
+ρ
+�
+;
+60c2
+3 =15
+4
+�
+c6 + 4c4νµ + 4c4ν2µ2 − 2c4νµ2 − 4c2ν2µ3 + c2ν2µ4
++4ν(c3 + 2cνµ − cνµ2)(1 − |µ|2)ρ + 4ν2(1 − |µ|2)2ρ2�
+;
+72c2c4 − 36c2
+1c4 =9
+2
+�
+c4
+1νµ + 3c2
+1ν2µ2 +
+�
+4 − 3c2
+1
+�
+ν2µ3 + c2
+1ν2µ4
++ 4ν2c1µ (1 − µ)
+�
+1 − |µ|2�
+ρ − 4ν2 �
+1 − |µ|2�
+|µ|2ρ2
++4ν2 �
+1 − |µ|2� �
+1 − |ρ|2�
+µψ
+�
+.
+(2.10)
+
+4
+B. RATH, K. S. KUMAR, D. V. KRISHNA
+Imputting the values from (2.10) in the expression (2.9), after simplifying, we get
+H3,1(f −1) =
+1
+8640
+�3
+4c2
+1ν2µ2 − 3c2
+1ν2µ3 + 3
+4c2
+1ν2µ4 − 11
+2 ν3µ3 + 18ν2µ3
+−
+�
+3c1ν2µ + 3c1ν2µ2� �
+1 − |µ|2�
+ρ − 3ν2 �
+5 + |µ|2� �
+1 − |µ|2�
+ρ2
++18ν2µ
+�
+1 − |µ|2�
+(1 − |ρ|2�
+ψ
+�
+.
+(2.11)
+Putting u := c1 and taking ν =
+�
+4 − u2�
+in (2.11), we obtain
+H3,1(f −1) =
+�
+4 − u2�2
+8640
+�3
+4u2µ2 + 3
+2u2µ3 + 3
+4u2µ4 − (4 − u2)µ3
+− 3uµ (1 + µ)
+�
+1 − |µ|2�
+ρ − 3
+�
+5 + |µ|2� �
+1 − |µ|2�
+ρ2
++18µ
+�
+1 − |µ|2�
+(1 − |ρ|2�
+ψ
+�
+.
+(2.12)
+Taking modulus on both sides of (2.12), using |µ| = v ∈ [0, 1], |ρ| = w ∈ [0, 1],
+c1 = u ∈ [0, 2] and |ψ| ≤ 1, we obtain
+(2.13)
+����H3,1(f −1)
+���� ≤ ϑ (u, v, w)
+8640
+,
+where ϑ : R3 → R is defined as
+ϑ (u, v, w) =
+�
+4 − u2�2 �3
+4u2v2 + 3
+2u2v3 + 3
+4u2v4 +
+�
+4 − u2�
+v3
++ 3uv (1 + v)
+�
+1 − v2�
+w + 3
+�
+5 + v2� �
+1 − v2�
+w2
++18v
+�
+1 − v2�
+(1 − w2��
+(2.14)
+Now, we are making an attempt to maximize the function ϑ (u, v, w) on
+Ω := [0, 2] × [0, 1] × [0, 1].
+A. On the vertices of Ω, from (2.14), we get
+ϑ (0, 0, 0) = ϑ (2, 0, 0) = ϑ (2, 1, 0) = ϑ (2, 0, 1) = ϑ (2, 1, 1) = 0,
+ϑ (0, 0, 1) = 240, ϑ (0, 1, 0) = ϑ (0, 1, 1) = 64.
+B. On the edges of Ω, from (2.14), we have
+(i) For the edge u = 0, v = 0, 0 < w < 1, we obtain.
+ϑ (0, 0, w) = 240w2 ≤ 240.
+(ii) For the edge u = 0, v = 1, 0 < w < 1, we obtain
+ϑ (0, 1, w) = 64.
+(iii) For u = 0, w = 0, 0 < v < 1,
+ϑ (0, v, 0) = 32v(9 − 7v2) ≤ 192
+�
+3
+7, , for v =
+√
+2.
+(iv) For u = 0, w = 1, 0 < v < 1,
+ϑ (0, v, 1) = 240 − 192v2 + 64v3 − 48v4 ≤ 240.
+(v) For v = 0, w = 1, 0 < u < 2,
+ϑ (u, 0, 1) = 15(4 − u2)2 ≤ 240.
+
+THE SHARP BOUND OF THE HANKEL DETERMINANT
+5
+(vi) For the edges: v = 1, w = 0, 0 < u < 2 or v = 1, w = 1, 0 < u < 2, we have
+ϑ (u, 1, w) = (4 − u2)2(4 + 2u2) ≤ 64.
+(vii) For the edges: u = 2, v = 0, 0 < w < 1 or u = 2, v = 1, 0 < w < 1 or
+u = 2, w = 0, 0 < v < 1 or c = 2, w = 1, 0 < v < 1 or v = 0, w = 0, 0 < u < 2,
+we obtain
+ϑ (2, v, w) = 0.
+C. Now, we consider the six faces of Ω.
+(i) On the face u = 2, from (2.14), we obtain
+ϑ (2, v, w) = 0.
+(ii) On the face u = 0, v ∈ (0, 1) and w ∈ (0, 1) from (2.14), we get
+ϑ (0, v, w) = 288v − 224v3 + (240 − 288v − 192v2 + 288v3 − 48v4)w2
+= 288v − 224v3 + 48(5 − v)(−1 + v)2(1 + v)w2
+≤ 288v − 224v3 + 48(5 − v)(−1 + v)2(1 + v)
+= 240 − 192v2 + 64v3 − 48v4 ≤ 240.
+(iii) On the face v = 0 u ∈ (0, 2), w ∈ (0, 1), from (2.14), we obtain
+ϑ (u, 0, w) = 15(4 − u2)2w2 ≤ 15(4 − u2)2 ≤ 240.
+(iv) On the face v = 1, u ∈ (0, 2), w ∈ (0, 1), from (2.14), we observe that the
+function ϑ (u, 1, w) is independent of w, from B(vi), we have ϑ (u, 1, w) ≤ 240.
+(v) On the face w = 0, u ∈ (0, 2), v ∈ (0, 1), from (2.14), we obtain
+ϑ (u, v, 0) = (4 − u2)2
+�3u2v2
+4
++ 3u2v3
+2
++ (4 − u2)v3 + 3u2v4
+4
++ 18v(1 − v2)
+�
+= (4 − u2)2
+�
+18v − 14v3 + u2
+�3v2
+4
++ v3
+2 + 3v4
+4
+��
+≤ (4 − u2)2
+�
+12
+�
+3
+7 + 2u2
+�
+≤ 192
+�
+3
+7, u ∈ (0, 2).
+(vi) On the face w = 1, in (2.14), we obtain
+ϑ (u, v, 1) = (4 − u2)2
+�3
+4u2v2 + 3
+2u2v3 + 3
+4u2v4 + (4 − u2)v3
++ 3uv(1 + v)(1 − v2) + 3(5 + v2)
+�
+1 − v2� �
+:= g3(u, v), with (u, v) ∈ R2.
+Note that all real solutions (u,v) of the system of equation
+∂g3
+∂u = 3
+2(−4 + u2)
+�
+8(−1 + v)v(1 + v)2 − 10u2(−1 + v)v(1 + v)2
++u3v2(3 + 2v + 3v2) − 4u(−10 + 9v2 − 2v3 + 3v4)
+�
+= 0
+and
+
+6
+B. RATH, K. S. KUMAR, D. V. KRISHNA
+∂g3
+∂v = 3
+2(−4 + u2)2(−8v(2 − v + v2) + u2v(1 + v + 2v2)
++ u(2 + 4v − 6v2 − 8v3)) = 0
+by a numerical computation are the following
+(0, 0), (−2.63625, −1.53087), (−1.0493, 1.14045) and (±2, x), x ∈ R.
+Therefore, g3 has no critical point in (0, 2) × (0, 1).
+D. Now, consider the interior portion of Ω i.e. (0, 2) × (0, 1) × (0, 1).
+Differentiating ϑ(u, v, w) partially with respect w, we obtain
+∂ϑ
+∂w = 1
+2(4 − u2)2 �
+60w2 + 3v2(u2 + 4uw − 16w2) + 12v(6 + uw − 6w2)
++3v4(u2 − 4uw − 4w2) + 2v3(−28 + u2 − 6uw + 36w2)
+�
+upon solving ∂ϑ
+∂w = 0, we get
+w0 = −
+uv(1 + v)
+2(5 − v)(1 − v) /∈ (0, 1) for (u, v) ∈ (0, 2) × (0, 1)
+Hence ϑ(u, v, w) has no critical point in the interior of Ω.
+In review of cases A, B, C and D, we obtained
+(2.15)
+max
+�
+ϑ(u, v, w) : u ∈ [0, 2], v ∈ [0, 1], w ∈ [0, 1]
+�
+= 240.
+From expression (2.13) and (2.15), we obtain
+(2.16)
+���H3,1(f −1)
+��� ≤ 1
+36.
+For p0 ∈ Sc, we obtain t2 = t3 = t5 = 0, t4 = 1/6, which follows the result.
+□
+Data Availability: My manuscript has no associate data
+References
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+
+THE SHARP BOUND OF THE HANKEL DETERMINANT
+7
+[9] O. S. Kwon, Y. J. Sim , The Sharp Bound of the Hankel Determinant of the Third Kind for
+Starlike Functions with Real Coefficients , Mathematics, 2019, doi:10.3390/math7080721.
+[10] A. Lecko, Y. J. Sim, B. Smiarowska, The Sharp Bound of the Hankel Determinant of the
+Third kind for starlike functions of order 1/2, Complex Anal. Oper. Theory, 13(5)(2019),
+2231–2238.
+[11] R. J. Libera, E. J. Zlotkiewicz, Early coefficients of the inverse of a regular convex function,
+Proc. Amer. Math. Soc. 85 (1982), no. 2, 225–230.
+[12] R. J. Libera and E. J. Zlotkiewicz, Coefficient bounds for the inverse of a function with
+derivative in P, Proc. Amer. Math. Soc., 87(2)(1983), 251–257.
+[13] S.
+Maharana,
+J.
+K.
+Prajapat
+and
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+Coefficient
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+direction,
+Honam
+Mathematical
+J.
+42
+(2020),
+781–794
+https://doi.org/10.5831/HMJ.2020.42.4.781 .
+[14] Ozaki, S. On the theory of multivalent functions. II. Sci. Rep. Tokyo Bunrika Daigaku. Sect.
+A 4 (1941), 45–87.
+[15] Ch. Pommerenke, Univalent functions, Vandenhoeck & Ruprecht, Gottingen 1975.
+[16] Ch. Pommerenke, On the coefficients and Hankel determinants of univalent functions, J.
+Lond. Math. Soc., 41(s-1)(1966), 111–122.
+[17] Y. J. Sim and P. Zaprawa, Third Hankel determinants for two classes of analytic functions
+with real coefficients, Forum Math., 33(4)(2021), 973-986, https://doi.org/10.1515/forum-
+2021-0014.
+[18] H.
+M.
+Srivastava,
+B.
+Khan,
+N.
+Khan,
+M.
+Tahir,
+S.
+Ahmad,
+NasirKhan,
+Up-
+per
+bound
+of
+the
+third
+Hankel
+determinant
+for
+a
+subclass
+of
+q-starlike
+func-
+tions
+associated
+with
+the
+q-exponential
+function,
+Bull.
+Sci.
+math.,
+167
+(2021),
+https://doi.org/10.1016/j.bulsci.2020.102942.
+[19] K. Ullah, H. M. Srivastava, A. Rafiq, M. Arif and S. Arjika, A study of sharp co-
+efficient bounds for a new subfamily of starlike functions, J. Inequal. Appl., (2021),
+https://doi.org/10.1186/s13660-021-02729-1.
+[20] P. Zaprawa, M. Obradovic and N. Tuneski, Third Hankel determinant for univalent star-
+like functions, RACSAM Rev. R. Acad. Cienc. Exactas F´ıs. Nat. Ser. A Mat. ,115(2021),
+https://doi.org/10.1007/s13398-020-00977-2.
+1.2.3Department of Mathematics, GITAM School of Science, GITAM (Deemed to be
+University), Visakhapatnam- 530 045, A.P., India
+Email address: brath@gitam.edu1∗,skarri9@gitam.in2,vamsheekrishna1972@gmail.com3
+

29AzT4oBgHgl3EQf9P5O/content/tmp_files/load_file.txt

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+filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AzT4oBgHgl3EQf9P5O/content/2301.01916v1.pdf,len=342
+page_content='arXiv:2301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AzT4oBgHgl3EQf9P5O/content/2301.01916v1.pdf'}
+page_content='01916v1  [math.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AzT4oBgHgl3EQf9P5O/content/2301.01916v1.pdf'}
+page_content='CV]  5 Jan 2023  THE SHARP BOUND OF THE THIRD HANKEL  DETERMINANT FOR INVERSE OF CONVEX FUNCTIONS  BISWAJIT RATH1, K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AzT4oBgHgl3EQf9P5O/content/2301.01916v1.pdf'}
+page_content=' SANJAY KUMAR 2, D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AzT4oBgHgl3EQf9P5O/content/2301.01916v1.pdf'}
+page_content=' VAMSHEE KRISHNA3  Abstract.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AzT4oBgHgl3EQf9P5O/content/2301.01916v1.pdf'}
+page_content=' The objective of this paper is to find the best possible upper bound  of the third Hankel determinant for inverse of convex functions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AzT4oBgHgl3EQf9P5O/content/2301.01916v1.pdf'}
+page_content='  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AzT4oBgHgl3EQf9P5O/content/2301.01916v1.pdf'}
+page_content=' Introduction  Denote H the family all analytic functions in unit disk D = {z ∈ C : |z| < 1}  and A be the subfamily of functions f normilized by the conditions  f(0) = f ′(0) − 1 = 0, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AzT4oBgHgl3EQf9P5O/content/2301.01916v1.pdf'}
+page_content='e, of the type  (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AzT4oBgHgl3EQf9P5O/content/2301.01916v1.pdf'}
+page_content='1)  f(z) =  ∞  �  n=1  anzn, a1 := 1,  and S be the subfamily of A, possessing univalent (schlicht) mappings.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AzT4oBgHgl3EQf9P5O/content/2301.01916v1.pdf'}
+page_content=' For f ∈ S,  has an inverse f −1 given by  (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AzT4oBgHgl3EQf9P5O/content/2301.01916v1.pdf'}
+page_content='2)  f −1(w) = w +  ∞  �  n=2  tnwn, |w| < ro(f);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AzT4oBgHgl3EQf9P5O/content/2301.01916v1.pdf'}
+page_content='  �  ro(f) ≥ 1  4  �  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AzT4oBgHgl3EQf9P5O/content/2301.01916v1.pdf'}
+page_content='  A typical problem in geometric function theory is to study a functional made up of  combination of the coefficients of the original functions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AzT4oBgHgl3EQf9P5O/content/2301.01916v1.pdf'}
+page_content=' For the positive integers  r, n, Pommerenke [16] characterized the rth- Hankel determinant of nth-order for  f given in (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AzT4oBgHgl3EQf9P5O/content/2301.01916v1.pdf'}
+page_content='1), defined as follows:  (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AzT4oBgHgl3EQf9P5O/content/2301.01916v1.pdf'}
+page_content='3)  Hr,n(f) =  an  an+1  · ·  an+r−1  an+1  an+2  · ·  an+r  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AzT4oBgHgl3EQf9P5O/content/2301.01916v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AzT4oBgHgl3EQf9P5O/content/2301.01916v1.pdf'}
+page_content='  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AzT4oBgHgl3EQf9P5O/content/2301.01916v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AzT4oBgHgl3EQf9P5O/content/2301.01916v1.pdf'}
+page_content='  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AzT4oBgHgl3EQf9P5O/content/2301.01916v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AzT4oBgHgl3EQf9P5O/content/2301.01916v1.pdf'}
+page_content='  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AzT4oBgHgl3EQf9P5O/content/2301.01916v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AzT4oBgHgl3EQf9P5O/content/2301.01916v1.pdf'}
+page_content='  an+r−1  an+r  · ·  an+2r−2  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AzT4oBgHgl3EQf9P5O/content/2301.01916v1.pdf'}
+page_content='  The problem of finding sharp estimates of the third Hankel determinant obtained  for r = 3 and n = 1 in (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AzT4oBgHgl3EQf9P5O/content/2301.01916v1.pdf'}
+page_content='2), given by  (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AzT4oBgHgl3EQf9P5O/content/2301.01916v1.pdf'}
+page_content='4)  H3,1(f) :=  a1 = 1  a2  a3  a2  a3  a4  a3  a4  a5  = 2a2a3a4 − a3  3 − a2  4 + a3a5 − a2  2a5,  is technically much tough than r = n = 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AzT4oBgHgl3EQf9P5O/content/2301.01916v1.pdf'}
+page_content='  In recent years, many authors are working on obtaining upper bounds (see [2, 8,  17, 18, 20]) and a few papers were devoted to the estimation of sharp upper bound  to H3,1(f), for certain subclasses of analytic functions (see [3, 6, 7, 9, 10, 19]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AzT4oBgHgl3EQf9P5O/content/2301.01916v1.pdf'}
+page_content='  2020 Mathematics Subject Classification.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AzT4oBgHgl3EQf9P5O/content/2301.01916v1.pdf'}
+page_content=' 30C45, 30C50.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AzT4oBgHgl3EQf9P5O/content/2301.01916v1.pdf'}
+page_content='  Key words and phrases.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AzT4oBgHgl3EQf9P5O/content/2301.01916v1.pdf'}
+page_content=' Holomorphic function, univalent function, Hankel determinant, Inverse  of Convex, Carath´eodory function.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AzT4oBgHgl3EQf9P5O/content/2301.01916v1.pdf'}
+page_content='  1  2  B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AzT4oBgHgl3EQf9P5O/content/2301.01916v1.pdf'}
+page_content=' RATH, K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AzT4oBgHgl3EQf9P5O/content/2301.01916v1.pdf'}
+page_content=' S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AzT4oBgHgl3EQf9P5O/content/2301.01916v1.pdf'}
+page_content=' KUMAR, D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AzT4oBgHgl3EQf9P5O/content/2301.01916v1.pdf'}
+page_content=' V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AzT4oBgHgl3EQf9P5O/content/2301.01916v1.pdf'}
+page_content=' KRISHNA  Recently Lecko et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AzT4oBgHgl3EQf9P5O/content/2301.01916v1.pdf'}
+page_content=' [6] obtained the sharp bound for the class of convex function  denoted as Sc, defined by  (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AzT4oBgHgl3EQf9P5O/content/2301.01916v1.pdf'}
+page_content='5)  Re  �  1 + zf ′′(z)  f ′(z)  �  > 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AzT4oBgHgl3EQf9P5O/content/2301.01916v1.pdf'}
+page_content='  Motivated by these results, in this paper we obtain sharp estimate for H3,1(f −1)  when f ∈ Sc as 1/36.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AzT4oBgHgl3EQf9P5O/content/2301.01916v1.pdf'}
+page_content='  The collection P, of all functions p, each one called as Carath´eodory function [5]  of the form,  (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AzT4oBgHgl3EQf9P5O/content/2301.01916v1.pdf'}
+page_content='6)  p(z) = 1 +  ∞  �  t=1  ctzt,  having a positive real part in D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AzT4oBgHgl3EQf9P5O/content/2301.01916v1.pdf'}
+page_content=' In view of (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AzT4oBgHgl3EQf9P5O/content/2301.01916v1.pdf'}
+page_content='4) and (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AzT4oBgHgl3EQf9P5O/content/2301.01916v1.pdf'}
+page_content='5), the coefficients of the  functions in Sc can be expressed in terms of coefficients of functions in P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AzT4oBgHgl3EQf9P5O/content/2301.01916v1.pdf'}
+page_content=' We then  obtain the upper bound of H3,1(f −1), buliding our analysis on the familiar formulas  of coefficients c2 (see, [15, p.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AzT4oBgHgl3EQf9P5O/content/2301.01916v1.pdf'}
+page_content=' 166]), c3 (see [11, 12]) and c4 can be found in [10].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AzT4oBgHgl3EQf9P5O/content/2301.01916v1.pdf'}
+page_content='  The foundation for proofs of our main results is the following lemma and we  adopt the procedure framed through Libera and Zlotkiewicz [12].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AzT4oBgHgl3EQf9P5O/content/2301.01916v1.pdf'}
+page_content='  Lemma 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AzT4oBgHgl3EQf9P5O/content/2301.01916v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AzT4oBgHgl3EQf9P5O/content/2301.01916v1.pdf'}
+page_content=' If p ∈ P, is of the form (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AzT4oBgHgl3EQf9P5O/content/2301.01916v1.pdf'}
+page_content='5) with c1 ≥ 0, such that c1 ∈ [0, 2] then  2c2 = c2  1 + νµ,  4c3 = c3  1 + 2c1νµ − c1νµ2 + 2ν  �  1 − |µ|2�  ρ,  and  8c4 = c4  1 + 3c2  1νµ +  �  4 − 3c2  1  �  νµ2 + c2  1νµ3 + 4ν  �  1 − |µ|2� �  1 − |ρ|2�  ψ  + 4ν  �  1 − |µ|2� �  c1ρ − cµρ − ¯µρ2�  ,  where ν := 4 − c2  1, for some µ, ρ and ψ such that |µ| ≤ 1, |ρ| ≤ 1 and |ψ| ≤ 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AzT4oBgHgl3EQf9P5O/content/2301.01916v1.pdf'}
+page_content='  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AzT4oBgHgl3EQf9P5O/content/2301.01916v1.pdf'}
+page_content=' Main result  Theorem 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AzT4oBgHgl3EQf9P5O/content/2301.01916v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AzT4oBgHgl3EQf9P5O/content/2301.01916v1.pdf'}
+page_content=' If f ∈ Sc, then  ��H3,1(f −1)  �� ≤ 1  36  and the inequality is sharp for p0(z) = (1 + z3)/(1 − z3).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AzT4oBgHgl3EQf9P5O/content/2301.01916v1.pdf'}
+page_content='  Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AzT4oBgHgl3EQf9P5O/content/2301.01916v1.pdf'}
+page_content=' For f ∈ Sc, there exists a holomorphic function p ∈ P such that  (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AzT4oBgHgl3EQf9P5O/content/2301.01916v1.pdf'}
+page_content='1)  �  1 + zf ′′(z)  f ′(z)  �  = p(z) ⇔ {f ′(z) + zf ′′(z)} = p(z)f ′(z)  Using the series representation for f and p in (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AzT4oBgHgl3EQf9P5O/content/2301.01916v1.pdf'}
+page_content='1), a simple calculation gives  a2 = c1  2 , a3 = c2  1 + c2  6  , a4 = 1  12  �1  2c3  1 + 3  2c1c2 + c3  �  and a5 = 1  20  �1  6c4  1 + c2  1c2 + 1  2c2  2 + 4  3c1c3 + c4  �  (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AzT4oBgHgl3EQf9P5O/content/2301.01916v1.pdf'}
+page_content='2)  Now from the defination (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AzT4oBgHgl3EQf9P5O/content/2301.01916v1.pdf'}
+page_content='2), we have  (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AzT4oBgHgl3EQf9P5O/content/2301.01916v1.pdf'}
+page_content='3)  w = f(f −1) = f −1(w) +  ∞  �  n=2  an(f −1(w))n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AzT4oBgHgl3EQf9P5O/content/2301.01916v1.pdf'}
+page_content='  THE SHARP BOUND OF THE HANKEL DETERMINANT  3  Further, we have  (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AzT4oBgHgl3EQf9P5O/content/2301.01916v1.pdf'}
+page_content='4)  w = f(f −1) = w +  ∞  �  n=2  tnwn +  ∞  �  n=2  an(w +  ∞  �  n=2  tnwn)n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AzT4oBgHgl3EQf9P5O/content/2301.01916v1.pdf'}
+page_content='  Upon simplification, we obtain  (t2 + a2)w2 + (t3 + 2a2t2 + a3)w3 + (t4 + 2a2t3 + a2t2  2 + 3a3t2 + a4)w4  +(t5 + 2a2t4 + 2a2t2t3 + 3a3t3 + 3a3t2  2 + 4a4t2 + a5)w5 + .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AzT4oBgHgl3EQf9P5O/content/2301.01916v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AzT4oBgHgl3EQf9P5O/content/2301.01916v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AzT4oBgHgl3EQf9P5O/content/2301.01916v1.pdf'}
+page_content='. = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AzT4oBgHgl3EQf9P5O/content/2301.01916v1.pdf'}
+page_content='  (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AzT4oBgHgl3EQf9P5O/content/2301.01916v1.pdf'}
+page_content='5)  Equating the coefficients of like power in (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AzT4oBgHgl3EQf9P5O/content/2301.01916v1.pdf'}
+page_content='5), upon simplification, we obtain  t2 = −a2;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AzT4oBgHgl3EQf9P5O/content/2301.01916v1.pdf'}
+page_content=' t3 = {−a3 + 2a2  2};' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AzT4oBgHgl3EQf9P5O/content/2301.01916v1.pdf'}
+page_content=' t4 = {−a4 + 5a2a3 − 5a3  2};' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AzT4oBgHgl3EQf9P5O/content/2301.01916v1.pdf'}
+page_content='  t5 = {−a5 + 6a2a4 − 21a2  2a3 + 3a2  3 + 14a4  2}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AzT4oBgHgl3EQf9P5O/content/2301.01916v1.pdf'}
+page_content='  (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AzT4oBgHgl3EQf9P5O/content/2301.01916v1.pdf'}
+page_content='6)  Using the values of an(n = 2, 3, 4, 5) from (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AzT4oBgHgl3EQf9P5O/content/2301.01916v1.pdf'}
+page_content='2) in (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AzT4oBgHgl3EQf9P5O/content/2301.01916v1.pdf'}
+page_content='6), upon simplification, we  obtain  t2 = −c1  2 , t3 = 1  6  �  2c2  1 − c2  �  , t4 = 1  24  �  −6c3  1 + 7c1c2 − 2c3  �  and t5 =  1  120  �  −6c4 + 22c1c3 − 46c2  1c2 + 7c2  2 + 24c4  1  �  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AzT4oBgHgl3EQf9P5O/content/2301.01916v1.pdf'}
+page_content='  (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AzT4oBgHgl3EQf9P5O/content/2301.01916v1.pdf'}
+page_content='7)  Now,  H3,1(f −1) =  t1 = 1  t2  t3  t2  t3  t4  t3  t4  t5  ,  (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AzT4oBgHgl3EQf9P5O/content/2301.01916v1.pdf'}
+page_content='8)  Using the values of tj, (j = 2, 3, 4, 5) from (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AzT4oBgHgl3EQf9P5O/content/2301.01916v1.pdf'}
+page_content='7) in (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AzT4oBgHgl3EQf9P5O/content/2301.01916v1.pdf'}
+page_content='8), it simplifies to give  H3,1(f −1) =  1  8640  �  4c6  1 − 24c4  1c2 + 12c3  1c3 + 39c2  1c2  2 − 44c3  2 + 36c1c2c3  −36c2  1c4 − 60c2  3 + 72c2c4  �  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AzT4oBgHgl3EQf9P5O/content/2301.01916v1.pdf'}
+page_content='  (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AzT4oBgHgl3EQf9P5O/content/2301.01916v1.pdf'}
+page_content='9)  In view of (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AzT4oBgHgl3EQf9P5O/content/2301.01916v1.pdf'}
+page_content='9), using the values of c2, c3 and c4 from lemma 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AzT4oBgHgl3EQf9P5O/content/2301.01916v1.pdf'}
+page_content='1, gives  24c4  1c2 =12  �  c6  1 + c4  1νµ  �  ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AzT4oBgHgl3EQf9P5O/content/2301.01916v1.pdf'}
+page_content='  12c3  1c3 =3  �  c6  1 + 2c4  1νµ − c4  1νµ2 + 2c3  1ν(1 − |µ|2)ρ  �  44c3  2 =11  2  �  c6  1 + 3c4  1νµ + 3c2  1ν2µ2 + ν3µ3�  ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AzT4oBgHgl3EQf9P5O/content/2301.01916v1.pdf'}
+page_content='  39c2  1c2  2 =39  4  �  c6  1 + 2c4  1νµ + c2  1ν2µ2�  ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AzT4oBgHgl3EQf9P5O/content/2301.01916v1.pdf'}
+page_content='  36c1c2c3 =9  2  �  c6  1 + 3c4  1νµ + 2c2  1ν2µ2 − c4  1νµ2 − c2  1ν2µ3  +2ν  �  c3  1 + c1νµ  � �  1 − |µ|2�  ρ  �  ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AzT4oBgHgl3EQf9P5O/content/2301.01916v1.pdf'}
+page_content='  60c2  3 =15  4  �  c6 + 4c4νµ + 4c4ν2µ2 − 2c4νµ2 − 4c2ν2µ3 + c2ν2µ4  +4ν(c3 + 2cνµ − cνµ2)(1 − |µ|2)ρ + 4ν2(1 − |µ|2)2ρ2�  ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AzT4oBgHgl3EQf9P5O/content/2301.01916v1.pdf'}
+page_content='  72c2c4 − 36c2  1c4 =9  2  �  c4  1νµ + 3c2  1ν2µ2 +  �  4 − 3c2  1  �  ν2µ3 + c2  1ν2µ4  + 4ν2c1µ (1 − µ)  �  1 − |µ|2�  ρ − 4ν2 �  1 − |µ|2�  |µ|2ρ2  +4ν2 �  1 − |µ|2� �  1 − |ρ|2�  µψ  �  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AzT4oBgHgl3EQf9P5O/content/2301.01916v1.pdf'}
+page_content='  (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AzT4oBgHgl3EQf9P5O/content/2301.01916v1.pdf'}
+page_content='10)  4  B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AzT4oBgHgl3EQf9P5O/content/2301.01916v1.pdf'}
+page_content=' RATH, K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AzT4oBgHgl3EQf9P5O/content/2301.01916v1.pdf'}
+page_content=' S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AzT4oBgHgl3EQf9P5O/content/2301.01916v1.pdf'}
+page_content=' KUMAR, D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AzT4oBgHgl3EQf9P5O/content/2301.01916v1.pdf'}
+page_content=' V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AzT4oBgHgl3EQf9P5O/content/2301.01916v1.pdf'}
+page_content=' KRISHNA  Imputting the values from (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AzT4oBgHgl3EQf9P5O/content/2301.01916v1.pdf'}
+page_content='10) in the expression (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AzT4oBgHgl3EQf9P5O/content/2301.01916v1.pdf'}
+page_content='9), after simplifying, we get  H3,1(f −1) =  1  8640  �3  4c2  1ν2µ2 − 3c2  1ν2µ3 + 3  4c2  1ν2µ4 − 11  2 ν3µ3 + 18ν2µ3  −  �  3c1ν2µ + 3c1ν2µ2� �  1 − |µ|2�  ρ − 3ν2 �  5 + |µ|2� �  1 − |µ|2�  ρ2  +18ν2µ  �  1 − |µ|2�  (1 − |ρ|2�  ψ  �  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AzT4oBgHgl3EQf9P5O/content/2301.01916v1.pdf'}
+page_content='  (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AzT4oBgHgl3EQf9P5O/content/2301.01916v1.pdf'}
+page_content='11)  Putting u := c1 and taking ν =  �  4 − u2�  in (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AzT4oBgHgl3EQf9P5O/content/2301.01916v1.pdf'}
+page_content='11), we obtain  H3,1(f −1) =  �  4 − u2�2  8640  �3  4u2µ2 + 3  2u2µ3 + 3  4u2µ4 − (4 − u2)µ3  − 3uµ (1 + µ)  �  1 − |µ|2�  ρ − 3  �  5 + |µ|2� �  1 − |µ|2�  ρ2  +18µ  �  1 − |µ|2�  (1 − |ρ|2�  ψ  �  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AzT4oBgHgl3EQf9P5O/content/2301.01916v1.pdf'}
+page_content='  (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AzT4oBgHgl3EQf9P5O/content/2301.01916v1.pdf'}
+page_content='12)  Taking modulus on both sides of (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AzT4oBgHgl3EQf9P5O/content/2301.01916v1.pdf'}
+page_content='12), using |µ| = v ∈ [0, 1], |ρ| = w ∈ [0, 1],  c1 = u ∈ [0, 2] and |ψ| ≤ 1, we obtain  (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AzT4oBgHgl3EQf9P5O/content/2301.01916v1.pdf'}
+page_content='13)  ����H3,1(f −1)  ���� ≤ ϑ (u, v, w)  8640  ,  where ϑ : R3 → R is defined as  ϑ (u, v, w) =  �  4 − u2�2 �3  4u2v2 + 3  2u2v3 + 3  4u2v4 +  �  4 − u2�  v3  + 3uv (1 + v)  �  1 − v2�  w + 3  �  5 + v2� �  1 − v2�  w2  +18v  �  1 − v2�  (1 − w2��  (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AzT4oBgHgl3EQf9P5O/content/2301.01916v1.pdf'}
+page_content='14)  Now, we are making an attempt to maximize the function ϑ (u, v, w) on  Ω := [0, 2] × [0, 1] × [0, 1].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AzT4oBgHgl3EQf9P5O/content/2301.01916v1.pdf'}
+page_content='  A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AzT4oBgHgl3EQf9P5O/content/2301.01916v1.pdf'}
+page_content=' On the vertices of Ω, from (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AzT4oBgHgl3EQf9P5O/content/2301.01916v1.pdf'}
+page_content='14), we get  ϑ (0, 0, 0) = ϑ (2, 0, 0) = ϑ (2, 1, 0) = ϑ (2, 0, 1) = ϑ (2, 1, 1) = 0,  ϑ (0, 0, 1) = 240, ϑ (0, 1, 0) = ϑ (0, 1, 1) = 64.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AzT4oBgHgl3EQf9P5O/content/2301.01916v1.pdf'}
+page_content='  B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AzT4oBgHgl3EQf9P5O/content/2301.01916v1.pdf'}
+page_content=' On the edges of Ω, from (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AzT4oBgHgl3EQf9P5O/content/2301.01916v1.pdf'}
+page_content='14), we have  (i) For the edge u = 0, v = 0, 0 < w < 1, we obtain.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AzT4oBgHgl3EQf9P5O/content/2301.01916v1.pdf'}
+page_content='  ϑ (0, 0, w) = 240w2 ≤ 240.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AzT4oBgHgl3EQf9P5O/content/2301.01916v1.pdf'}
+page_content='  (ii) For the edge u = 0, v = 1, 0 < w < 1, we obtain  ϑ (0, 1, w) = 64.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AzT4oBgHgl3EQf9P5O/content/2301.01916v1.pdf'}
+page_content='  (iii) For u = 0, w = 0, 0 < v < 1,  ϑ (0, v, 0) = 32v(9 − 7v2) ≤ 192  �  3  7, , for v =  √  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AzT4oBgHgl3EQf9P5O/content/2301.01916v1.pdf'}
+page_content='  (iv) For u = 0, w = 1, 0 < v < 1,  ϑ (0, v, 1) = 240 − 192v2 + 64v3 − 48v4 ≤ 240.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AzT4oBgHgl3EQf9P5O/content/2301.01916v1.pdf'}
+page_content='  (v) For v = 0, w = 1, 0 < u < 2,  ϑ (u, 0, 1) = 15(4 − u2)2 ≤ 240.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AzT4oBgHgl3EQf9P5O/content/2301.01916v1.pdf'}
+page_content='  THE SHARP BOUND OF THE HANKEL DETERMINANT  5  (vi) For the edges: v = 1, w = 0, 0 < u < 2 or v = 1, w = 1, 0 < u < 2, we have  ϑ (u, 1, w) = (4 − u2)2(4 + 2u2) ≤ 64.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AzT4oBgHgl3EQf9P5O/content/2301.01916v1.pdf'}
+page_content='  (vii) For the edges: u = 2, v = 0, 0 < w < 1 or u = 2, v = 1, 0 < w < 1 or  u = 2, w = 0, 0 < v < 1 or c = 2, w = 1, 0 < v < 1 or v = 0, w = 0, 0 < u < 2,  we obtain  ϑ (2, v, w) = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AzT4oBgHgl3EQf9P5O/content/2301.01916v1.pdf'}
+page_content='  C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AzT4oBgHgl3EQf9P5O/content/2301.01916v1.pdf'}
+page_content=' Now, we consider the six faces of Ω.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AzT4oBgHgl3EQf9P5O/content/2301.01916v1.pdf'}
+page_content='  (i) On the face u = 2, from (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AzT4oBgHgl3EQf9P5O/content/2301.01916v1.pdf'}
+page_content='14), we obtain  ϑ (2, v, w) = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AzT4oBgHgl3EQf9P5O/content/2301.01916v1.pdf'}
+page_content='  (ii) On the face u = 0, v ∈ (0, 1) and w ∈ (0, 1) from (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AzT4oBgHgl3EQf9P5O/content/2301.01916v1.pdf'}
+page_content='14), we get  ϑ (0, v, w) = 288v − 224v3 + (240 − 288v − 192v2 + 288v3 − 48v4)w2  = 288v − 224v3 + 48(5 − v)(−1 + v)2(1 + v)w2  ≤ 288v − 224v3 + 48(5 − v)(−1 + v)2(1 + v)  = 240 − 192v2 + 64v3 − 48v4 ≤ 240.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AzT4oBgHgl3EQf9P5O/content/2301.01916v1.pdf'}
+page_content='  (iii) On the face v = 0 u ∈ (0, 2), w ∈ (0, 1), from (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AzT4oBgHgl3EQf9P5O/content/2301.01916v1.pdf'}
+page_content='14), we obtain  ϑ (u, 0, w) = 15(4 − u2)2w2 ≤ 15(4 − u2)2 ≤ 240.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AzT4oBgHgl3EQf9P5O/content/2301.01916v1.pdf'}
+page_content='  (iv) On the face v = 1, u ∈ (0, 2), w ∈ (0, 1), from (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AzT4oBgHgl3EQf9P5O/content/2301.01916v1.pdf'}
+page_content='14), we observe that the  function ϑ (u, 1, w) is independent of w, from B(vi), we have ϑ (u, 1, w) ≤ 240.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AzT4oBgHgl3EQf9P5O/content/2301.01916v1.pdf'}
+page_content='  (v) On the face w = 0, u ∈ (0, 2), v ∈ (0, 1), from (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AzT4oBgHgl3EQf9P5O/content/2301.01916v1.pdf'}
+page_content='14), we obtain  ϑ (u, v, 0) = (4 − u2)2  �3u2v2  4  + 3u2v3  2  + (4 − u2)v3 + 3u2v4  4  + 18v(1 − v2)  �  = (4 − u2)2  �  18v − 14v3 + u2  �3v2  4  + v3  2 + 3v4  4  ��  ≤ (4 − u2)2  �  12  �  3  7 + 2u2  �  ≤ 192  �  3  7, u ∈ (0, 2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AzT4oBgHgl3EQf9P5O/content/2301.01916v1.pdf'}
+page_content='  (vi) On the face w = 1, in (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AzT4oBgHgl3EQf9P5O/content/2301.01916v1.pdf'}
+page_content='14), we obtain  ϑ (u, v, 1) = (4 − u2)2  �3  4u2v2 + 3  2u2v3 + 3  4u2v4 + (4 − u2)v3  + 3uv(1 + v)(1 − v2) + 3(5 + v2)  �  1 − v2� �  := g3(u, v), with (u, v) ∈ R2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AzT4oBgHgl3EQf9P5O/content/2301.01916v1.pdf'}
+page_content='  Note that all real solutions (u,v) of the system of equation  ∂g3  ∂u = 3  2(−4 + u2)  �  8(−1 + v)v(1 + v)2 − 10u2(−1 + v)v(1 + v)2  +u3v2(3 + 2v + 3v2) − 4u(−10 + 9v2 − 2v3 + 3v4)  �  = 0  and  6  B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AzT4oBgHgl3EQf9P5O/content/2301.01916v1.pdf'}
+page_content=' RATH, K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AzT4oBgHgl3EQf9P5O/content/2301.01916v1.pdf'}
+page_content=' S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AzT4oBgHgl3EQf9P5O/content/2301.01916v1.pdf'}
+page_content=' KUMAR, D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AzT4oBgHgl3EQf9P5O/content/2301.01916v1.pdf'}
+page_content=' V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AzT4oBgHgl3EQf9P5O/content/2301.01916v1.pdf'}
+page_content=' KRISHNA  ∂g3  ∂v = 3  2(−4 + u2)2(−8v(2 − v + v2) + u2v(1 + v + 2v2)  + u(2 + 4v − 6v2 − 8v3)) = 0  by a numerical computation are the following  (0, 0), (−2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AzT4oBgHgl3EQf9P5O/content/2301.01916v1.pdf'}
+page_content='63625, −1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AzT4oBgHgl3EQf9P5O/content/2301.01916v1.pdf'}
+page_content='53087), (−1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AzT4oBgHgl3EQf9P5O/content/2301.01916v1.pdf'}
+page_content='0493, 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AzT4oBgHgl3EQf9P5O/content/2301.01916v1.pdf'}
+page_content='14045) and (±2, x), x ∈ R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AzT4oBgHgl3EQf9P5O/content/2301.01916v1.pdf'}
+page_content='  Therefore, g3 has no critical point in (0, 2) × (0, 1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AzT4oBgHgl3EQf9P5O/content/2301.01916v1.pdf'}
+page_content='  D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AzT4oBgHgl3EQf9P5O/content/2301.01916v1.pdf'}
+page_content=' Now, consider the interior portion of Ω i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AzT4oBgHgl3EQf9P5O/content/2301.01916v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AzT4oBgHgl3EQf9P5O/content/2301.01916v1.pdf'}
+page_content=' (0, 2) × (0, 1) × (0, 1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AzT4oBgHgl3EQf9P5O/content/2301.01916v1.pdf'}
+page_content='  Differentiating ϑ(u, v, w) partially with respect w, we obtain  ∂ϑ  ∂w = 1  2(4 − u2)2 �  60w2 + 3v2(u2 + 4uw − 16w2) + 12v(6 + uw − 6w2)  +3v4(u2 − 4uw − 4w2) + 2v3(−28 + u2 − 6uw + 36w2)  �  upon solving ∂ϑ  ∂w = 0, we get  w0 = −  uv(1 + v)  2(5 − v)(1 − v) /∈ (0, 1) for (u, v) ∈ (0, 2) × (0, 1)  Hence ϑ(u, v, w) has no critical point in the interior of Ω.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AzT4oBgHgl3EQf9P5O/content/2301.01916v1.pdf'}
+page_content='  In review of cases A, B, C and D, we obtained  (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AzT4oBgHgl3EQf9P5O/content/2301.01916v1.pdf'}
+page_content='15)  max  �  ϑ(u, v, w) : u ∈ [0, 2], v ∈ [0, 1], w ∈ [0, 1]  �  = 240.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AzT4oBgHgl3EQf9P5O/content/2301.01916v1.pdf'}
+page_content='  From expression (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AzT4oBgHgl3EQf9P5O/content/2301.01916v1.pdf'}
+page_content='13) and (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AzT4oBgHgl3EQf9P5O/content/2301.01916v1.pdf'}
+page_content='15), we obtain  (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AzT4oBgHgl3EQf9P5O/content/2301.01916v1.pdf'}
+page_content='16)  ���H3,1(f −1)  ��� ≤ 1  36.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AzT4oBgHgl3EQf9P5O/content/2301.01916v1.pdf'}
+page_content='  For p0 ∈ Sc, we obtain t2 = t3 = t5 = 0, t4 = 1/6, which follows the result.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AzT4oBgHgl3EQf9P5O/content/2301.01916v1.pdf'}
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+page_content='org/10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AzT4oBgHgl3EQf9P5O/content/2301.01916v1.pdf'}
+page_content='1007/s13398-020-00977-2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AzT4oBgHgl3EQf9P5O/content/2301.01916v1.pdf'}
+page_content='  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AzT4oBgHgl3EQf9P5O/content/2301.01916v1.pdf'}
+page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AzT4oBgHgl3EQf9P5O/content/2301.01916v1.pdf'}
+page_content='3Department of Mathematics, GITAM School of Science, GITAM (Deemed to be  University), Visakhapatnam- 530 045, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AzT4oBgHgl3EQf9P5O/content/2301.01916v1.pdf'}
+page_content='P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AzT4oBgHgl3EQf9P5O/content/2301.01916v1.pdf'}
+page_content=', India  Email address: brath@gitam.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AzT4oBgHgl3EQf9P5O/content/2301.01916v1.pdf'}
+page_content='edu1∗,skarri9@gitam.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AzT4oBgHgl3EQf9P5O/content/2301.01916v1.pdf'}
+page_content='in2,vamsheekrishna1972@gmail.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AzT4oBgHgl3EQf9P5O/content/2301.01916v1.pdf'}
+page_content='com3' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AzT4oBgHgl3EQf9P5O/content/2301.01916v1.pdf'}

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+Excitations of Ising Strings on a Lattice
+Andreas Athenodoroua,b, Sergei Dubovskyc, Conghuan Luoc,
+and Michael Teperd
+a Computation-based Science and Technology Research Center,
+The Cyprus Institute, Cyprus
+b Dipartimento di Fisica, Universit´a di Pisa and INFN,
+Sezione di Pisa, Largo Pontecorvo 3, 56127 Pisa, Italy
+cCenter for Cosmology and Particle Physics,
+Department of Physics, New York University
+New York, NY, 10003, USA
+dRudolf Peierls Centre for Theoretical Physics,
+Clarendon Laboratory, University of Oxford,
+Parks Road, Oxford OX1 3PU, UK
+and
+All Souls College, University of Oxford,
+High Street, Oxford OX1 4AL, UK
+Abstract
+The 3d Ising model in the low temperature (ferromagnetic) phase describes dynam-
+ics of two-dimensional surfaces—domain walls between clusters of parallel spins. The
+Kramers–Wannier duality maps these surfaces into worldsheets of confining strings in
+the Wegner’s Z2 gauge theory. We study the excitation spectrum of long Ising strings by
+simulating the Z2 gauge theory on a lattice. We observe a strong mixing between string
+excitations and the lightest glueball state and do not find indications for light massive
+resonances on the string worldsheet.
+arXiv:2301.00034v1  [hep-lat]  30 Dec 2022
+
+Contents
+1
+Introduction
+1
+2
+Ising Model and Z2 Gauge Theory
+3
+3
+Effective String Theory
+5
+4
+Review of Lattice Techniques
+7
+4.1
+Lattice gauge theory and Monte-Carlo simulations . . . . . . . . . . . . . . . .
+7
+4.2
+Extracting spectra
+. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
+9
+4.3
+Constructing flux tube operators
+. . . . . . . . . . . . . . . . . . . . . . . . .
+11
+5
+Results
+15
+5.1
+The absolute ground state and the string tension
+. . . . . . . . . . . . . . . .
+16
+5.2
+Glueball States . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
+19
+5.3
+Excited states . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
+19
+5.4
+Finite volume corrections . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
+24
+5.5
+Including multitrace operators . . . . . . . . . . . . . . . . . . . . . . . . . . .
+32
+6
+Concluding Remarks
+34
+A Compilation of energy spectra
+36
+References
+43
+1
+Introduction
+The Ising model has been a fruitful area of research since its discovery in 1920’s [1]. The
+3d Ising universality class is realized in a number of physical systems such as 3d uni-axial
+magnets [2] and liquid-vapor critical points [3]. On the theoretical side, a lot of work has
+been devoted over the years to the physics of the 3d Ising model and to calculations of
+its observables, such as critical exponents. A celebrated example of a successful approach
+is provided by the ϵ-expansion [4]. Over the last decade, an impressive progress has been
+achieved by the numerical conformal bootstrap [5–7], which fixes critical exponents and OPE
+coefficients of the 3d Ising model to the greatest precision. Monte-Carlo simulations also give
+very precise results for the critical exponents of the 3d Ising model (see, e.g., [8]).
+Still this leaves one wondering whether a better analytical control is possible over the
+3d Ising model, especially given that the 2d Ising model is exactly solvable. A particularly
+intriguing set of ideas [9,10] is related to the possibility of rewriting the 3d Ising model as a
+theory of (super)strings. In this description the string worldsheet corresponds to a boundary
+between clusters of positive and negative spins.
+In the 2d Ising model the corresponding
+boundaries describe worldlines of free Majorana particles, which gives rise to an expectation
+1
+
+for fermionic excitations to be present on the string worldsheet in the 3d case. This idea has
+been realized explicitly in the lattice phase of the Ising model [11], however, the continuum
+description of the Ising strings is still missing. The corresponding string theory is expected
+to be strongly coupled, however see [12] for an interesting recent proposal towards a weakly
+coupled description.
+Given this state of affairs it is natural to explore the structure of the Ising strings exper-
+imentally, where by experiments we mean lattice Monte–Carlo simulations. For this purpose
+it is convenient to use the 3d version of the Kramers–Wannier duality, which maps the low
+energy ferromagnetic phase of the 3d Ising model into a confining phase of the Z2 lattice
+gauge theory [13]. Under this duality, Ising domain walls are mapped into worldsheets of Z2
+confining strings. To gain insight into the worldsheet dynamics it is natural to focus on the
+so-called long strings (or torelons). These are strings wrapped around one of the compact
+spatial dimensions. The ground state energy and the first few lowest-lying states of Ising
+strings in the long string sector have been previously studied in [14–16].
+In this work, we aim to extend these results with a more precise spectrum calculation
+and to determine energies of a larger number of excited states. Excitations of closed flux
+tubes wrapped around one of the spatial dimensions are characterized by their longitudinal
+momentum q along the flux tube. In addition, one may also define two parity transformations.
+The longitudinal parity Pl corresponds to a reflection along the string and maps q to −q. The
+transverse parity Pt corresponds to a reflection in the transverse direction. The main goal of
+our study is to check whether Ising strings carry massive resonant states on their worldsheet.
+Our initial results seem to indicate the presence of a massive resonance in the parity (++)
+sector (at q = 0). The same state is also present at the lowest non-vanishing q1. However, a
+careful analysis shows that this state is a bulk glueball rather than a new worldsheet state.
+Similar string spectrum computations were previously performed in the 3d U(1) gauge
+theory [17] and in the 3d and 4d SU(N) Yang-Mills theories [18–21]. In these studies, massive
+resonances are observed in some cases, such as for the fundamental 4d SU(N) confining string
+and confining strings in higher representations. Quite surprisingly though, fundamental con-
+fining strings in 3d SU(N) gluodynamics don’t show any sign of additional massive resonant
+modes on the string worldsheet.
+We see that Ising strings are in some sense in between these two options. On one side,
+we observe a well-pronounced resonant state in the spectrum of torelon excitations. On the
+other hand, this is not a new state, but rather a bulk glueball. This strong mixing between
+torelon excitations and glueballs is possible due to the absence of large N suppression in the
+Ising case.
+The rest of the paper is organized as follows. In section 2, we review properties of the
+3d Ising model and its duality to the Z2 lattice gauge theory. In section 3 we review the
+basics of the effective string theory, which provides a good approximation for the lowest-lying
+spectrum. In section 4 we summarize the basics of the lattice gauge theory and of the Monte-
+Carlo simulations. We describe the algorithm for computing the closed flux tube spectrum,
+and discuss how we reduce the systematic and statistical errors and improve the projection
+1Recall that the values of q are quantized as a result of a compactification on a circle.
+2
+
+onto low-lying states. In section 5 we present our results for some of the basic parameters
+such as the string tension and the lightest glueball mass. We present and analyze the closed
+flux tube spectra in 3d Z2 gauge theory for a wide range of string lengths. We start with the
+absolute ground state and continue onto excited states in different sectors. In particular, we
+identify a massive resonance state that is not described by the Nambu-Goto theory. Then
+we describe the checks which we performed, which indicate that the observed state is not in
+fact a novel worldsheet state but rather a scattering state of a long string with an additional
+unbound glueball. In section 6, we present our conclusions and discuss future directions.
+2
+Ising Model and Z2 Gauge Theory
+The 3d Ising model is one of the simplest spin models of (anti-)ferromagnetism. Its partition
+function is given by
+Z =
+�
+si
+e
+−H(si)
+T
+,
+(1)
+where the Ising Hamiltonian is given by
+H(si) = −J
+�
+⟨i,j⟩
+sisj − h
+�
+i
+si .
+(2)
+Here the first sum runs over all neighboring pairs of spins si = ±1 on a cubic lattice. In the
+present paper we are interested in the Ising model with a vanishing external magnetic field
+h = 0 .
+Then the theory enjoys a global Z2 symmetry, which flips signs of all spins. Positive val-
+ues of the coupling constant J correspond to ferromagnetism and negative ones to anti-
+ferromagnetism. Indeed, for positive J the Hamiltonian is smaller for spins pointing in the
+same direction making it energetically favorable for spins to be aligned. On the other hand,
+thermal fluctuations tend to randomize the spins. Which effect wins depends on the temper-
+ature, so the model exhibits a (second order) phase transition at a critical temperature Tc.
+As a consequence of the bipartite property of the square lattice the ferromagnetic and anti-
+ferromagnetic models are equivalent at h = 0. Namely, they can be mapped into each other
+by taking J → −J and flipping half of the spins, which correspond to one of the sublattices.
+In what follows we assume
+J > 0 .
+At a critical temperature T = Tc the spins develop long range correlations which are described
+by a conformal field theory. At temperatures below the critical one the global Z2 symmetry
+is spontaneously broken and a typical spin configuration describes clusters of positive and
+negative spins separated by domain walls of positive tension. In the vicinity of the critical
+temperature,
+T ≲ Tc
+3
+
+this phase is described by a continuous gapped Ising field theory. As reviewed in the intro-
+duction, it is a longstanding question whether it is possible to rewrite the Ising dynamics as a
+tractable continuum string theory, where the string worldsheet describes the dynamics of the
+domain walls. Our goal here is to study the structure of the Ising strings through the lattice
+Monte-Carlo simulation.
+To study the string dynamics it is instructive to map the Ising model into a Z2 gauge theory.
+This map has been constructed by Wegner [13] and can be considered as a generalization of
+the Kramers–Wannier duality of the 2d Ising model (see, e.g., [22] for a review). Unlike in the
+2d Ising model which is self-dual, the duality maps the 3d Ising model into a different theory
+defined by the following partition function
+Zgauge(β) =
+�
+{σl=±1}
+exp
+�
+β
+�
+□
+σ□
+�
+.
+(3)
+Here σl variables define a Z2 gauge connection which lives on the links of the dual lattice. The
+coupling constant β of the dual theory is related to the Ising model parameters via
+β = −1
+2 log tanh J
+T .
+(4)
+This Abelian gauge theory exhibits a number of properties characteristic of the non-Abelian
+SU(N) Yang–Mills theory.
+First, it enjoys a global 1-form Z2 center symmetry (see [23]
+for a modern introduction). Similarly to the SU(N) case, upon compactification on a circle
+the Z2 center symmetry is realized by (pseudo)gauge transformations with twisted boundary
+conditions. A Polyakov loop operator, defined as a Wilson loop wound around the circle,
+carries a negative Z2 charge.
+As a result, in the phase with unbroken center symmetry
+a sector with a Polyakov loop insertion is orthogonal to a trivial sector with no operators
+wound around the circle. Analogously to the SU(N) case we will refer to the states created
+by topologically trivial operators as glueballs. Deformed Polyakov loops acting on a vacuum
+produce “long” flux tube states, which are the main target of our study.
+The phase with unbroken center symmetry, which describes the confined phase of the Z2
+gauge theory, is realized at [24]
+β < βc ≈ 0.7614133(22) ,
+where the critical value β = βc corresponds to the conformal Ising point. In addition, Ising
+strings exhibit a roughening transition at [25]
+β = βr = 0.47542(1) ,
+so we are interested in the range βr < β < βc, where the string dynamics is described by a
+continuum theory in the scaling limit β → βc.
+The deconfining phase transition at β = βc needs to be distinguished from the one that
+happens when the circumference R of the spatial circle gets sufficiently small, namely at [14]
+R = Rc ≈ 0.82ℓs ,
+(5)
+4
+
+where ℓ−2
+s
+is the tension of a confining string. The latter transition corresponds to the finite
+temperature deconfining phase transition of the Z2 gauge theory understood as a (2 + 1)-
+dimensional quantum field theory. The parameter β is a coupling constant of this theory,
+which also has an interpretation as the inverse temperature, if one understands the Z2 gauge
+theory as a 3-dimensional classical statistical model. The Polyakov loop plays a role of the
+order parameter for both phase transitions.
+In principle, both Ising and Z2 descriptions can be used for Monte-Carlo studies of Ising
+strings (see, e.g., [14–16,26] for some previous work). In the Ising description this is achieved
+by introducing “interfaces”, i.e., by flipping the sign of the coupling J on the links which
+intersect the string worldsheet. To study the spectrum of string excitations, which is our main
+goal here, the gauge theory description appears more convenient. Indeed, in this description
+excited strings states are created by deformed Polyakov loops. As reviewed in section 4 this
+makes it straightforward to produce a large basis of excited states by changing the shape of
+the Polyakov loop. Furthermore, a precision mass determination requires a good overlap of
+the operator basis with the low lying string states. The gauge theory formulation allows this
+to be achieved by the well-developed techniques of blocking and smearing.
+For future reference, note that in addition to the string tension ℓ−2
+s , the Z2 gauge theory
+in the confining phase has another characteristic energy scale—the inverse correlation length
+ξ−1, which is set by the lightest glueball mass. Given that the parity invariant Ising model
+has a single relevant deformation, in the scaling limit the ratio of the two scales is universal.
+Its numerical value is [27]
+ξ2
+ℓ2
+s
+≈ 0.1056(19) .
+(6)
+3
+Effective String Theory
+In the absence of additional symmetries confining strings are not expected to carry any mass-
+less states on the worldsheet apart from the (D − 2) gapless translational Goldstone bosons
+describing transverse oscillations of a string. Here D is the total number of space-time dimen-
+sions. In particular, one expects to find a single massless mode on the worldsheet of D = 3
+Ising strings. Then the spectrum of low lying long string excitations is strongly constrained by
+the non-linearly realized target space Poincar´e symmetry and can be calculated using the ef-
+fective string theory (see, e.g., [28,29] for a review). Effective string theory provides a natural
+reference point to be compared with the actual string spectrum, so let us brie��y summarize
+properties of the effective string spectrum.
+The most straightforward approach for calculating the effective string theory predictions
+is based on the perturbative expansion which uses the ratio ℓs/R as a small parameter. As
+a consequence of the non-linearly realized Poincar´e symmetry all terms in this expansion up
+to (and including) O(1/R5) are universal. This means that those terms are insensitive to the
+microscopic theory as soon as no additional massless degrees of freedom are present on the
+worldsheet. This universality provides a powerful self-consistency check for lattice results. On
+the other hand it makes it quite challenging to probe the underlying microscopic theory by
+5
+
+high precision measurements of the string ground state for which the ℓs/R expansion has good
+convergence properties.
+Furthermore, the ℓs/R expansion exhibits poor convergence for excited string states. An
+efficient technique to calculate the effective string theory predictions for these states is based
+on the Thermodynamic Bethe Ansatz [30,31], which can also be reformulated as an undressing
+method based on the T ¯T deformation [32]. In this approach one calculates perturbatively the
+worldsheet S-matrix, and then makes use of a non-perturbative relation between the S-matrix
+and the finite volume spectrum to predict the latter. This technique is a close cousin of the
+familiar L¨uscher method [33] combined with the TBA method [34] for calculating the leading
+order winding corrections, which is possible due to an approximate integrability of the effective
+string theory. The leading order TBA string spectrum is given by
+EGGRT(Nl, Nr) =
+�
+4π2(Nl − Nr)2
+R2
++ R2
+ℓ4
+s
++ 4π
+ℓ2
+s
+�
+Nl + Nr − D − 2
+12
+�
+,
+(7)
+which is nothing but the Goddard–Goldstone–Rebbi–Thorne (GGRT) spectrum [35] of a
+bosonic string in a winding sector. Here Nl and Nr are non-negative integers called levels,
+which count the total left- and right-moving momenta along the string. The total longitudinal
+momentum is given by
+p = 2π(Nl − Nr)
+R
+.
+(8)
+In what follows it will be instructive to compare the Ising string spectrum with the GGRT
+one.
+Note that at D = 26 the GGRT spectum (7) coincides with the exact spectrum of
+critical bosonic strings. At D ̸= 3, 26 this spectrum is not compatible with the D-dimensional
+Poincar´e symmetry and should be considered as a leading order approximation only. The
+D = 3 case is somewhat special, and an integrable theory of a single massless boson with the
+spectrum given by (7) appears to be a consistent candidate for the worldsheet theory of a long
+D = 3 string. Motivated by the lattice data, the confining string of D = 3 Yang–Mills theory
+was conjectured to describe a single massless bosons, however, the corresponding spectrum
+deviates from the D = 3 GGRT formula. As we will see, for the Ising string the deviations
+are even more pronounced.
+The GGRT states are completely characterized by the occupation numbers nl(k), nr(k),
+where k is a positive integer labeling longitudinal momenta.
+These string excitations are
+generated by creation operators ak and a−k2.
+We will denote the corresponding state as
+|nl(k), nr(k)⟩, which is a shorthand notation of |nl(k), nr(k); k = 1, 2, . . . ⟩. Given such a state
+its levels can be computed as
+Nl =
+�
+k
+nl(k)k,
+Nr =
+�
+k
+nr(k)k .
+(9)
+In what follows we will refer to effective string excitations as phonons.
+For instance, the
+N = ˜N = 2 GGRT level corresponds to two degenerate states.
+One of these states is a
+2For convenience we omit the †. Because the annihilation operators will not appear in this paper, it should
+cause no confusion.
+6
+
+two-phonon excitation with
+nl(2) = nr(2) = 1 ,
+and another a four-phonon excitation with
+nl(1) = nr(1) = 2 ,
+where in both cases all other phonon occupation numbers vanish.
+As discussed in the Introduction, the long string spectrum is invariant under longitudinal
+and transverse parity transformations Pl and Pt. It is straightforward to determine the cor-
+responding transformation properties of the GGRT states. Namely, as far as the transverse
+parity is concerned, its action depends only on the total number of excitations and all GGRT
+state are eigenvalues of Pt,
+Pt|nl(k), nr(k)⟩ = (−1)
+�
+k(nl(k)+nr(k))|nl(k), nr(k)⟩ .
+(10)
+On the other hand, the longitudinal parity acts by exchanging the left- and right-moving
+excitations,
+Pl|nl(k), nr(k)⟩ = |nr(k), nl(k)⟩ .
+(11)
+Finally, note that in our discussion of the GGRT spectrum we implicitly set the total
+transverse momentum pt to zero. By restoring the pt dependence we obtain the full set of the
+GGRT states |nl(k), nr(k), pt⟩, with the energies given by the conventional relativistic formula,
+E(pt) =
+�
+p2
+t + E(0)2 .
+For convenience in Table 1 we present the states created by phonon creation operators in
+different sectors with q = 0, 1 and Nl + Nr ≤ 6. We will discuss more about the quantum
+numbers that define the sectors in section 4.3.
+4
+Review of Lattice Techniques
+4.1
+Lattice gauge theory and Monte-Carlo simulations
+A general lattice gauge theory (LGT) is described by a set of fields associated with the links
+of a lattice. Lattice links may be labeled by a pair (n, µ), where n labels the lattice site, and
+µ is a direction. Each lattice link is then mapped to an element Uµ(n) of the gauge group.
+For Z2 gauge theory these elements are simply ±1. For a cubic lattice the action is given by
+S = β
+�
+plaq
+{1 − Re(Tr Uplaq)} ,
+(12)
+where the sum is over elementary squares (“plaquettes”) of the lattice which may be labeled
+as (n, µ, ν) and
+Uplaq(n, µ, ν) = Uµ(n) · Uν(n + ˆµ) · U †
+µ(n + ˆν) · U †
+ν(n) ,
+7
+
+q = 0
+Nl, Nr
+Pt, Pr
+GGRT States
+Nl = Nr = 0
+++
+|0⟩
+Nl = Nr = 1
+++
+a1a−1|0⟩
+Nl = Nr = 2
+++
+a2a−2|0⟩
+++
+a1a1a−1a−1|0⟩
+−+
+(a2a−1a−1 + a1a1a−2)|0⟩
+−−
+(a2a−1a−1 − a1a1a−2)|0⟩
+Nl = Nr = 3
+++
+a3a−3|0⟩
+++
+a2a1a−2a−1|0⟩
+++
+a1a1a1a−1a−1a−1|0⟩
+++
+(a1a1a1a−3 + a3a−1a−1a−1)|0⟩
++−
+(a1a1a1a−3 − a3a−1a−1a−1)|0⟩
+−+
+(a3a−2a−1 + a2a1a−3)|0⟩
+−−
+(a3a−2a−1 − a2a1a−3)|0⟩
+−+
+(a2a1a−1a−1a−1 + a1a1a1a−2a−1)|0⟩
+−−
+(a2a1a−1a−1a−1 − a1a1a1a−2a−1)|0⟩
+q = 1
+Nl, Nr
+Pt
+GGRT States
+Nl = 1, Nr = 0
+−
+a1|0⟩
+Nl = 2, Nr = 1
++
+a2a−1|0⟩
+−
+a1a1a−1|0⟩
+Nl = 3, Nr = 2
++
+a3a−2|0⟩
++
+a2a1a−1a−1|0⟩
++
+a1a1a1a−2|0⟩
+−
+a3a−1a−1|0⟩
+−
+a2a1a−2|0⟩
+−
+a1a1a1a−1a−1|0⟩
+Table 1: Table with the states of the lowest GGRT levels with q = 0, 1 and Nl +Nr ≤ 6.
+8
+
+is an ordered product of gauge fields around a plaquette. The action (12) is gauge invariant
+and can be used to generate Monte-Carlo simulations.
+Periodic lattices are used in this
+work. In principle, one can generate millions of configurations using Markov Chain Monte-
+Carlo algorithms. After achieving thermalization, we compute statistical quantities through
+importance sampling.
+Different algorithms may have different thermalization speeds and
+different step sizes between configurations. In this paper we only use the Metropolis algorithm.
+For each lattice system, we created 200000 configurations to perform measurements, with 25
+sweeps between two measurements in order to reduce auto-correlation.
+Statistical quantities calculated in this work are correlation functions
+⟨φi(U)φj(U) · · · ⟩ =
+� �
+dUφi(U)φj(U) · · · e−S ,
+(13)
+of gauge invariant operators φi(U)’s. Two-point correlators calculated at different times can
+be used to extract the spectrum of different physical states such as glueballs and flux tubes.
+The corresponding procedure is further discussed in section 4.2.
+The lattice spacing a has units of length, but in numerical simulations we only deal with
+numbers, so we have to choose units where everything is dimensionless. A common choice is to
+use lattice units, which sets a = 1. This choice is implicitly assumed in the action expression
+(12). This choice is convenient during the simulations, but the cost is that the continuum limit
+becomes obscure. So it is also common to express physical observables using the units defined
+by a certain characteristic energy scale of interest. In this work we are mostly interested in
+confining strings, so we will use string units which set the string tension to one, ℓs = 1.
+Independently of the units, the continuum limit is achieved when
+a2
+ℓ2
+s
+→ 0 .
+(14)
+Of course, in practice this is impossible to achieve on a finite lattice. At the fixed lattice size
+the quality of the continuum limit is controlled by the difference between the Z2 coupling
+constant β and its critical value βc = 0.7614133(22). In order to stay in the confined phase we
+need to keep β < βc. Note that we cannot take the difference β − βc too small, because the
+string width ℓs then becomes larger than an overall size of the lattice, making it impossible
+to observe confinement.
+4.2
+Extracting spectra
+In this work we use the framework of [18,20,36] to measure the spectrum. Namely we construct
+a set of operators φi in a sector characterized by certain quantum numbers and acting on
+constant time slices3. Then a two-point correlator of two operators separated by nt lattice
+units in the time direction, which corresponds to the physical time t = ant, can be written in
+the following form
+Cij(t) = ⟨φ†
+i(t)φj(0)⟩ =
+�
+k
+⟨v|φ†
+ie−Ht|k⟩⟨k|φj|v⟩ =
+�
+k
+cikc∗
+kje−Ekt,
+(15)
+3Of course, we work on an Euclidean lattice, so a choice of the “time” direction is a matter of convention.
+9
+
+where the sum goes over a complete set |k⟩ of energy eigenstates with the chosen quantum
+numbers, |v⟩ is the absolute vacuum state and cik’s are the overlap coefficients
+cik = ⟨v|φ†
+i|k⟩ .
+As the time separation increases, higher energy contributions decay faster and only lowest
+energy states survive. It can be shown [37] that at large times the eigenvalues λa(t) of the
+matrix C−1(0)C(t) are given by the spectrum,
+λa(t) ≈ e−tEa,
+t → ∞ ,
+(16)
+if the basis of operators is large enough. To determine the energies in practice one first con-
+structs the approximate eigenstates Φi by diagonalizing the correlation matrix C−1(0)C(t = a)
+at early times, and then extracts the corresponding energy eigenvalues from the exponential
+falloff of the diagonal correlation functions ⟨Φ†
+i(t)Φi(0)⟩. To illustrate this procedure, let us
+consider the simplest case of a single operator, which allows to determine the ground state
+energy in the corresponding sector. In this case the diagonalization is trivial, so one simply
+studies the correlator
+⟨φ†(t)φ(0)⟩ =
+�
+n
+|⟨v|φ|n⟩|2e−Ent →
+t→∞ |⟨v|φ|0⟩|2e−E0t.
+(17)
+To analyze its behavior it is convenient to define an effective mass
+ameff(t) = − ln
+�
+⟨φ†(t)φ(0)⟩
+⟨φ†(t − a)φ(0)⟩
+�
+.
+(18)
+In the limit of an infinite statistics it decreases monotonically over time and asymptotes to
+the actual ground state energy in the φ sector,
+ameff(t) →
+t→∞ aE0.
+(19)
+In practice one plots the effective mass as a function of time and extracts E0 from the position
+of a plateau, which is followed by statistical fluctuations. For the ground state, the effective
+mass sets an upper bound on the actual energy and it is possible to observe the plateau up
+to rather late times.
+A general strategy for measuring energies of excited states is similar, but the practicalities
+become more and more challenging for highly excited states. Indeed, statistical noise in the
+measured effective mass is an unavoidable feature of the Monte-Carlo simulations using the
+importance sampling to compute correlators. The amplitude of the noise stays constant in
+time, while correlators exhibit an exponential decay.
+Inevitably, at large enough time tn
+statistical noise becomes larger than the signal and the effective mass needs to be measured
+before this happens. Correlators corresponding to heavier excited states decay faster, so that
+the critical time tn is shorter for them.
+Clearly, this implies that one needs to achieve a maximal possible overlap of the approxi-
+mate eigenstates Φi with the true energy eigenstates, so that the plateau can be measured as
+10
+
+early as possible. On the other hand, given that we perform a diagonalization in an artificially
+truncated finite dimensional Hilbert space, every approximate eigenstate necessarily has an
+admixture of heavier states which needs to decay before the plateau can be observed. This
+problem becomes more and more severe for highly excited states.
+To overcome this problem one needs to maximize the projection of an approximate eigen-
+state on the true energy eigenstates. This projection can be estimated by the gap between
+the value of the effective mass at t = a and the plateau. Typically, for us this projection
+drops below ∼ 0.5 around level Nl, Nr = 3, so we do not expect the corresponding energy
+determinations to be reliable.
+There are several ways to improve a quality of the plateau. First, one may try to minimize
+the measured energies in lattice units. This can be achieved by choosing the values of the
+parameters such that the string tension is smaller in the lattice units. In the Ising model this
+can be achieved by picking the value of β close to the critical point. However, other issues arise
+as one approaches the critical point. First, as one does this, one needs to take a larger lattice
+to model a system of the same physical size (i.e., as measured in string units). Given that we
+work on a three-dimensional lattice, the simulation time grows as a cube of the lattice size.
+Also, close to the critical point, correlations between gauge field configurations created by the
+Metropolis algorithm become higher. To overcome this one needs to increase the sampling
+interval, which also results in a longer simulation time. All in all, a limited computing power
+prevents one from approaching the critical point too closely.
+The second way to reduce statistical errors is by creating a larger size of samples. This is
+also limited by the computing resource.
+Finally, one can improve the quality of the operators, so that the overlaps of the approxi-
+mate eigenstates to the exact ones are closer to unity. This can be achieved both by starting
+with a larger set of operators, and also by suppressing the overlap of the operators with the
+highly energetic microscopic states using blocking and smearing techniques. We will discuss
+this more in section 4.3.
+4.3
+Constructing flux tube operators
+In this paper we work in the confining phase of the Z2 gauge theory. Equivalently, this is
+the phase with an unbroken center symmetry. Recall that given a gauge theory compacti-
+fied on a circle, the center symmetry may be defined4 by making use of the “twisted gauge
+transformation” generated by gauge functions satisfying
+g(R) = Λg(0) ,
+(20)
+where Λ is a center element of the gauge group. The Yang-Mills action functional is invariant
+under such a transformation. However, given that the gauge function (20) is not periodic, this
+transformation defines a global (rather than a gauge) symmetry of the theory. On the other
+hand, any two transformations satisfying (20) with the same Λ can be related to each other by
+4A modern definition of the center symmetry as a 1-form symmetry does not require to consider a com-
+pactification [23]. A traditional and less general discussion presented here is enough for our purposes.
+11
+
+a conventional gauge transformations. Hence, after dividing out over the conventional gauge
+transformations, one obtains a global symmetry transformation which is isomorphic to the
+center subgroup of the gauge group. For SU(N) gauge theory it is the ZN center symmetry,
+and Λ = e
+2πik
+N . For the Z2 gauge theory the center symmetry is Z2 itself.
+This definition makes it clear that an arbitrary Wilson loop
+WC = Tr
+�
+P exp(i
+�
+C
+Aµ(x)dxµ)
+�
+,
+(21)
+corresponding to the contour C with a trivial winding along the chosen compact direction is
+neutral under the center symmetry. Indeed, such a loop necessarily crosses any transverse slice
+an equal number of times from both sides and all factors of Λ cancel out. On the other hand,
+a Polyakov loop is wound around the periodic dimension, so it crosses any transverse slice in
+one direction one time more than in the opposite direction. As a result, it is charged under
+the center symmetry transformation. This also shows that its vacuum expectation value(vev)
+plays a role of the order parameter for the center symmetry. In the confining phase Polyakov
+loops have zero vev, and a long string sector is generated by acting on the vacuum by (an
+arbitrarily deformed) Polyakov loop. Of course, in addition one may add also any number of
+topologically trivial Wilson loops creating additional glueball states. The center symmetry
+ensures that this sector does not mix with the topologically trivial one, which is generated by
+the glueball operators only.
+Before describing the set of operators which we used to probe long strings, let us describe
+conserved quantum numbers in these sector. First, there is a longitudinal momentum p along
+the flux tube. Flux tubes are wound around a circle of a circumference R, so the longitudinal
+momentum is quantized
+p = 2πq
+R ,
+with q being an integer. The ground state is translationally invariant, which corresponds to
+q = 0.
+In addition, there are two parity transformations Pt and Pl, which we already introduced
+in our discussion of the GGRT spectrum in section 3. It is straightforward to describe how
+they act on the gauge theory operators, without any reference to effective strings. Let us
+consider a long string winding around the x direction. Then the transverse parity is a mirror
+transformation acting on the transverse y direction,
+(x, y)
+Pt
+−→ (x, −y) .
+Similarly, the longitudinal parity Pl acts as a mirror transformation of the longitudinal x-
+direction,
+(x, y)
+Pl
+−→ (−x, y) .
+Note that in general the longitudinal parity does not commute with the longitudinal momen-
+tum,
+Pl p Pl = −p ,
+12
+
+Figure 1: Increasing the blocking level of a link by one.
+so that only q = 0 states may be simultaneous eigenstates of p and Pl.
+Finally, long string states may also carry a non-vanishing transverse momentum pt. It does
+not convey any useful information about the worldsheet dynamics and we will always set it
+to zero by averaging over transverse positions of all operators.
+Let us describe now the set of operators, which we use to probe the long string sector. The
+simplest operator charged under the center symmetry associated to the compact x direction
+is the straight Polyakov loop
+φP(x, t) =
+R/a
+�
+n=1
+Ux(x + na, y, t) .
+(22)
+where R = La is the string length. In principle, this operator can be used to measure the
+ground state energy of a long flux tube. However, its overlap with the ground state of the flux
+tube is quite poor. Indeed, the Polyakov loop (22) creates a string with a width of order the
+lattice spacing a. On the other hand, a physical string close to its ground state is expected
+to have width of order the characteristic string scale ℓs.
+The overlap can be improved by applying a combination of smearing and blocking proce-
+dures [38]. One starts with the usual link field, which corresponds to blocking level Nbl = 1.
+Then one replaces an original link with a sign of a weighted average over the link itself and
+two staples attached to it (see Fig. 1). In our simulations we chose the averaging weight to be
+0.75. Finally, one constructs a twice longer link by multiplying two consecutive smeared links.
+The result is what one calls a level 2 blocked link. To construct the links at Nbl-th blocking
+level one applies the same procedure using the blocking level Nbl − 1 links as an input.
+Using the blocked links we can now create a basis of Polyakov loop operators of different
+shapes. In Fig. 3 we present the shapes used in our simulation. Note that some of these
+operators look like creating a flux tube and an additional glueball rather than just a flux
+tube excitation. Equivalently, using the SU(N) language, they look like multi trace opera-
+tors. However, for the Z2 theory there is no sharp distinction between single trace and multi
+trace operators, because any operator can be formally presented in the single trace form by
+connecting different components by going back and forward along some path between them
+(see Fig. 2), given the Abelian nature.
+Finally, to obtain operators with a definite set of quantum numbers one performs averaging
+over the action of the corresponding symmetry transformation.
+For example, in order to
+13
+
+For an Abelian gauge group
+Figure 2: For an Abelian gauge group there is no sharp distinction between string
+excitations and additional glueballs.
+Figure 3: The set of operators used in our simulation.
+construct an operator with a definite longitudinal momentum p, one sums over all longitudinal
+translations with a phase
+φ(p) =
+L
+�
+k=1
+φ(x + ak)eipak .
+(23)
+In the same way one constrains pt = 0 by summing over all the translations in the transverse
+y direction without a phase.
+Similarly, one may obtain operators with definite value of transverse and longitudinal
+parities (Pt, Pl). For example, as we discussed, at p = 0 both parities can be be assigned,
+so we get four different sectors (++), (+−), (−+) and (−−). To construct the corresponding
+operators one starts with a Wilson line operator UC corresponding to a certain path C, and
+14
+
+defines the following eigenstate combination
+˜UC = (UC ± UPlC) ± (UPtC ± UPtPlC) .
+(24)
+Here the signs inside the brackets correspond to the eigenvalue of Pl, and the sign in the
+middle corresponds to the eigenvalue of Pt.
+5
+Results
+Let us now present results of our simulations. In this work we performed Z2 lattice gauge
+theory simulations at β = 0.756321. This value corresponds to the rough and confining phase.
+It is sufficiently close to the critical value βc = 0.7614133(22) [24], to allow for sufficiently long
+and clear plateaux in the effective mass. Namely, as follows from the results presented later,
+for this value of β the correlation length ξ (which is set by the inverse mass of the lightest
+glueball ξ = m−1
+G ) is equal to
+ξ = 4.631(8)a .
+Unless specified otherwise, the results presented are obtained on lattices of a size
+l⊥ = lt = 70a ,
+in the transverse and time directions, and the lattice size along the string is varied in the
+range
+R ∈ [20a, 80a] ,
+which corresponds to the range
+R ∈ [1.38ℓs, 5.53ℓs] ,
+in string units, where the string length is obtained by fitting the absolute ground state energy
+of the flux tube to the GGRT formula. Recall that the finite temperature deconfinement
+transition corresponds to R ∼ 0.82ℓs. In order to estimate finite volume corrections and for
+some other checks we also used lattices with other transverse sizes in the range from 55a to
+300a. These values of lattice parameters and the corresponding basic physical observables are
+summarized in Table 2.
+β
+βc
+R/a
+Rc/a
+a/ℓs
+amG
+0.756321
+0.7614133(22)
+[20,80]
+∼ 11.8
+0.0691(1)
+0.2159(4)
+Table 2: Basic parameters of our simulation: the value of the coupling and its critical
+value, the range of the string circumference and its critical value, the string tension and
+the lightest glueball mass.
+Let us now present results of simulations with these parameters. We start with the absolute
+flux tube ground state, and continue to excited states in different sectors. Comparing the result
+15
+
+to the GGRT spectrum we find that the most pronounced qualitative difference is the presence
+of an extra state in the parity (++) sector at q = 0. This state can naturally be interpreted
+as a massive scalar resonance on the string worldsheet. We identify the corresponding state
+also in the q = 1 sector. Later we present results of an additional dedicated analysis which
+indicates that this resonance is actually caused by the bulk glueball rather than by a genuine
+worldsheet state.
+5.1
+The absolute ground state and the string tension
+The flux tube ground state is translationally invariant, has q = 0, and belongs to the (++)
+sector. Understandably, of all the string states this one is the most straightforward to identify.
+As illustrated in Fig. 4, the corresponding effective mass exhibits a well pronounced plateau
+even for the longest string circumference R = 80a considered in our simulations, which allows
+for a high precision determination of the ground state energy as a function of R. very well
+In Fig. 5 we present the ground state energy as a function of circumference R. The solid
+line shows the GGRT ground state energy. These results are plotted in string units with the
+string length parameter ℓs determined by fitting the data to the GGRT ground state energy.
+For the ℓs extraction we used the data in the range R ∈ [25a, 80a], where the quality of the
+GGRT fit is the best. The resulting value of ℓs in lattice units is presented in Table 2. We
+observe that the GGRT approximation reproduces very well the ground state energy of the
+Ising string all the way down to R ∼ 1.4ℓs. On the other hand, the measured ground state
+energy significantly deviates from the GGRT formula at shorter values of R. In particular,
+the GGRT ground state energy vanishes at R ≈ 1.02ℓs, while the Ising ground state energy
+stays positive (and approximately linear) down to a smaller critical value given by (5).
+To quantify the agreement of the measured ground state energy with the GGRT approxi-
+mation, we also fitted the observed energies at the short string regime [1.4ℓs, 2.8ℓs] using the
+following ansatz
+E0(R) = EGGRT(R) + cγ
+ℓs
+�ℓs
+R
+�γ
+,
+(25)
+for different values of γ and using the string length ℓs and the coefficient cγ as the fitting
+parameters. To interpret the results it is instructive to compare the obtained values of cγ with
+the corresponding coefficients of the ℓs/R expansion of the GGRT ground state energy itself,
+E0(R) = R
+ℓ2
+s
+− π
+6
+1
+R − π2
+72
+ℓ2
+s
+R3 − π3
+432
+ℓ4
+s
+R5 + O(ℓ6
+s) ,
+(26)
+where we listed all the universal terms in the ℓs/R expansion. For γ = 1, the best fit value of
+c1 is negligible compared to the value of the corresponding term in (26)
+c1 ≈ 0.024(13) ≪ π
+6 ≈ 0.52 ,
+so we conclude that our results provide a quite precise determination of the first universal
+term in the ℓs/R expansion (also known as the L¨uscher term). On the other hand for γ = 3
+16
+
+0
+2
+4
+6
+8
+10
+12
+0.0
+0.1
+0.2
+0.3
+0.4
+0.5
+aE(t)
+t/a
+Figure 4: The effective mass computed as in formula (18) as a function of time for the
+absolute ground states at string circumference R/a = 20, 40, 60, 80, represented as blue,
+yellow, green and red dots. The horizontal solid lines are the resulting fitted values
+of the state’s energies. The shaded bands represent the corresponding 1σ uncertainty
+intervals.
+17
+
+0
+1
+2
+3
+4
+5
+0
+1
+2
+3
+4
+5
+Eℓs
+R/ℓs
+Figure 5: The absolute ground state energy at different string lengths in string units.
+The solid line is the GGRT approximation for the ground state energy.
+we obtain
+c3 ≈ 0.074(28) ≲ π2
+72 ≈ 0.14 ,
+so that our results are consistent with the 1/R3 universal term, but cannot be considered as
+a high precision test of the universality at this order.
+As an additional crosscheck of our simulation we also determined the mass of the lightest
+glueball mG. When expressed in string units it reads
+mG ≈ 3.124(10)ℓ−1
+s ,
+(27)
+which agrees well with earlier measurements (cf. (6)).
+It is instructive to take a look at the ground state energy for even shorter strings: R ≲ 1.2ℓs,
+as also shown in Figure 5. Here one observes a large deviation from the GGRT formula. Clearly
+in this regime the ℓs/R expansion does not converge, so that it cannot be used to measure
+the perturbative non-universal corrections to the GGRT formula. It is worth noting that
+these data do seem to extrapolate towards the deconfining point and exhibit scaling behavior,
+which indicates that it is a second order phase transition. According to the Svetitsky-Yaffe
+conjecture [39], this deconfining transition is described by the 2d Ising universality class, of
+which the scaling behavior is linear
+E0(R)
+R→Rc
+∝
+(R − Rc) .
+(28)
+From our measurements, it is plausible to be linear. But to really determine the exponent, we
+need results of higher precision and more data points. One difficulty around the critical point
+is that the ground state energy goes to zero, so that a larger lattice is needed to perform its
+accurate determination.
+18
+
+5.2
+Glueball States
+As a cross-check for our results, we also calculated the low-lying spectrum of Z2 glueballs
+in the 0+ sector, which is summarized in Table 3. Here we can observe the finite volume
+corrections for low-lying glueball states.
+For example, for the lightest glueball, the finite
+volume correction becomes observable for R ≤ 30a. One may also wonder whether we can
+observe the state corresponding to two parallel flux tubes, which also has the same quantum
+numbers. It has the mass of two ground state flux tubes. We do not observe such a state here,
+which indicates that the local operators we use for glueball states have poor overlap on these
+states. Comparing our results with that in [27], our measurements have higher precision, and
+they agree well. The largest deviation is found for the second excited state, for which our
+mass is somewhat lower, but still within a 2σ interval.
+(ly/a) × (lt/a)
+lx/a
+aE; 0+
+70 × 70
+25
+0.1978(28)
+0.2531(91)
+0.3519(75)*
+30
+0.2075(30)
+0.2992(87)
+0.3726(110)*
+35
+0.2163(16)
+0.3635(51)
+0.4528(117)*
+40
+0.2170(15)
+0.3804(81)
+0.5097(95)
+45
+0.2144(17)
+0.3896(54)
+0.5329(100)
+47
+0.2118(14)
+0.3865(96)
+0.5319(115)
+50
+0.2159(20)
+0.3742(62)
+0.5019(166)
+52
+0.2131(22)
+0.3920(52)
+0.5237(93)
+54
+0.2182(12)
+0.3899(89)
+0.5141(93)
+55
+0.2141(20)
+0.3990(40)
+0.5326(108)
+56
+0.2169(18)
+0.3953(44)
+0.5462(59)
+58
+0.2152(22)
+0.3849(66)
+0.4947(158)
+60
+0.2178(20)
+0.3998(52)
+0.5080(154)
+65
+0.2138(20)
+0.3906(64)
+0.5153(182)
+70
+0.2168(11)
+0.3984(61)
+0.5497(67)
+75
+0.2159(17)
+0.4025(44)
+0.5541(66)
+80
+0.2175(17)
+0.3886(73)
+0.5216(144)
+Fitted masses
+0.2159(4)
+0.3937(16)
+0.5359(27)
+Table 3: The spectrum of Z2 glueballs in the 0+ sector at β = 0.756321 for different
+lattice sizes.
+5.3
+Excited states
+Let us now present our results for the excited state’s energies of the Ising string. We start
+with zero momentum states, q = 0. As discussed before, these states split into four subsectors
+19
+
+with different transverse and longitudinal parities,
+(Pt, Pl) = (++), (+−), (−+), (−−) .
+(29)
+In Fig. 6 we presented the energy differences between the first three excited states in the
+(++) sector and the ground state energy. As we will see later, restricting to these three states
+somewhat oversimplifies the overall picture. Nevertheless, it provides a good strating point
+for interpreting our results. The numerical values of the corresponding energies (and also of
+higher excited states) can be found in Table 4 in the Appendix. In addition to two levels,
+which are naturally associated with the (1, 1) and (2, 2) GGRT states5, we observe on this
+plot an additional level, which is not associated with any of the GGRT states. Given that the
+energy gap between this exotic level and the absolute ground state is approximately constant
+over the large range of R, it is natural to associate this state with a massive (++) resonance
+on a string worldsheet. The resonance mass can be estimated by fitting the energy gap to a
+constant, which results in
+mℓs = 3.825(50) ,
+(30)
+where we performed the fit at the intermediate values of string circumference, R/ℓs ∈ [2.4, 4.2]
+to reduce possible effects related to level crossing and winding corrections. The latter can be
+incorporated by applying the TBA technique (cf. [31]); we will present results of this analysis
+in a separate publication.
+There are two subtleties worth mentioning here. First, the resonance exhibits two level
+crossings with the GGRT states in the range of R covered by our data. Namely, it crosses
+the (1, 1) level at R ∼ 2ℓs , and the (2, 2) level at R ∼ 5ℓs. In the GGRT spectrum the
+(2, 2) level corresponds to two degenerate states—a two-phonon and a four-phonon states. By
+inspecting Table 4 one indeed observes two nearly degenerate states close to the (2, 2) level at
+R ≲ 4ℓs. However, one of these states disappears as one approaches the second level crossing
+at R ≳ 4ℓs. The explanation for this is not clear at this point. As follows from the data
+presented in Table 4, the energy of the second (2, 2) GGRT state starts to increase away from
+the GGRT spectrum at around R ≳ 3.8ℓs. As we will see later the (++) resonance is actually
+a glueball state mixed with the flux tube. It is possible that these large deviations from the
+GGRT formula appear above the glueball threshold, due to interactions between the unbound
+glueball and the flux tube.
+The second subtlety, which is likely related to the first one, is that the energy gap (30) is
+larger than the mass of the lightest glueball (27) in the infinite volume theory. This implies
+that (30) is not a strictly localized worldsheet state, but rather a metastable bound state
+between a flux tube and a glueball. In particular, in addition to decaying into a two-phonon
+flux tube excitation it may also decay into a flux tube and a glueball state. Note that the
+Ising model does not have a parameter which would suppress mixing between genuine flux
+tube excitations and flux tube states with additional glueballs. This is different from the
+Yang–Mills case, where such a mixing is suppressed in the ’t Hooft large-N limit. As a result,
+one may doubt whether the state (30) is really due to intrinsic worldsheet dynamics. Perhaps,
+5In the following, for convenience we denote the GGRT levels of states in the format (Nl, Nr).
+20
+
+2
+3
+4
+5
+0
+1
+2
+3
+4
+5
+6
+∆Eℓs
+R/ℓs
+Figure 6: Energy differences with the ground state for q = 0 excited states in the (++)
+parity sector as a function of string circumference at different string lengths. Blue curves
+are the (1, 1) and (2, 2) GGRT levels. The red horizontal line is the fitted resonance mass.
+this state should be considered instead as an admixture of the flux tube and an unbound bulk
+glueball. On the other hand, our basis of operators was designed to have a good overlap with
+states localized in the vicinity of the flux tube, so a priori one could expect that it is not
+sensitive to the states with additional unbound glueballs.
+We performed several checks to clarify the proper interpretation of this state. First, if the
+exotic state (30) were due to an additional unbound glueball, then one would expect to find
+a state with similar properties also in the (−+) sector. Indeed, in infinite volume adding a
+glueball to a flux tube ground state leads to a continuum of states labeled by the asymptotic
+transverse momentum.
+In a finite volume this continuum turns into a “discretuum”.
+In
+the absence of interactions between the flux tube and the glueball this discretuum would
+correspond to the ground state (++) and a series of degenerate doublets with (++) and (−+)
+parities. However, the interaction with the flux tube breaks the degeneracy, so one obtains a
+series of alternating (++) and (−+) eigenstates.
+Furthermore, energies of all these states, possibly apart from the lowest one, have a rather
+strong dependence on the transverse size l⊥, due to the momentum quantization. This depen-
+dence may be used to distinguish between strongly bound flux tube excitations and unbound
+states from the discretuum.
+To probe these states, one may enlarge the set of operators in Fig. 3 by adding operators
+which are expected to have a good overlap with unbound flux tube/glueball states, to see
+21
+
+whether additional states indeed appear.
+We will describe the results of this analysis in
+section 5.5. As we will see there, our overall conclusion is that the state (30) should indeed
+be interpreted as a state with an additional unbound glueball.
+Let us turn now to excited states in other sectors. For the q = 0 (+−) sector the effective
+string theory predicts that the lowest energy state appears at the (3, 3) GGRT level and
+corresponds to a Pl odd linear combination of nl(3) = 1, nr(1) = 3 and nr(3) = 1, nl(1) = 3
+states. Indeed, our analysis does not reveal any low lying states in this sector. We provide the
+measured energies of the lightest (+−) state in Table 5 in the Appendix. At R/ℓs ≳ 4 these
+energies are in between the (3, 3) and (4, 4) GGRT levels and become significantly heavier at
+shorter R. Given how heavy these states are we expect that their energy determinations are
+likely to be subject to significant systematic uncertainties. The only robust conclusion one
+can draw from these results at the moment is that no anomalous light states appear in this
+sector.
+Let us discuss now Pt odd states, which are the states with an odd number of phonons.
+For both (−+) and (−−) sectors the lowest GGRT states appear at the (2, 2) level, and they
+correspond to even and odd linear combinations of nl(2) = 1, nr(1) = 2 and nr(2) = 1,
+nl(1) = 2 states. We plot the measured energies of the lightest states in these sectors in
+Fig. 7, and present the numerical values of these energies and those of the heavier states
+in Tables 6, 7 in the Appendix. We observe that at R ≳ 4ℓs these two states are nearly
+degenerate, as expected for the GGRT spectrum. In this range of R their energies are quite
+close to the expected (2, 2) GGRT value, with a minor systematic disagreement. It is most
+likely due to an overestimate of these rather heavy energies due to an admixture of higher
+excited states.
+At R ≲ 4ℓs the two states are split, and this splitting becomes very large at R ≲ 3ℓs,
+mostly due to a rather dramatic increase in the energy of the (−−) state. Interestingly, the
+energy of the lightest (+−) states discussed earlier exhibits a similar feature in the same range
+of R. At the moment it is hard to tell what is the cause of this effect. Note that, as discussed
+in a similar context in [32] for the SU(N) data from [19], the splitting between three-phonon
+(−+) and (−−) cannot be explained by a correction to the two-phonon phase shift. Instead, it
+is indicative of a strong inelastic multi-phonon scattering. Interestingly, this splitting appears
+to be much more dramatic in the Ising case as compared to the SU(N) flux tubes.
+Finally, let us discuss states with nonzero longitudinal momentum q = 1, which are plotted
+in Fig. 8 and tabulated in Tables 8, 9. The ground state in this sector, which is parity odd,
+agrees exceptionally well with the GGRT (1, 0) prediction. This is expected, given that the
+(1, 0) GGRT state corresponds to adding an essentially free (modulo winding corrections)
+phonon to the ground state of a flux tube. The first excited parity odd state also agrees very
+well with the (2, 1) GGRT level.
+To interpret the two lowest energy parity even q = 1 states it is instructive to compare
+their energies to the (2, 1) GGRT level and also to the free approximation for the energy of
+the boosted resonance state,
+∆E =
+�
+m2 + p2 ,
+(31)
+where p = 2π
+R . We observe from Fig. 8 that the two low lying states naturally correspond to
+22
+
+2
+3
+4
+5
+4
+6
+8
+10
+∆Eℓs
+R/ℓs
+Figure 7: Energy differences with the ground state for q = 0 excited states in the (−+)
+(blue dots) and (−−) (brown dots) parity sectors as a function of string circumference
+at different string lengths. The blue curve is the energy of the (2, 2) GGRT level.
+a level crossing between the (2, 1) GGRT level and a boosted resonance state.
+To illustrate how statistical fluctuations influence our results, especially for higher level
+states, it is instructive to take a look at the effective mass plateaux behaviour for different
+states and at the corresponding effective mass fits. In Fig. 4 we plotted the effective mass as
+a function of time separation for the absolute ground states at different string lengths. As
+expected, we see that as the string length increases, which corresponds to the heavier ground
+state energy, statistical fluctuations become larger and the uncertainty in the effective mass
+determination grows. A generic behavior observed for each of the states is that the effective
+mass exhibits a drop at early times and then stabilizes on a plateau. The rate of the initial
+drop characterizes the quality of the overlap of our operator basis onto the corresponding
+state. Statistical fluctuations increase at larger with time and dominate the measurement at
+late times.
+All these features are even more pronounced for excited states as illustrated in Fig. 9. Here
+we chose the string length such that the non-universal corrections to the GGRT spectrum is
+small, and at the same time the resonant state is also well pronounced. As compared to the
+ground state we observe that statistical fluctuations start to dominate the plateau at earlier
+times. At the energy of around 0.67a−1, which corresponds to the second excited state in the
+parity (−+) sector at R = 60a, this effect reaches the point when the position of the plateau
+is hard to determine. Also, because statistical fluctuations here dominate so early, they are
+23
+
+2
+3
+4
+5
+5
+6
+7
+8
+9
+10
+∆Eℓs
+R/ℓs
+Figure 8: Energy differences with the ground state for q = 1 excited states in the (+)
+(blue dots) and (−) (brown dots) parity sectors as a function of string circumference at
+different string lengths. Blue curves show energies of the (1, 0), (2, 1) and (3, 2) GGRT
+levels. A red curve shows an estimate for the resonance state using the resonance mass
+(30).
+likely to prevent us from observing the point of the plateau stabilization, leading to a possible
+overestimation of the energy. Consequently, a reliable spectrum calculation in this energy
+range requires a larger sample size.
+5.4
+Finite volume corrections
+Let us discuss the finite size dependence of the presented results. To be more precise, in our
+simulation we have a finite size lattice system with periodic boundary conditions: R × l⊥ × lt.
+The main goal of the simulation is to measure the dependence of string energy levels on the
+longitudinal size R. Instead, in this section we will discuss the sensitivity of the presented
+results to l⊥ and lt. Our goal is twofold. On the one hand the (in)sensitivity of the measured
+string energy levels to l⊥ and lt provide a consistency check for the extrapolation of the
+measured energy levels to infinite volume. On the other hand, as was already mentioned,
+the scattering states containing additional glueball(s) states are expected to exhibit a strong
+dependence on l⊥, which can be used to probe the nature of a massive resonance state observed
+in the (++) sector.
+In more detail, the spatial finite volume dependence of a single particle or string state
+24
+
+*
+*
+*
+*
+*
+*
+*
+*
+*
+*
+*
+*
+*
+*
+*
+*
+*
+*
+*
+*
+*
+*
+*
+*
+*
+*
+*
+0
+2
+4
+6
+8
+10
+12
+0.0
+0.2
+0.4
+0.6
+0.8
+aE(t)
+t/a
+Figure 9: The effective mass computed as in formula (18) as a function of time, for the
+first, second and third excited states in the q = 0 (++) sector and compactification
+length R = 40a, represented as blue, yellow, green dots, and for the ground state, first
+and second excited states in the q = 0 (−+) sector and compactification length R = 60a,
+represented as blue, yellow and green “∗”. The horizontal solid lines in dark colors are
+the fitted value of the mass of the corresponding states. The shaded bands in light colors
+represents ±1 standard deviations.
+25
+
+with zero momentum in the transverse direction is related to winding corrections associated
+to (virtual) particles propagating around the spatial circle. For massive states, which is always
+the case for us6, these corrections are of order O(e−caml⊥), where the constant c depends on
+the theory [40]. These corrections are exponentially suppressed, so as we take the transverse
+size to be moderately large, it will disappear very quickly.
+The story is similar for corrections associated to the finite size of the temporal circle. To
+partially account for these corrections we used exponents associated with both directions in
+time to fit the two-point correlators instead of a single exponential as written in (17). This
+still neglects all the time evolutions that wind around the time circle for more than one round,
+but these effects are further exponentially suppressed.
+Clearly, winding corrections are most prominent for the lightest states. In particular, l⊥
+and lt need to be sufficiently large for a high precision determination of the low lying string
+states at small R.
+For multiparticle scattering states there are larger finite volume corrections that go like
+O(1/(ml⊥)). These are associated with finite momenta of individual particles in a multiparticle
+state. In particular, the infinite volume energy spectrum of multiparticle states is continuous.
+Instead, in a finite spatial volume one expects to find a discretuum of states which becomes
+more and more dense as the lattice size increases.
+To probe the size of finite volume effects in our results we performed simulations at dif-
+ferent lattices and compare the corresponding energy spectra. We do not find a significant
+dependence of the measured flux tube spectrum on the temporal lattice size, as follows from
+the data summarized in Appendix A. These data describe low-lying flux tube spectra mea-
+sured on 40 × 55 × 55 and 40 × 55 × 70 lattices. The difference is well within error bars. So
+in what follows we fix lt = 70a, where the time windings can be safely ignored.
+Let us discuss now a set of plots illustrating how energies of low-lying states depend
+on the transverse size. We do not discuss states in the (+−) sector because their energy
+determinations are not very reliable due to large statistical uncertainties. In this section we
+fix the size of the longitudinal direction to R = 40a = 2.77ls.
+The transverse size dependence of the (++) states is illustrated in Fig. 10. Blue, yellow,
+green and red dots are natural to identify with the GGRT states. They match the correspond-
+ing GGRT energies fairly well, and do not exhibit strong finite volume dependence. This is
+also true for the resonance state, which is represented by brown dots. However, there is an
+extra state represented by purple dots, which exhibits a very pronounced volume dependence
+at smaller values of l⊥. As follows from our earlier discussion, this volume dependence suggests
+that this state belongs to a discretuum of scattering states describing a string with an addi-
+tional glueball with non-vanishing relative momentum. This suggests that also the resonance
+state should be zero relative momentum at the bottom of the string-glueball discretuum rather
+than a genuine string excitation. In the next section we present further evidence supporting
+this conclusion.
+6Note that what matters here is the mass of a string as a whole as it move in the transverse direction.
+This should not be confused with the mass of longitudinal string excitations, which is of course zero for the
+Goldstone modes.
+26
+
+0.05
+0.10
+0.15
+0.20
+0.25
+0
+2
+4
+6
+8
+10
+E/√σ
+1/l⊥
+√σ
+Figure 10: Energies in the q = 0 (++) sector at R = 40a = 2.76ls as a function of the
+inverse transverse size. Horizontal lines of different colors represent the GGRT spectrum
+starting with N = ˜N = 0. The brown dashed line represents the resonance mass.
+The transverse size dependence of the (−+) states is illustrated in Fig. 11. These states
+are quite a bit heavier than the lightest ones observed in the (++) sector and it is harder
+to interpret what happens here. It looks natural to associate blue and yellow dots with the
+proper string excitations. Their agreement with the GGRT predictions is not so good, and
+the lightest (blue) state appears to exhibit some volume dependence at the small values of
+the transverse size l⊥
+√σ ≲ 4.5. In any case, one also observes two additional states (green
+and red) which exhibit a very pronounced volume dependence. As in the (++) case this is
+suggestive of the scattering states interpretation.
+The transverse size dependence of (−−) states is presented in Fig. 12.
+There are no
+recognizable scattering states among the low-lying states with Eℓs ≲ 10. This is expected.
+Indeed to construct a Pl = − scattering states one can either take a Pl = − flux tube or
+glueball state, or consider a state where both flux tube and a glueball carry a non-vanishing
+longitudinal momentum. In all cases the resulting state is expected to be quite heavy.
+For completeness we also presented the transverse volume dependence of q = 1 states in
+Figs. 13, 14. The corresponding scattering states can be obtained by boosting a glueball in
+the q = 0 states, so these states can be used as consistency check. We expect to find scattering
+states for both Pt = + and Pt = − sectors among q = 1 states. These states with strong finite
+27
+
+0.05
+0.10
+0.15
+0.20
+0.25
+6
+7
+8
+9
+10
+E/√σ
+1/l⊥
+√σ
+Figure 11: Energies in the q = 0 (−+) sector at R = 40a = 2.76ls as a function of the
+inverse transverse size. Horizontal lines of different colors represent the GGRT spectrum
+starting with N = ˜N = 2.
+volume dependence are indeed present and represented by purple dots in Fig. 13 and by red
+dots in Fig. 14. The green dots in Fig. 13 represent a resonance state, which can be plausibly
+reinterpreted as string-glueball discretuum with zero relative momentum.
+We conclude that for the coupling β = 0.756321, which we use, a lattice with lt = l⊥ = 70a
+is large enough to ignore finite size effects for the GGRT states at the current level of precision
+at values of R which is not too close to the deconfining value Rc = 0.82ℓs.
+We do see strong finite volume corrections associated both with lt and l⊥ dependence as
+we approach the deconfinement transition Rc = 0.82ℓs. A much larger lattice size is needed
+to perform accurate measurements in the vicinity of that point. Also, we see evidence for the
+existence of the flux tube-glueball scattering states at large transverse size for both values of
+the transverse parity Pt. This indicates that our set of operators have a sizable overlap with
+these states and calls for a more rigorous look on the nature of the massive state in the (++)
+sector. This will be the goal of the next section.
+28
+
+0.05
+0.10
+0.15
+0.20
+0.25
+8
+9
+10
+11
+12
+E/√σ
+1/l⊥
+√σ
+Figure 12: Energies in the q = 0 (−−) sector at R = 40a = 2.76ls as a function of inverse
+transverse size. Horizontal lines of different colors represent the GGRT spectrum starting
+with N = ˜N = 2.
+29
+
+0.05
+0.10
+0.15
+0.20
+0.25
+7
+8
+9
+10
+E/√σ
+1/l⊥
+√σ
+Figure 13: Energies in the q = 1 (+) sector at R = 40a = 2.76ls as a function of the
+inverse transverse size. Horizontal lines of different colors represent the GGRT spectrum
+starting from N = 2, ˜N = 1.
+30
+
+0.05
+0.10
+0.15
+0.20
+0.25
+0
+2
+4
+6
+8
+10
+E/√σ
+1/l⊥
+√σ
+Figure 14: Energies in the q = 1 (−) sector at R = 40a = 2.76ls as a function of the
+inverse transverse size. Horizontal lines of different colors represent the GGRT spectrum
+starting from N = 1, ˜N = 0.
+31
+
+5.5
+Including multitrace operators
+A sizable mixing between flux tube and scattering states is an interesting peculiarity of the
+Ising model, not present in the non-Abelian Yang–Mills theory. In the Yang–Mills case the
+scattering states are created by multitrace operators whose overlap on the flux tube states
+produced by single trace operators is suppressed even at moderately large number of colors N.
+As discussed before, in the Ising case there is no distinction between multitrace and single trace
+operators. We just saw, this leads to a substantial overlap of our operator basis (which was
+intended to create pure flux tube states) on the scattering states. On the other hand, this basis
+is definitely not very well suited for an accurate identification and separation of the scattering
+states, because one still expects that the corresponding overlap is somewhat suppressed as
+a consequence of locality. Hence, it should be instructive to enlarge the operator basis by
+introducing additional operators with a good overlap on the scattering states. This will allow
+us to better probe the nature of the (++) resonance and to confirm its interpretation as a zero
+momentum scattering state. The additional (pseudo) multi trace operators can be constructed
+by considering a product of (smeared and blocked) plaquette operators φG producing glueball
+states with the straight Polyakov loop (22),
+φscattering =
+l⊥/a
+�
+n,m=1
+φP(y + na)φG(y + ma)e
+2πiq⊥(n−m)a
+l⊥
+.
+(32)
+The double sum in (32) is needed to project on a state with a vanishing total transverse mo-
+mentum, which is also characterized by a relative momentum q⊥7. We include such operators
+with q⊥ = 0, 1, 2, 3, 4 and Pt = ± (q⊥ = 0 state only appears in the Pt = + sector). On the
+other hand, for these operators Pl = + because this holds for the φG and φP that we use, and
+no relative longitudinal momentum is introduced.
+We now repeat the analysis of the transverse volume dependence of the spectrum using
+this extended basis of operators. This should allow a more thorough determination of the
+low-lying spectrum including also the discretuum of scattering states. If the (++) resonance
+is a genuine string state, one expects to find two low-lying massive states that don’t receive
+pronounced finite volume corrections.
+One of these states would then correspond to the
+lowest lying glueball scattering state and another to the string excitation (which can also be
+interpreted as a bound state of a string and a glueball).
+The results for the (++) sector are presented in Fig. 15. Here we chose the compactification
+radius R = 55a to ensure that the lowest scattering state is well separated from the GGRT
+states. We clearly see that beneath the (2, 2) GGRT level, there is only one non-GGRT state
+(represented by cyan dots) whose energy exhibits only a moderate dependence on a transverse
+size. In addition, there is a series of non-GGRT states with a strong volume dependence
+(represented by yellow, brown, purple and mauve-blue dots) which become very dense at
+large transverse size and accumulate around the expected threshold for the continuum of the
+scattering states. It is natural to interpret these levels as fluxtube-glueball scattering states
+7Note that q⊥ is only an approximate quantum number.
+32
+
+0.05
+0.10
+0.15
+0.20
+0.25
+0
+2
+4
+6
+8
+10
+E/√σ
+1/l⊥
+√σ
+Figure 15: Energies in the q = 0 (++) sector at R = 55a = 3.80ls as a function of inverse
+transverse size determined using an extended operator basis. Horizontal solid lines of
+di��erent colors represent the GGRT spectrum starting from N = ˜N = 0. The lower
+dashed blue line represents the energy of the absolute ground state plus the glueball mass.
+The upper dashed blue line represents the absolute ground state plus the resonance mass
+as given by (30).
+33
+
+with q⊥ = 1, 2, 3, 4 and the level represented by the cyan dots as a q⊥ = 0 state at the bottom
+of the continuum.
+Interestingly, this candidate q⊥ = 0 state still exhibits a noticeable transverse size depen-
+dence in the range of ℓ⊥ presented in Fig. 15. The corresponding energy gap at the shortest
+values of ℓ⊥ is significantly higher than the glueball mass. This is indicative of a considerable
+repulsive interaction between the glueball and the flux tube.
+These interactions appear to be important also for the states with non-zero relative mo-
+mentum q. In particular, a priori one could have expected that the ℓ⊥ dependence of the
+corresponding energies can be captured by the free dispersion relation,
+E =
+�
+m2
+flux + p2
+⊥ +
+�
+m2
+glue + p2
+⊥ ,
+(33)
+with p⊥ = 2πq⊥/ℓ⊥. However, we find that this ansatz does not provide a very good fit,
+indicative of considerable interactions with the flux tube. These interactions are expected
+also to affect the GGRT states above the continuum threshold. This may actually resolve one
+of the puzzles encountered earlier. Namely, one expects to find two states at the (2, 2) GGRT
+level. However, only one such state is present in Fig. 15 (the one labeled by green dots). This
+phenomenon is also observed in Table 4, where we find out that one of the (2, 2) GGRT states
+start to deviate from GGRT spectrum at R ≳ 3.8ℓs. It appears that a strong mixing between
+the GGRT and scattering states may provide an explantation for this effect.
+We observe similar new states with a strong volume dependence also in the (−+) sector.
+The corresponding flux tube spectrum as a function of the transverse size is presented in
+Fig. 16.
+Here blue and orange dots are plausible candidates for GGRT (2, 2) and (3, 3)
+states given that their volume dependence is relatively mild. In addition we find four states
+with a strong and monotonic volume dependence, which makes them natural candidates for
+q⊥ = 1, 2, 3, 4 states in the discretuum. There is no analogue of the q⊥ = 0 state in this sector.
+As in the (++) sector the complicated pattern of the corresponding energies, suggests that a
+considerable mixing between flux tube and glueball states is present.
+To summarize, we believe that the analysis presented here strongly disfavors the existence
+of light massive excitations on the worldsheet of the Z2 confining flux tube. In particular, the
+state which appears as a massive resonance in the (++) sector corresponds to the glueball
+scattering state. In addition, our results indicate the presence of a significant mixing between
+flux tube excitations and scattering glueball states.
+6
+Concluding Remarks
+To summarize, we calculated the low-lying spectrum of closed flux tube excitations up to the
+N = ˜N = 3 GGRT level in the Z2 gauge theory, at a coupling β = 0.756321 which is close to
+the critical point βc = 0.7614133(22) [24]. The compactification radius covers a wide range
+1.38ℓs ≤ R ≤ 5.53ℓs from moderately short strings to very long ones, but still above the
+deconfinement transition at Rc ∼ 0.82ℓs [41]. The resulting spectrum agrees with the GGRT
+predictions for N = ˜N ≤ 1 states within most of the range of R, and also for N = ˜N = 2
+states for moderately long strings.
+34
+
+0.05
+0.10
+0.15
+0.20
+0.25
+7
+8
+9
+10
+E/√σ
+1/l⊥
+√σ
+Figure 16: Energies in the q = 0 (−+) sector at R = 55a = 3.80ls as a function of the
+inverse transverse size determined using an extended operator basis. Horizontal solid
+lines of different colors represent the GGRT spectrum starting from N = ˜N = 2. The
+dashed green line represents the energy of absolute ground state plus glueball mass.
+.
+35
+
+Somewhat surprisingly, our analysis did not reveal any massive excitations on the world-
+sheet of the Ising string. A heuristic argument suggesting the presence of a resonance is based
+on realizing the critical Ising model as an IR fixed point of the φ4 theory. Then one may
+attempt to study the properties of the Ising strings by analyzing domain walls in a mass-
+deformed φ4 theory. Even though this approach is not based on a well-controlled perturbative
+expansion at d = 3, it was argued [42] to provide a decent approximation to the ratio of the
+lighest glueball mass to the string tension. A domain wall in φ4 theory does support a massive
+localized resonance [43], so based on this logic one might have expected to find one also in
+the Ising case. It will be interesting to study what happens to this resonance using a more
+systematic approach, based on the ϵ-expansion rather than a direct study of the d = 3 φ4
+model.
+We did observe a state in the 0++ sector which has an appearance of a massive reso-
+nance. However, a detailed analysis revealed that this is a multitrace scattering state with an
+additional glueball rather than a genuine flux tube excitation. This is related to another in-
+teresting (and expected) aspect of the observed spectra. Namely, they indicate the presence of
+a significant mode mixing between string excitations and glueball scattering states related to
+the repulsive glueball/string interaction. It will be interesting to perform an analytical anal-
+ysis of these spectra using an appropriate generalization of the TBA method and to extract
+the scattering amplitudes describing glueball/string interactions. It will also be interesting
+to connect this data to the properties of the line defect in the Ising model at the conformal
+point, which has been studied in [44].
+Another possible direction to extend this work is to study the dynamics of strings in the
+ZN gauge theory. In particular, it will be interesting to study how the 3d U(1) gauge theory
+(studied, e.g., in [45–47]) is recovered in the N → ∞ limit. It is natural to expect that strong
+glueball/string interactions should be present in this whole family of theories.
+Acknowledgements.
+We thank Ofer Aharony, Victor Gorbenko, Michele Caselle, Nabil
+Iqbal and Yifan Wang for fruitful discussions. This work is supported in part by the NSF grant
+PHY-2210349, by the BSF grant 2018068 and by the Simons Collaboration on Confinement
+and QCD Strings. The work of CL is partly supported by funding resources from NYU physics
+department, and the simulation is run on NYU Greene cluster.
+A
+Compilation of energy spectra
+In this appendix we list all the closed flux tube spectra we’ve computed for the Z2 gauge theory
+at the coupling β = 0.756321, with different lattice sizes and different quantum numbers. The
+convention for denoting sectors follows (29).
+36
+
+R/a
+l⊥ × lt/a2
+aE(R) ; q = 0 (++)
+20
+70 × 70
+0.0668(8)
+0.2384(46)
+0.3460(49)
+0.4893(47)
+0.4882(69)
+0.4996(199)*
+0.6339(109)
+25
+0.0966(11)
+0.3022(50)
+0.3664(58)
+0.4873(87)
+0.5895(290)
+0.5649(175)
+0.5338(138)
+30
+0.1211(17)
+0.3251(65)
+0.3917(82)
+0.5151(65)
+0.5884(62)
+0.4929(117)
+0.5838(164)*
+35
+0.1506(13)
+0.3456(134)
+0.4167(126)
+0.5460(60)
+0.5479(135)
+0.5542(117)
+0.5416(161)
+40
+0.1785(14)
+0.3766(60)
+0.4318(124)
+0.5439(80)
+0.5123(161)
+0.6392(117)
+0.5854(80)*
+45
+0.2037(19)
+0.3827(97)
+0.4444(208)
+0.5436(87)
+0.5533(242)
+0.6279(130)
+0.5464(177)*
+47
+0.2143(12)
+0.3969(35)
+0.4737(101)
+0.5448(69)
+0.5596(147)
+0.6539(64)
+0.6010(69)
+50
+0.2255(34)
+0.4002(58)
+0.4997(108)
+0.5599(52)
+0.5850(154)
+0.6507(96)
+0.6282(109)
+52
+0.2339(19)
+0.4118(65)
+0.5009(92)
+0.5634(59)
+0.5813(198)
+0.6407(139)
+0.6327(67)*
+54
+0.2492(27)
+0.4239(61)
+0.5285(66)
+0.5537(92)
+0.6113(122)
+0.6650(62)
+0.6441(57)*
+55
+0.2500(29)
+0.4106(100)
+0.4715(280)
+0.5600(61)
+0.5612(242)
+0.6571(100)
+0.6259(72)
+56
+0.2571(26)
+0.4214(66)
+0.5043(117)
+0.5583(45)
+0.6008(169)
+0.6671(46)*
+58
+0.2686(19)
+0.4409(33)
+0.5278(106)
+0.5609(70)
+0.6136(139)
+0.6375(113)*
+60
+0.2819(29)
+0.4404(72)
+0.5412(138)
+0.5701(75)
+0.6386(207)
+0.6543(88)
+0.7048(96)
+65
+0.3085(31)
+0.4640(60)
+0.5683(116)
+0.5850(83)
+0.6663(122)
+0.6567(137)*
+0.6680(199)*
+70
+0.3238(45)
+0.4678(61)
+0.5612(149)
+0.5935(85)
+0.6718(202)*
+0.6483(144)*
+0.6858(174)*
+75
+0.3586(38)
+0.5012(68)
+0.6167(96)
+0.6058(77)
+0.7401(114)
+0.6921(121)*
+0.7561(118)*
+80
+0.3745(64)
+0.5093(75)
+0.6069(132)
+0.6197(157)
+0.7463(152)*
+0.7849(126)*
+40
+55 × 55
+0.1731(19)
+0.3514(57)
+0.4407(90)
+0.5443(86)
+0.5715(177)
+0.5694(118)
+0.6240(146)
+40
+55 × 70
+0.1766(14)
+0.3678(43)
+0.4517(82)
+0.5497(72)
+0.5847(159)
+0.5800(81)
+40
+65 × 70
+0.1772(17)
+0.3662(70)
+0.4162(133)
+0.5506(69)
+0.5665(136)
+0.6653(87)
+0.5415(176)
+40
+80 × 70
+0.1780(17)
+0.3817(50)
+0.4540(130)
+0.5556(40)
+0.4818(108)
+0.6378(96)
+0.6385(98)
+40
+160 × 70
+0.1768(15)
+0.3772(44)
+0.4660(61)
+0.5607(41)
+0.4972(94)
+0.6469(57)
+0.5515(104)*
+40
+300 × 70
+0.1770(13)
+0.3767(21)
+0.4557(63)
+0.5449(48)
+0.4928(96)
+0.6332(122)
+0.5611(103)*
+Table 4: The energies, E(R), of the lightest seven flux tube states with length R in the
+sector q = 0 (++).
+37
+
+R/a
+l⊥ × lt/a2
+aE(R) ; q = 0 (+−)
+20
+70 × 70
+0.9337(259)
+25
+0.8790(60)
+30
+0.8055(221)
+35
+0.7678(83)
+40
+0.7155(172)
+45
+0.7316(80)*
+47
+0.7392(99)*
+50
+0.7520(115)
+52
+0.7487(105)
+54
+0.6905(208)
+55
+0.7501(123)*
+56
+0.7272(85)
+58
+0.7644(55)*
+60
+0.7189(112)*
+65
+0.7264(132)*
+70
+0.7528(53)*
+75
+0.7401(159)*
+80
+0.7242(126)*
+40
+55 × 55
+0.7458(198)
+40
+55 × 70
+0.7513(183)
+40
+65 × 70
+0.7518(79)
+40
+80 × 70
+0.7478(80)
+40
+160 × 70
+0.7215(185)
+40
+300 × 70
+0.7389(79)
+Table 5: The energies, E(R), of the lightest flux tube state with length R in the sector
+q = 0 (+−).
+38
+
+R/a
+l⊥ × lt/a2
+aE(R) ; q = 0 (−+)
+20
+70 × 70
+0.4497(43)
+0.6023(116)
+0.6516(268)
+0.9297(72)
+25
+0.4572(88)
+0.5693(87)
+0.7823(110)
+0.8336(357)
+30
+0.4634(65)
+0.5525(107)
+0.7294(197)
+0.7506(230)*
+35
+0.4784(45)
+0.5851(41)
+0.7281(41)
+0.7736(329)
+40
+0.4686(65)
+0.5666(73)
+0.7019(137)
+0.7166(222)*
+45
+0.4925(54)
+0.5682(177)
+0.6753(100)
+0.7935(123)
+47
+0.5095(46)
+0.5961(98)
+0.6698(121)
+0.7487(241)*
+50
+0.5261(52)
+0.5991(87)
+0.6866(71)
+0.7826(102)
+52
+0.5364(59)
+0.6139(58)*
+0.6904(88)
+0.7770(115)
+54
+0.5310(64)
+0.6063(100)*
+0.6897(69)
+0.8138(53)
+55
+0.5405(56)
+0.6393(63)
+0.6994(91)
+0.7418(258)*
+56
+0.5561(57)
+0.6405(58)
+0.6923(84)
+0.7685(137)*
+58
+0.5467(39)
+0.6314(62)
+0.6979(83)
+0.7007(83)
+60
+0.5717(44)
+0.6331(78)
+0.6604(148)*
+0.8186(61)
+65
+0.5795(60)
+0.6652(71)*
+0.6964(108)*
+70
+0.6059(40)
+0.7006(330)
+0.7084(104)*
+75
+0.6259(38)
+0.6874(218)*
+0.7141(98)*
+80
+0.6177(125)
+0.7305(118)*
+0.7384(73)*
+40
+55 × 55
+0.5278(42)
+0.6653(115)
+0.7093(101)
+40
+55 × 70
+0.5308(62)
+0.6988(136)
+0.7007(83)
+40
+65 × 70
+0.5052(53)
+0.5939(80)
+0.7149(90)
+0.7571(208)
+40
+80 × 70
+0.4801(83)
+0.5430(71)
+0.7136(66)
+0.6882(161)
+40
+160 × 70
+0.4848(53)
+0.4733(55)
+0.5400(80)*
+0.7147(142)
+0.5930(135)*
+40
+300 × 70
+0.4836(32)
+0.4694(82)
+0.5325(92)*
+0.6608(146)
+Table 6: The energies, E(R), of the lightest four flux tube states (for 40 × 160 × 70 it is
+five) with length R in the sector q = 0 (−+).
+39
+
+R/a
+l⊥ × lt/a2
+aE(R) ; q = 0 (−−)
+20
+70 × 70
+0.7911(92)
+0.8396(220)
+0.8715(63)
+25
+0.6850(68)
+0.7588(50)
+0.7775(99)
+30
+0.6349(27)
+0.7458(84)
+0.9011(484)
+35
+0.6008(74)
+0.6935(170)
+40
+0.5763(71)
+0.6459(125)
+0.6780(171)
+45
+0.5709(35)
+0.6772(161)
+0.7331(108)
+47
+0.5615(41)
+0.6669(127)
+0.7235(166)
+50
+0.5664(64)
+0.6833(121)
+0.7018(168)*
+52
+0.5707(69)
+0.6595(131)
+0.7488(163)
+54
+0.5704(51)
+0.6867(87)
+0.7153(130)*
+55
+0.5784(56)
+0.6894(150)
+0.7056(159)*
+56
+0.5827(51)
+0.6697(186)
+0.7709(76)*
+58
+0.5731(61)
+0.7214(104)
+0.7735(71)*
+60
+0.5838(62)
+0.6984(169)
+0.8113(54)*
+65
+0.5877(70)
+0.7686(62)
+0.7783(365)*
+70
+0.6121(77)
+0.7115(210)*
+75
+0.6286(72)
+0.6914(222)*
+80
+0.6424(80)
+0.7357(326)*
+40
+55 × 55
+0.5763(50)
+0.6136(100)
+0.7731(133)
+40
+55 × 70
+0.5597(112)
+0.6315(82)
+0.7454(217)
+40
+65 × 70
+0.5768(47)
+0.6218(129)
+0.6967(191)
+40
+80 × 70
+0.5706(131)
+0.6211(192)
+0.7599(241)
+40
+160 × 70
+0.5805(47)
+0.6568(103)
+0.7938(270)
+40
+300 × 70
+0.5875(34)
+0.6772(116)
+0.8248(115)
+Table 7: The energies, E(R), of the lightest three flux tube states with length R in the
+sector q = 0 (−−).
+40
+
+R/a
+l⊥ × lt/a2
+aE(R) ; q = 1 (+)
+20
+70 × 70
+0.4899(45)
+0.5612(88)
+0.6680(73)
+0.6693(192)
+0.8066(195)
+25
+0.4722(37)
+0.5236(92)
+0.6141(132)
+0.6804(73)
+0.7865(307)
+30
+0.4644(29)
+0.4919(111)
+0.6029(64)
+0.6480(153)
+0.7514(132)
+35
+0.4585(41)
+0.5169(85)
+0.5594(197)
+0.6343(83)
+0.7561(34)
+0.7460(141)
+40
+0.4778(37)
+0.4953(95)
+0.5904(106)
+0.6632(94)
+0.6684(45)
+0.7484(61)
+45
+0.4820(34)
+0.5359(51)
+0.6110(73)
+0.6414(69)
+0.6353(95)
+0.7416(71)
+47
+0.4801(32)
+0.5323(65)
+0.6249(42)
+0.6377(44)
+0.6633(53)
+0.7538(74)
+50
+0.4919(29)
+0.5509(61)
+0.6414(48)
+0.6300(35)
+0.6583(131)
+0.7675(98)
+52
+0.4898(41)
+0.5536(75)
+0.6269(80)
+0.6236(78)
+0.6246(110)
+0.7402(138)
+54
+0.4996(41)
+0.5856(62)
+0.6359(64)
+0.6267(51)
+0.6441(232)
+0.7397(84)
+55
+0.4938(35)
+0.5478(98)
+0.6345(95)
+0.6167(105)
+0.6546(156)*
+0.7532(84)
+56
+0.5016(51)
+0.5732(75)
+0.6255(70)
+0.6550(48)
+0.6540(58)*
+58
+0.5071(37)
+0.5524(118)
+0.6527(44)
+0.6361(57)
+0.6942(82)
+0.7587(69)
+60
+0.5228(36)
+0.5882(79)
+0.6614(143)
+0.6439(66)
+0.6799(72)*
+65
+0.5337(57)
+0.5905(93)
+0.6771(77)
+0.6370(103)*
+70
+0.5452(44)
+0.6166(94)
+0.6734(97)*
+75
+0.5595(72)
+0.6462(121)
+0.7031(59)*
+80
+0.5865(47)
+0.6628(157)
+0.6883(73)*
+40
+55 × 55
+0.4621(31)
+0.5323(61)
+0.6214(79)
+0.6969(58)
+0.6851(124)
+0.7858(88)
+40
+55 × 70
+0.4615(53)
+0.5312(56)
+0.6243(29)
+0.6811(99)
+0.6838(75)
+0.7738(57)
+40
+65 × 70
+0.4724(25)
+0.5049(53)
+0.6065(52)
+0.6560(93)
+0.6999(53)
+0.7679(71)
+40
+80 × 70
+0.4745(33)
+0.5076(48)
+0.5712(97)
+0.5977(127)
+0.6489(77)
+0.6982(99)
+40
+160 × 70
+0.4737(27)
+0.5317(77)
+0.6076(64)
+0.5532(106)*
+0.6793(85)
+40
+300 × 70
+0.4701(26)
+0.5394(46)
+0.6040(62)
+0.5587(79)*
+0.6958(42)
+0.6950(121)
+Table 8: The energies, E(R), of the lightest six flux tube states with length R in the
+sector q = 1 (+).
+41
+
+R/a
+l⊥ × lt/a2
+aE(R) ; q = 1 (−)
+20
+70 × 70
+0.4025(14)
+0.5260(73)
+0.6537(90)
+0.6300(52)
+0.6813(202)
+0.7666(70)
+25
+0.3621(16)
+0.4946(58)
+0.6232(78)
+0.6326(65)
+0.6182(84)
+0.7189(66)
+30
+0.3448(16)
+0.4861(50)
+0.5896(73)
+0.5737(180)
+0.6070(159)
+0.6968(108)
+35
+0.3382(19)
+0.4886(37)
+0.5600(89)
+0.6152(71)
+0.5959(140)*
+0.6828(114)
+40
+0.3403(14)
+0.4855(52)
+0.5741(107)
+0.6100(70)
+0.6154(54)
+0.7201(63)
+45
+0.3467(23)
+0.4857(53)
+0.5562(93)
+0.6007(69)
+0.6247(107)
+0.6813(228)
+47
+0.3524(21)
+0.4953(31)
+0.5891(51)
+0.6019(50)
+0.6478(58)
+0.6982(81)
+50
+0.3577(24)
+0.4911(59)
+0.5963(74)
+0.5999(59)
+0.6490(71)
+0.6966(58)
+52
+0.3665(23)
+0.5012(58)
+0.6073(41)
+0.6179(61)
+0.6777(79)
+0.6595(143)*
+54
+0.3700(17)
+0.5075(33)
+0.6171(53)
+0.6229(55)
+0.6671(73)
+0.6819(52)
+55
+0.3681(35)
+0.5033(59)
+0.6040(70)
+0.6162(47)
+0.6852(65)
+0.7166(110)*
+56
+0.3741(24)
+0.5116(38)
+0.6230(35)*
+0.6196(68)
+0.6727(58)
+0.6947(70)
+58
+0.3840(22)
+0.5163(41)
+0.6030(72)
+0.6296(59)
+0.6812(64)
+0.6810(68)*
+60
+0.3899(20)
+0.5221(54)
+0.5940(135)
+0.6431(77)
+0.6850(56)
+0.6740(113)*
+65
+0.4058(23)
+0.5346(49)
+0.6342(53)
+0.6606(68)
+0.6921(83)
+0.6810(170)*
+70
+0.4230(26)
+0.5555(52)
+0.6418(222)
+0.6895(86)
+0.7063(61)*
+75
+0.4357(42)
+0.5623(97)
+0.6660(86)
+0.7014(84)*
+80
+0.4572(45)
+0.5701(84)
+0.7048(69)
+0.7088(93)*
+40
+55 × 55
+0.3424(17)
+0.4991(64)
+0.6044(67)
+0.6363(98)
+0.6713(104)
+0.7215(61)
+40
+55 × 70
+0.3429(14)
+0.4994(45)
+0.6061(59)
+0.6152(137)
+0.6743(124)
+0.6909(95)
+40
+65 × 70
+0.3420(19)
+0.4876(47)
+0.5947(123)
+0.6052(27)
+0.6154(93)
+0.7075(118)
+40
+80 × 70
+0.3375(23)
+0.4890(31)
+0.5483(89)
+0.6091(53)
+0.6414(87)
+0.7307(51)
+40
+160 × 70
+0.3426(11)
+0.4855(31)
+0.5294(73)
+0.5975(59)
+0.5975(32)
+0.7095(98)
+40
+300 × 70
+0.3375(17)
+0.4843(34)
+0.5386(47)
+0.5863(45)
+0.5760(66)
+0.7030(81)
+Table 9: The energies, E(R), of the lightest six flux tube states with length R in the
+sector q = 1 (−).
+42
+
+References
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+

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+arXiv:2301.01466v1  [math.PR]  4 Jan 2023
+A Bayesian Perspective on Feller, Pollard and the
+Complete Monotonicity of the Mittag-Leffler Function
+Nomvelo Karabo Sibisi
+sbsnom005@myuct.ac.za
+January 5, 2023
+Abstract
+Pollard used contour integration to show that the Mittag-Leffler function is the
+Laplace transform of a positive function, thereby proving that it is completely
+monotone. He also cited personal communication by Feller of a discovery of the
+result by “methods of probability theory”. Feller used the two-dimensional Laplace
+transform of a bivariate distribution to derive the result. We prove the result by a
+Bayesian approach. We proceed to prove the complete monotonicity of the multi-
+parameter Mittag-Leffler function, thereby generalising the Pollard result by meth-
+ods of Bayesian probability theory.
+Keywords—
+Bayesian reasoning; complete monotonicity; stable & gamma distributions;
+Mittag-Leffler function; Prabhakar function.
+1
+Introduction
+The problem of interest in this paper is the study of the complete monotonicity of the Mittag-
+Leffler function.
+Complete monotonicity is an analytic property of functions.
+Accordingly,
+Pollard [18] used analytic methods to prove the property in the instance of the Mittag-Leffler
+function. Pollard also cited personal communication by Feller of a discovery of the result by
+“methods of probability theory”. However, Pollard’s comment notwithstanding, the published
+proof by Feller [7] (XIII.8) is also analytic rather than probabilistic (we discuss both approaches
+later in this section). This prompted us to ask the following:
+1. What might constitute a “method of probability theory” in proving an analytic property of
+a function, at least in the context of proving that the Mittag-Leffler function is completely
+monotone?
+2. What additional or complementary insight, if any, might the method of probability theory
+offer relative to an analytic method?
+
+The approach of this paper is to assign appropriate probability distributions and use the sum
+and product rules of probability theory to explore analytic attributes of associated functions.
+This is an instance of Bayesian reasoning for analytic purposes, without the Bayesian inference
+step associated with data analysis. We do not have cause to invoke Bayes’ rule to generate a
+posterior distribution from an assigned prior distribution and a prescribed likelihood. Beyond
+reproducing known analytic results due to Pollard and Feller, we discuss the generalisation that
+flows from adopting such Bayesian reasoning. We start with definitions of complete monotonicity
+and the Mittag-Leffler function.
+1.1
+Definitions
+An infinitely differentiable function ϕ(x) on x > 0 is completely monotone if its derivatives
+ϕ(n)(x) satisfy (−1)nϕ(n)(x) ≥ 0, n ≥ 0. Bernstein’s theorem states that ϕ(x) is completely
+monotone iff it may be expressed as
+ϕ(x) =
+� ∞
+0
+e−xt dF(t) =
+� ∞
+0
+e−xtf(t)dt
+(1)
+for a non-decreasing distribution function F(t) with density f(t), i.e. F(t) =
+� t
+0 f(u)du. The
+first integral in (1) is formally called the Laplace-Stieltjes transform of F and the latter the
+(ordinary) Laplace transform of f. For bounded F(t), ϕ(x) is defined on x ≥ 0. Integrating (1)
+by parts in this case gives ϕ(x) in terms of the ordinary Laplace transform of F:
+ϕ(x) = x
+� ∞
+0
+e−xtF(t) dt =
+� ∞
+0
+e−tF(t/x) dt
+(2)
+The Mittag-Leffler function Eα(x) is defined by the infinite series
+Eα(x) =
+∞
+�
+k=0
+xk
+Γ(αk + 1)
+α ≥ 0
+(3)
+For later reference, the Laplace transform of Eα(−λxα) (λ > 0) is
+� ∞
+0
+e−sxEα(−λxα) dx = sα−1
+λ + sα
+Re(s) ≥ 0
+(4)
+We may turn to the problem of proving the complete monotonicity of Eα(−x). We discuss the
+approaches due to Pollard and Feller in turn before turning to the Bayesian perspective.
+1.2
+Pollard’s Method
+In a 1948 paper, Pollard [18] led with the opening remark:
+“W. Feller communicated to me his discovery – by the methods of probability theory
+– that if 0 ≤ α ≤ 1 the function Eα(−x) is completely monotonic for x ≥ 0. This
+means that it can be written in the form
+Eα(−x) =
+� ∞
+0
+e−xtdPα(t)
+where Pα(t) is nondecreasing and bounded. In this note we shall prove this fact
+directly and determine the function Pα(t) explicitly.”
+[we use Pα where Pollard used Fα, which we reserve for another purpose]
+2
+
+Having dispensed with E0(−x) = 1/(1 + x) and E1(−x) = e−x since “there is nothing to be
+proved in these cases”, Pollard used a contour integral representation of Eα(−x):
+Eα(−x) =
+1
+2πi
+�
+C
+sα−1es
+x + sα ds =
+1
+2πiα
+�
+C′
+ez
+1
+α
+x + z dz
+(5)
+to prove that
+pα(t) ≡ P ′
+α(t) = 1
+α fα(t−1/α) t−1/α−1
+0 < α < 1
+(6)
+where fα(t) is defined by
+e−sα =
+� ∞
+0
+e−stfα(t) dt
+0 < α < 1
+(7)
+Pollard [17] had earlier proved that fα(t) > 0, so that pα(t) ≥ 0, thereby completing his proof
+that Eα(−x) is completely monotone for 0 ≤ α ≤ 1. Pollard stopped at the point of deriving (6),
+the density pα(t) ≡ P ′
+α(t). As per initial task, we proceed to discuss Pα(t) explicitly. We first
+recognise fα(t) as the density of the stable distribution Fα on [0, ∞)
+Fα(t) =
+� t
+0
+fα(u) du
+0 < α < 1
+(8)
+with normalisation Fα(∞) = 1. In turn, the Pollard distribution Pα(t) is
+Pα(t) =
+� t
+0
+pα(u) du = 1
+α
+� t
+0
+fα(u−1/α) u−1/α−1 du
+(9)
+Janson [13] derived Pα(t) as a limiting distribution of a P´olya urn scheme. Pα(t) is known as
+the Mittag-Leffler distribution in the probabilistic literature (one of two distributions bearing
+the same name as discussed later).
+Setting y = u−1/α in (9) gives a simple relation between Pα and Fα:
+Pα(t) =
+� ∞
+t−1/α fα(y) dy = 1 −
+� t−1/α
+0
+fα(y) dy ≡ 1 − Fα(t−1/α)
+(10)
+This ‘duality’ between the Mittag-Leffler and stable distributions is key to the discussion that
+follows. The Pollard result may accordingly be written in several equivalent forms:
+Eα(−x) =
+� ∞
+0
+e−xtdPα(t) =
+� ∞
+0
+e−tPα(t/x) dt
+or
+Eα(−xα) =
+� ∞
+0
+e−tPα(x−αt) dt =
+� ∞
+0
+e−t(1 − Fα(xt−1/α)) dt
+(11)
+Another representation arising from change of variable in Pollard’s original result is
+αEα(−xα) =
+� ∞
+0
+e−xαu fα(u−1/α) u−1/α−1 du
+= x
+� ∞
+0
+e−t fα(xt−1/α) t−1/α−1 dt
+(12)
+Setting aside Pollard’s contour integral proof, it is hard to evaluate directly any of the equivalent
+integral representations above to demonstrate that they do indeed generate Eα(−x), Eα(−xα).
+A method that may be convenient to prove one representation effectively proves all other rep-
+resentations because they are interchangeable ways of stating the Pollard result. In particular,
+Feller followed an indirect route to prove the representation (11), discussed next.
+3
+
+1.3
+Feller’s Method
+In an illustration of the use of the two-dimensional Laplace transform, Feller [7](p453) considered
+1 − Fα(xt−1/α) as a bivariate distribution over x > 0, t > 0. The Laplace transform over x,
+followed by that over t gives
+� ∞
+0
+e−sx(1 − Fα(xt−1/α)) dx = 1
+s − e−tsα
+s
+(13)
+1
+s
+� ∞
+0
+e−λt �
+1 − e−tsα�
+dt = 1
+λ
+sα−1
+λ + sα
+(14)
+By reference to (4), the right hand side of (14) is the Laplace transform of Eα(−λxα)/λ. Since
+the two-dimensional Laplace transform equivalently can be evaluated first over t then over x,
+Feller concluded that
+Eα(−λxα) = λ
+� ∞
+0
+e−λt(1 − Fα(xt−1/α)) dt
+(15)
+which, for λ = 1, is the Pollard result in the form (11). Feller’s proof is based on the interchange
+of the order of integration (Fubini’s theorem) and the uniqueness of Laplace transforms. We
+represent it by the commutative diagram below, where Ls|t denotes the one-dimensional Laplace
+transform of a bivariate source function at fixed t, to give a bivariate function of (s, t) where s
+is the Laplace variable.
+1 − Fα(xt−1/α)
+1
+s − e−tsα
+s
+1
+λEα(−λxα)
+1
+λ
+sα−1
+λ + sα
+Ls|t
+easy
+Lλ|s
+easy
+Lλ|x
+hard
+L −1
+x|λ
+easy
+(16)
+The desired proof is the “hard” direct path, which is equivalent to the “easy” indirect path.
+We will return to commutative diagram representation in a different context later in the paper
+when we discuss infinite divisibility.
+Feller’s concise proof uses “methods of probability theory”, as cited by Pollard, only to the extent
+of choosing the bivariate distribution as input to the two-dimensional Laplace transform. Short
+of any further insight, the methods by both Pollard and Feller might be described as analytic
+rather than probabilistic. We may now turn to an approach that may indeed be described as a
+“method of probability theory” in the context of the Pollard problem.
+1.4
+Purpose and Scope of Paper
+As stated earlier, the approach is this paper is that of Bayesian reasoning, involving strict use of
+the sum and product rules of probability theory. The assignment of appropriate distribution in
+our context is guided by the task of proving that Eα(−x) is completely monotone. We first cast
+Feller’s argument in such terms before proceeding to a more general probabilistic discussion.
+The Mittag-Leffler function is of growing interest in probability theory and physics, with a
+diversity of applications, notably fractional calculus. A comprehensive study of the properties
+4
+
+and applications of the Mittag-Leffler function and its numerous generalisations is beyond the
+scope of this paper. We consciously restrict the scope to the theme of complete monotonicity
+and Mittag-Leffler functions, underpinned by Bayesian reasoning.
+Other studies that explicitly discuss complete monotonicity and Mittag-Leffler functions build
+upon complex analytic approaches similar to Pollard’s rather than the probabilistic underpin-
+ning discussed here. For example, de Oliviera et al. [5] and Mainardi and Garrappa [14] studied
+the complete monotonicity of xβ−1Eγ
+α,β(−xα), whereas G´orska [10] explored the complete mono-
+tonicity of Eγ
+α,β(−x). Eγ
+α,β(x) is the three-parameter variant of the Mittag-Leffler function, also
+known as the Prabhakar function. These papers comment on the fundamental importance of the
+complete monotonicity of Mittag-Leffler functions used in the modelling of physical phenomena,
+such as anomalous dielectric relaxation and viscoelasticity.
+Finally, we are keenly aware that there are other views on the interpretation of “methods of
+probability theory”. We comment on this before discussing the Bayesian approach in detail.
+1.5
+Probabilistic Perspectives
+The phrase ‘methods of probability theory’ used by Pollard may suggest an experiment with
+random outcomes as a fundamental metaphor. As noted earlier, Pα is derived as a limiting
+distribution of a P´olya urn scheme in the probabilistic literature.
+Diversity of approach is commonplace in probability theory and mathematics more generally.
+For example, in a context of nonparametric Bayesian analysis, Ferguson [8] constructed the
+Dirichlet process based on the gamma distribution as the fundamental probabilistic concept,
+without invoking a random experiment. Blackwell and MacQueen [3] observed that the Ferguson
+approach “involves a rather deep study of the gamma process” as they proceeded to give an
+alternate construction based on the metaphor of a generalised P´olya urn scheme. Adopting the
+one approach is not to deny or diminish the other, but to bring attention to the diversity of
+thinking in probability theory, even when the end result is the same mathematical object. We
+look upon this as healthy complementarity rather than undesirable contestation.
+We discuss complete monotonicity by methods of probability theory in the sense of Bayesian
+reasoning.
+For the purpose at hand, we have no need to invoke an underlying random ex-
+periment or indeed an explicit random variable, while not denying the latter as an alternative
+probabilistic approach. Hence, for example, we shall continue to express the Laplace transform
+of a distribution as an explicit integral rather than as an expectation E
+�
+e−sX�
+for a random
+variable X.
+2
+A Bayesian Method
+First, we note that the scale change s → t1/αs (t > 0) in (7) gives
+e−tsα =
+� ∞
+0
+e−sxfα(x t−1/α)t−1/α dx ≡
+� ∞
+0
+e−sxfα(x|t) dx
+(17)
+5
+
+where fα(x|t) ≡ fα(x t−1/α)t−1/α is the stable density conditioned on the scale parameter t, with
+fα(x) ≡ fα(x|1). Correspondingly, the stable distribution conditioned on t is
+Fα(x|t) =
+� x
+0
+fα(u|t) du =
+� xt−1/α
+0
+fα(u) du ≡ Fα(xt−1/α)
+(18)
+with Laplace transform e−tsα/s.
+We then assign a distribution G(t) to the scale parameter t of Fα(x|t). Then, by the sum and
+product rules of probability theory, the unconditional or marginal distribution Mα(x) over x is
+Mα(x) =
+� ∞
+0
+Fα(x|t)dG(t)
+(19)
+with Laplace transform
+� ∞
+0
+e−sxMα(x) dx = 1
+s
+� ∞
+0
+e−tsα dG(t)
+(20)
+Mα is also referred to as a mixture distribution, arising from randomising or mixing the parame-
+ter t in Fα(x|t) with G(t). This has the same import as saying that we assign a prior distribution
+G(t) on t and we shall continue to use the latter language.
+G may depend on one or more parameters. A notable example is the gamma distribution G(µ, λ)
+with shape and scale parameters µ > 0, λ > 0 respectively:
+dG(t|µ, λ) =
+λµ
+Γ(µ) tµ−1e−λt dt
+(21)
+λ is not fundamental and may be set to λ = 1 by change of scale t → λt, while µ controls the
+shape of G(t|µ, λ). The marginal (19) becomes Mα(x|µ, λ), with Laplace transform
+� ∞
+0
+e−sxMα(x|µ, λ) dx = 1
+s
+�
+λ
+λ + sα
+�µ
+= 1
+s
+�
+1 −
+sα
+λ + sα
+�µ
+(22)
+We may now state Feller’s approach from a Bayesian perspective.
+2.1
+A Bayesian View of Feller’s Approach
+The case µ = 1 in (21) gives the exponential distribution dG(t|λ) = λe−λtdt. Then Mα(x|λ) ≡
+Mα(x|µ = 1, λ) is
+Mα(x|λ) =
+� ∞
+0
+Fα(x|t)dG(t|λ) = λ
+� ∞
+0
+Fα(x|t)e−λt dt
+(23)
+The Laplace transform of Mα(x|λ), read from (22) with µ = 1, is
+� ∞
+0
+e−sxMα(x|λ) dx = 1
+s − sα−1
+λ + sα
+(24)
+=⇒
+Mα(x|λ) = 1 − Eα(−λxα)
+(25)
+=⇒ Eα(−λxα) = 1 − Mα(x|λ) = λ
+� ∞
+0
+(1 − Fα(x|t))e−λt dt
+(26)
+This reproduces Feller’s result (15) from a Bayesian perspective.
+The difference is purely a
+matter of conceptual outlook:
+6
+
+Feller: Study the two-dimensional Laplace transform of the bivariate distribution 1−Fα(xt−1/α),
+where Fα is the stable distribution. Deduce that Eα(−λxα)/λ is the Laplace transform
+of 1 − Fα(xt−1/α) over t at fixed x, where λ is the Laplace variable.
+Bayes: Assign an exponential prior distribution G(t|1, λ) to the scale factor t of Fα(x|t) ≡
+Fα(xt−1/α), where G(t|µ, λ) is the gamma distribution. Marginalise over t to generate the
+Feller result directly.
+Feller himself might also have established the result by the latter reasoning. Under subordination
+of processes, Feller [7](p451) discussed mixture distributions but he did not specifically discuss
+the Mittag-Leffler function in this context in his published work. The task fell on Pillai [15] to
+study Mα(x|µ) ≡ Mα(x|µ, λ = 1), including its infinite divisibility and the corresponding Mittag-
+Leffler stochastic process. He also proved that Mα(x|1) = 1 − Eα(−xα) (as discussed above),
+which he referred to as the Mittag-Leffler distribution. There are thus two distributions bearing
+the name “Mittag-Leffler distribution”: Mα(x) = 1 − Eα(−xα) and Pα(t) = 1 − Fα(t−1/α).
+The natural question arising from the Bayesian approach is whether there might be other choices
+of µ in G(µ, λ) (or indeed other choices of G altogether) that yield the Pollard result and, if so,
+what insight they might offer. At face value, there would appear to be nothing further to be
+said since other choices of µ can be expected to lead to different results, beyond the study of
+the Mittag-Leffler function. The main contribution of this paper is that, in fact, there is a limit
+relationship that generates the Pollard result for any µ > 0, as discussed next.
+We first note, given the definition of the conditional stable density
+fα(x|t) ≡ fα(x t−1/α)t−1/α =⇒ fα(1|t) ≡ fα(t−1/α)t−1/α
+that we may write Pα(t) of (9) and the representation (12) of the Pollard result as
+Pα(t) =
+� t
+0
+pα(u) du = 1
+α
+� t
+0
+fα(1|u) u−1 du
+(27)
+αEα(−λxα) = x
+� ∞
+0
+fα(x|t) t−1e−λt dt
+0 < α < 1
+(28)
+u = x−αt :
+Eα(−λxα) =
+� ∞
+0
+e−λxαu dPα(u)
+(29)
+The intent is to generate this representation using the general G(µ, λ) prior distribution, i.e.
+without reference to Pollard’s analytic method and without explicit restriction to the G(µ = 1, λ)
+case that is equivalent to Feller’s approach, as demonstrated above.
+3
+Main Contribution
+We first state Theorem 1, which warrants dedicated discussion, even though it is actually a
+special case of the more general Theorem 3 stated later. We note first that the density of the
+marginal distribution Mα(x|µ, λ) of Section 2 is
+mα(x|µ, λ) =
+� ∞
+0
+fα(x|t) dG(t|µ, λ)
+µ > 0, λ > 0
+=
+λµ
+Γ(µ)
+� ∞
+0
+fα(x|t) tµ−1e−λt dt
+=
+µλµ
+Γ(µ + 1)
+� ∞
+0
+fα(x|t) tµ−1e−λt dt
+(30)
+7
+
+where the latter expression follows from the identity µΓ(µ) = Γ(µ + 1).
+Theorem 1. The limit
+lim
+n→∞
+n
+µ x mα(x|µ
+n, λ) = lim
+n→∞
+n
+µ x
+� ∞
+0
+fα(x|t) dG(t|µ
+n, λ)
+(31)
+is finite and independent of µ for any µ > 0. This limit yields the following integral representa-
+tion of the Mittag-Leffler function Eα(−λxα)
+αEα(−λxα) = x
+� ∞
+0
+fα(x|t) t−1e−λt dt
+(32)
+u = x−αt :
+Eα(−λxα) =
+� ∞
+0
+e−λxαu dPα(u)
+(33)
+where Pα(t) is the (one-parameter) Pollard distribution
+Pα(t) = 1
+α
+� t
+0
+fα(1|u) u−1 du
+= 1
+α
+� t
+0
+fα(u−1/α) u−1/α−1 du
+Hence Eα(−x) is completely monotone.
+Proof of Theorem 1. The Laplace transform of x mα(x|µ, λ) is
+� ∞
+0
+e−sxx mα(x|µ, λ) dx =
+� ∞
+0
+e−sxx
+� ∞
+0
+fα(x|t) dG(t|µ, λ) dx
+= − d
+ds
+� ∞
+0
+� ∞
+0
+e−sxfα(x|t) dx dG(t|µ, λ)
+= − d
+ds
+� ∞
+0
+e−tsα dG(t|µ, λ)
+= αsα−1
+� ∞
+0
+t e−tsα dG(t|µ, λ)
+= αsα−1 λµ
+Γ(µ)
+� ∞
+0
+tµ e−(λ+sα)t dt
+= αsα−1 λµ
+Γ(µ)
+Γ(µ + 1)
+(λ + sα)µ+1
+= λµµα
+sα−1
+(λ + sα)µ+1
+=⇒
+lim
+n→∞
+n
+µ
+� ∞
+0
+e−sxx mα(x|µ
+n, λ) dx = α sα−1
+λ + sα
+which is the Laplace transform of αEα(−λxα). With the aid of (30), it also readily follows that
+the limit (31) is
+lim
+n→∞
+n
+µ x mα(x|µ
+n, λ) = x
+� ∞
+0
+fα(x|t) t−1e−λt dt
+The integral representations (32) and (33) of Eα(−λxα) follow, hence the conclusion that Eα(−x)
+is completely monotone.
+Pursuing the Bayesian theme, we turn next to Laplace convolution to demonstrate the complete
+monotonicity of the two and three parameter Mittag-Leffler functions.
+8
+
+4
+A Convolution Representation
+Toward a more general discussion, we first present an alternative representation of xfα(x|t) using
+Laplace convolution. The convolution {ρ ⋆ f}(x) of ρ(x), f(x) is given by
+{ρ ⋆ f}(x) =
+� x
+0
+ρ(x − u)f(u) du
+(34)
+The convolution theorem states that the Laplace transform of {ρ⋆f} is a product of the Laplace
+transforms of ρ, f.
+4.1
+One Parameter Case
+Proposition 1. Let ρα(x) = x−α/Γ(1 − α), 0 < α < 1 with Laplace transform sα−1.
+Let
+{ρα ⋆ fα(·|t)}(x) be the convolution of ρα(x) and fα(x|t) with Laplace transform sα−1e−tsα.
+Then
+x fα(x|t) = α t{ρα ⋆ fα(·|t)}(x) = α {ρα ⋆ fα}(xt−1/α)
+(35)
+where {ρα ⋆ fα}(x) is the convolution of ρα(x) and fα(x) ≡ fα(x|1). For compatibility with later
+discussion, we also use the name wα(x|t) defined by αwα(x|t) ≡ x fα(x|t).
+Proof of Proposition 1. By the convolution theorem, {ρα ⋆ fα(·|t)}(x) has Laplace transform
+sα−1e−tsα = − 1
+αt
+d
+dse−tsα = 1
+αt
+� ∞
+0
+e−sxxfα(x|t) dx
+=⇒
+α t {ρα ⋆ fα(·|t)}(x) = xfα(x|t)
+The convolution {ρα ⋆ fα(·|t)}(x) takes the explicit form:
+{ρα ⋆ fα(·|t)}(x) =
+� x
+0
+ρα(x − u)fα(u|t) du
+=
+� x
+0
+ρα(x − u)fα(ut−1/α)t−1/α du
+y = ut−1/α :
+=
+� xt−1/α
+0
+ρα(x − yt1/α)fα(y) dy
+=
+� xt−1/α
+0
+ρα(t1/α(xt−1/α − y))fα(y) dy
+= t−1
+� xt−1/α
+0
+ρα(xt−1/α − y)fα(y) dy
+= t−1{ρα ⋆ fα}(xt−1/α)
+so that αwα(x|t) ≡ x fα(x|t) = α t{ρα ⋆ fα(·|t)}(x) = α {ρα ⋆ fα}(xt−1/α).
+Hence the following are equivalent representations of the Pollard distribution Pα(t):
+Pα(t) =
+� t
+0
+wα(1|t) u−1 du ≡ 1
+α
+� t
+0
+fα(1|u) u−1 du
+=
+� t
+0
+{ρα ⋆ fα(·|u)}(1) du
+=
+� t
+0
+{ρα ⋆ fα}(u−1/α) u−1 du
+(36)
+9
+
+The motivation for the convolution representation is to facilitate generalisation. Specifically,
+the Laplace transform αtsα−1e−tsα of xfα(x|t) is the derivative of −e−tsα. However, a more
+general term like tsα−βe−tsα cannot arise from simple derivatives of e−tsα for non-integer β. It
+might be interpreted as a fractional derivative, as can be represented instead by convolutions.
+Accordingly, we proceed to consider more general convolutions than the convolution form (35)
+for xfα(x|t).
+4.2
+Two Parameter Case
+First, we introduce the two-parameter Mittag-Leffler function
+Eα,β(x) =
+∞
+�
+k=0
+xk
+Γ(αk + β)
+(37)
+The Laplace transform of xβ−1Eα,β(−λxα) is
+� ∞
+0
+e−sxxβ−1Eα,β(−λxα) dx = sα−β
+λ + sα
+(38)
+We may now proceed to prove that Eα,β(−x) is completely monotone by showing that it is the
+Laplace transform of a two-parameter variant Pα,β(t) of the Pollard distribution. We follow
+a corresponding two-parameter variant of the convolution argument presented above for the
+one-parameter case.
+Proposition 2. Let ρα,β(x) = xβ−α−1/Γ(β − α) β > α, with Laplace transform sα−β. Let
+{ρα,β ⋆ fα(·|t)}(x) be the convolution of ρα,β(x) and fα(x|t). Then
+wα,β(x|t) ≡ t {ρα,β ⋆ fα(·|t)}(x) = t(β−1)/α {ρα,β ⋆ fα}(xt−1/α)
+(39)
+(the name wα,β(x|t) is a shorthand adopted for convenience).
+Proof of Proposition 2.
+{ρα,β ⋆ fα(·|t)}(x) =
+� x
+0
+ρα,β(x − u)fα(u|t) du
+=
+� xt−1/α
+0
+ρα,β(t1/α(xt−1/α − u))fα(u) du
+= t(β−1)/α−1
+� xt−1/α
+0
+ρα,β(xt−1/α − u)fα(u) du
+= t(β−1)/α−1{ρα,β ⋆ fα}(xt−1/α)
+Thus wα,β(x|t) ≡ t {ρα,β ⋆ fα(·|t)}(x) = t(β−1)/α{ρα,β ⋆ fα}(xt−1/α).
+Theorem 2. The two-parameter Mittag-Leffler function Eα,β(−λxα) has the integral represen-
+tation
+Eα,β(−λxα) =
+� ∞
+0
+e−λxαt dPα,β(t)
+(40)
+10
+
+where Pα,β(t), which we refer to as the two-parameter Pollard distribution, is
+Pα,β(t) =
+� t
+0
+wα,β(1|u) u−1 du
+≡
+� t
+0
+{ρα,β ⋆ fα(·|u)}(1) du
+=
+� t
+0
+{ρα,β ⋆ fα}(u−1/α) u(β−1)/α−1 du
+(41)
+Hence Eα,β(−x) is completely monotone.
+Proof of Theorem 2. The theorem is a particular case of the more general Theorem 3 below,
+hence the current proof is deferred to that of the latter theorem.
+4.3
+Three Parameter Case
+The three-parameter Mittag-Leffler function, also known as the Prabhakar function, is given by
+Eγ
+α,β(x) =
+1
+Γ(γ)
+∞
+�
+k=0
+Γ(γ + k)
+k! Γ(αk + β) xk
+(42)
+The Laplace transform of xβ−1Eγ
+α,β(−λxα) is
+� ∞
+0
+e−sxxβ−1Eγ
+α,β(−λxα) dx =
+sαγ−β
+(λ + sα)γ
+(43)
+We may now proceed to prove that Eγ
+α,β(−x) is completely monotone by showing that it is the
+Laplace transform of a three-parameter variant P γ
+α,β(t) of the Pollard distribution. In principle,
+we need only have discussed the three-parameter case from the outset because the two and one-
+parameter instances are the special cases γ = 1 and γ = β = 1 respectively. We chose instead
+to present in sequential order for clarity of exposition.
+We devote a separate section to the three-parameter case, which subsumes all prior discussion,
+by restating Theorem 1 in the three-parameter context.
+5
+Main Theorem
+We start with a proposition required for the general theorem that follows:
+Proposition 3. Let ργ
+α,β(x) = xβ−αγ−1/Γ(β − αγ) (0 < α < 1, γ > 0, β > αγ) and let {ργ
+α,β ⋆
+fα(·|t)}(x) be the convolution of ργ
+α,β(x) and the stable density fα(x|t). Then
+wγ
+α,β(x|t) ≡ tγ {ργ
+α,β ⋆ fα(·|t)}(x) = t(β−1)/α{ργ
+α,β ⋆ fα}(xt−1/α)
+(44)
+11
+
+Proof of Proposition 3.
+{ργ
+α,β ⋆ fα(·|t)}(x) =
+� x
+0
+ργ
+α,β(x − u)fα(u|t) du
+=
+� xt−1/α
+0
+ργ
+α,β(t1/α(xt−1/α − u))fα(u) du
+= t(β−1)/α−γ
+� xt−1/α
+0
+ργ
+α,β(xt−1/α − u)fα(u) du
+= t(β−1)/α−γ{ργ
+α,β ⋆ fα}(xt−1/α)
+Thus wγ
+α,β(x|t) ≡ tγ {ργ
+α,β ⋆ fα(·|t)}(x) = t(β−1)/α{ργ
+α,β ⋆ fα}(xt−1/α).
+Theorem 3. Let ργ
+α,β(x), wγ
+α,β(x|t) (0 < α < 1, γ > 0, β > αγ) be as defined in Proposition 3 and
+let G(µ, λ) be the gamma distribution with shape and scale parameters µ > 0, λ > 0 respectively.
+Let the distribution Mγ
+α,β(x|µ, λ) have density
+mγ
+α,β(x|µ, λ) =
+� ∞
+0
+wγ
+α,β(x|t) dG(t|µ, λ)
+=
+λµ
+Γ(µ)
+� ∞
+0
+wγ
+α,β(x|t) tµ−1e−λt dt
+(45)
+≡
+λµ
+Γ(µ)
+� ∞
+0
+{ργ
+α,β ⋆ fα(·|t)}(x) tγ+µ−1e−λt dt
+(46)
+=
+λµ
+Γ(µ)
+� ∞
+0
+{ργ
+α,β ⋆ fα}(xt−1/α) t(β−1)/α+µ−1e−λt dt
+(47)
+where the latter two forms follow from Proposition 3. Then the following limit is finite and
+independent of µ for any µ > 0
+lim
+n→∞
+n
+µ mγ
+α,β(x|µ
+n, λ)
+(48)
+This limit yields the following integral representation of the three-parameter Mittag-Leffler or
+Prabhakar function Eγ
+α,β(−λxα)
+Eγ
+α,β(−λxα) =
+� ∞
+0
+wγ
+α,β(x|t) t−1e−λt dt =
+� ∞
+0
+e−λxαt dP γ
+α,β(t)
+(49)
+where P γ
+α,β(t), which we refer to as the three-parameter Pollard distribution, is
+P γ
+α,β(t) =
+� t
+0
+wγ
+α,β(1|u) u−1 du
+≡
+1
+Γ(γ)
+� t
+0
+{ργ
+α,β ⋆ fα(·|u)}(1) uγ−1 du
+=
+1
+Γ(γ)
+� t
+0
+{ργ
+α,β ⋆ fα}(u−1/α) u(β−1)/α−1 du
+(50)
+Hence Eγ
+α,β(−x) is completely monotone.
+12
+
+Proof of Theorem 3. The Laplace transform �mγ
+α,β(s|µ, λ) of (45) is
+�mγ
+α,β(s|µ, λ) ≡
+� ∞
+0
+e−sx mγ
+α,β(x|µ, λ) dx
+= sαγ−β λµ
+Γ(µ)
+� ∞
+0
+tγ+µ−1e−(λ+sα)t dt
+= λµ Γ(γ + µ)
+Γ(µ)
+sαγ−β
+(λ + sα)γ+µ
+(51)
+=⇒
+lim
+n→∞
+n
+µ
+� ∞
+0
+e−sxmγ
+α,β(x|µ
+n, λ) dx = Γ(γ)
+sαγ−β
+(λ + sα)γ
+(52)
+By (43), the right hand side is the Laplace transform of Γ(γ) xβ−1Eγ
+α,β(−λxα). Given (46) and
+(47), it also readily follows that the limit (48) is
+� ∞
+0
+tγ{ργ
+α,β ⋆ fα(·|t)}(x)t−1e−λtdt =
+� ∞
+0
+{ργ
+α,β ⋆ fα}(xt−1/α) t(β−1)/α−1e−λtdt
+=⇒
+Eγ
+α,β(−λxα) = x1−β
+Γ(γ)
+� ∞
+0
+{ργ
+α,β ⋆ fα}(xt−1/α) t(β−1)/α−1e−λt dt
+u = x−αt : =
+1
+Γ(γ)
+� ∞
+0
+e−λxαu {ργ
+α,β ⋆ fα}(u−1/α) u(β−1)/α−1 du
+=
+� ∞
+0
+e−λxαu dP γ
+α,β(u)
+Hence Eγ
+α,β(−x) is completely monotone.
+Theorem 3 may be visually represented by the following commutative diagram, where mγ
+α,β(x|µ, λ)
+and its Laplace transform �mγ
+α,β(s|µ, λ) are given by (45) and (51) respectively. The equivalence
+of the two routes from the top left node to the bottom left node induces the integral represen-
+tation of the Mittag-Leffler function.
+mγ
+α,β(x|µ, λ)
+�mγ
+α,β(s|µ, λ)
+Γ(γ)xβ−1Eγ
+α,β(−λxα)
+Γ(γ)
+sαγ−β
+(λ + sα)γ
+L
+lim
+n→∞
+n
+µ �mγ
+α,β(s|µ
+n, λ)
+lim
+n→∞
+n
+µ mγ
+α,β(x|µ
+n, λ)
+L −1
+(53)
+The representation (49) of Eγ
+α,β(x), with P γ
+α,β(t) given by (50), is equivalent to equation (2.4)
+in G´orska et al. [10]. The difference is one of approach.
+This paper offers a fundamentally
+probabilistic argument, while G´orska et al. [10] follows a complex analytic route inspired by
+Pollard [18]. The balance of G´orska et al. [10] is devoted to finding an explicit formula for a
+function f γ
+α,β(x) featuring in the paper in terms of the Meijer G function and associated confluent
+Wright function. In turns out that f γ
+α,β(x) in G´orska et al. [10] is identical to {ργ
+α,β ⋆ fα}(x) in
+13
+
+this paper. We are content to leave it in the conceptually simple convolution form:
+{ργ
+α,β ⋆ fα}(x) =
+� x
+0
+ργ
+α,β(x − u)fα(u) du
+=
+1
+Γ(β − αγ)
+� x
+0
+(x − u)β−αγ−1fα(u) du
+(54)
+rather than express it in terms of special functions. In our context, we have actually worked
+with the conditional density
+wγ
+α,β(x|t) ≡ tγ {ργ
+α,β ⋆ fα(·|t)}(x) = t(β−1)/α{ργ
+α,β ⋆ fα}(xt−1/α)
+where we assigned a gamma prior distribution to the scale parameter t. The density wγ
+α,β(x|t)
+reduces to (54) for the particular choice t = 1.
+We have completed the task of proving that the three-parameter Mittag-Leffler function Eγ
+α,β(−x)
+is completely monotone by methods of probability theory, using Bayesian reasoning to derive an
+explicit form for P γ
+α,β(t), whose Laplace transform is Eγ
+α,β(−x). Beyond that, we draw conclu-
+sions on the complete monotonicity of related functions, notably xβ−1Eγ
+α,β(−xα) and Eγ
+α,β(−xα)
+in isolation. First, we discuss xβ−1Eγ
+α,β(−xα), the bottom left node of the commutative dia-
+gram (53), in the Bayesian context of Theorem 3. The discussion involves an alternative repre-
+sentation of the fundamental probabilistic object – the convolution density {ργ
+α,β ⋆ fα(·|t)}(x).
+6
+An Alternative Representation
+For xβ−1Eγ
+α,β(−λxα) to be completely monotone, there must exist a distribution Rγ
+α,β(u|λ)
+defined by the Laplace transform
+xβ−1Eγ
+α,β(−λxα) =
+� ∞
+0
+e−xu dRγ
+α,β(u|λ)
+(55)
+In turn, the Laplace transform of (55) is the Stieltjes transform (or iterated Laplace transform)
+of Rγ
+α,β(u|λ):
+sαγ−β
+(λ + sα)γ =
+� ∞
+0
+1
+s + u dRγ
+α,β(u|λ)
+(56)
+Then, as de Oliviera et al. [5], Mainardi and Garrappa [14] show, the Stieltjes inversion formula
+(Titchmarsh [22](11.8, p318), Widder [23](VIII.7, p342)) gives
+dRγ
+α,β(u|λ) = 1
+π Im
+�
+(e−iπu)αγ−β
+(λ + (e−iπu)α)γ
+�
+du
+(57)
+The expression in braces on the RHS of (57) is (56) at s = e−iπu. In particular, for γ = β = 1,
+(57) reduces to
+dRα(u|λ) = 1
+π
+λ uα−1 sin πα
+λ2 + 2λ uα cos πα + u2α du
+(58)
+which has been discussed in various contexts in the fractional calculus and probabilistic literature
+(e.g. James [12] in the latter context).
+14
+
+We have mentioned (55) for completeness but it was not the core of our probabilistic discussion,
+whose focus was to determine P γ
+α,β(t), with Laplace transform Eγ
+α,β(−x). That said, we can
+offer a ‘hybrid’ derivation of (55) that combines the core of the probabilistic argument in the
+form of the convolution density {ργ
+α,β ⋆ fα(·|t)}(x) with the complex analytic Stieltjes inversion
+argument presented above.
+Assume {ργ
+α,β ⋆ fα(·|t)}(x) to be the Laplace transform of a distribution Sγ
+α,β(u|t):
+{ργ
+α,β ⋆ fα(·|t)}(x) =
+� ∞
+0
+e−xu dSγ
+α,β(u|t)
+(59)
+In turn, the Laplace transform of (59) is the Stieltjes transform of Sγ
+α,β(u|t):
+sαγ−βe−tsα =
+� ∞
+0
+1
+s + u dSγ
+α,β(u|t)
+(60)
+By the Stieltjes inversion formula:
+dSγ
+α,β(u|t) = 1
+π Im
+�
+(ue−iπ)αγ−βe−t(ue−iπ)α�
+du
+(61)
+Hence, using the representation (59) in the proof of Theorem 3:
+Γ(γ) xβ−1Eγ
+α,β(−λxα) =
+� ∞
+0
+tγ{ργ
+α,β ⋆ fα(·|t)}(x) t−1e−λt dt
+=
+� ∞
+0
+dt tγ−1e−λt
+� ∞
+0
+e−xu dSγ
+α,β(u|t)
+= 1
+π Im
+� ∞
+0
+du e−xu(ue−iπ)αγ−β
+� ∞
+0
+tγ−1e−(λ+(ue−iπ)α)t dt
+= Γ(γ)
+π
+Im
+� ∞
+0
+e−xu
+(e−iπu)αγ−β
+(λ + (e−iπu)α)γ du
+= Γ(γ)
+� ∞
+0
+e−xu dRγ
+α,β(u|λ)
+(62)
+thereby reproducing (55).
+The Stieltjes transform and its complex analytic inverse are not unfamiliar in probability theory.
+In his study of a family of distributions known as generalised gamma convolutions, Bondesson [4]
+used the concept under the guise of Pick functions (also known as Nevanlinna functions).
+We turn next to the complete monotonicity of Eγ
+α,β(−λxα).
+7
+A Further Consequence
+There is a well-known property of completely monotone functions (e.g. Schilling et al. [20]) that
+we state without proof in Proposition 4. We start with a definition:
+Definition 1. A Bernstein function is a nonnegative function η(x), x ≥ 0 with a completely
+monotone derivative, i.e. η(x) ≥ 0 and (−1)k−1η(k)(x) ≥ 0, k ≥ 1. For example, η(x|λ) = λxα
+(0 ≤ α ≤ 1, λ > 0) is a Bernstein function.
+15
+
+Proposition 4. If ϕ(x) is completely monotone and η is a Bernstein function, ϕ(η) is completely
+monotone.
+Theorem 4. Given a Bernstein function η, the Mittag-Leffler function Eγ
+α,β(−η) is completely
+monotone. For example, Eγ
+α,β(−λxα) is completely monotone.
+Proof of Theorem 4. We have already shown that Eγ
+α,β(−x) is completely monotone. Hence,
+by Proposition 4, Eγ
+α,β(−η) is completely monotone for a Bernstein function η. Specifically,
+η(x|λ) = λxα (0 ≤ α ≤ 1, λ > 0) is a Bernstein function, hence Eγ
+α,β(−λxα) is completely
+monotone.
+The complete monotonicity of Eγ
+α,β(−λxα) implies that there exists a distribution Qγ
+α,β(t|λ)
+whose Laplace transform is Eγ
+α,β(−λxα):
+Eγ
+α,β(−λxα) =
+� ∞
+0
+e−xt dQγ
+α,β(t|λ)
+(63)
+Qγ
+α,β(t|λ) is to Eγ
+α,β(−λxα) what P γ
+α,β(t) is to Eγ
+α,β(−x). However, determining Qγ
+α,β(t|λ) appears
+to be a challenging problem, whether the approach is analytic or probabilistic.
+Clearly, (63) and (57) are identical for β = 1, i.e. Qγ
+α,1(t|λ) ≡ Rγ
+α,1(t|λ). But, to our awareness,
+determining Qγ
+α,β(t|λ) for β ̸= 1 is an open problem. We shall not pursue it further here. Our
+primary purpose in this section was to bring attention to Theorem 4 and hence the existence of
+a distribution Qγ
+α,β(t|λ) defined by (63).
+8
+A Different Generalisation
+As mentioned in Section 1.5, the Pollard distribution Pα is known as the Mittag-Leffler distri-
+bution in probabilistic literature. For completeness, we briefly discuss a different generalisation
+of Pα that features extensively in such literature. It is known as the generalised Mittag-Leffler
+distribution Pα,θ (Pitman [16], p70 (3.27)), also denoted by ML(α, θ) (Goldschmidt and Haas [9],
+Ho et al. [11]).
+Despite its name, Pα,θ(t) is different from the two-parameter Pollard distribution Pα,β(t) dis-
+cussed above, whose Laplace transform is the Mittag-Leffler function Eα,β(−x). Janson [13]
+showed that Pα,θ may be constructed as a limiting distribution of a P´olya urn scheme.
+It
+is also intimately linked to a concept known as ‘polynomial tilting’.
+For some parameter θ,
+fα,θ(x) ∝ x−θfα(x) is said to be a polynomially tilted variant of fα(x) (e.g. Arbel et al. [1], De-
+vroye [6], James [12]). Here, we consider the polynomially tilted density fα,θ(x|t) ∝ x−θfα(x|t)
+conditioned on a scale factor t > 0. Normalisation gives
+fα,θ(x|t) =
+Γ(θ + 1)
+Γ(θ/α + 1)tθ/α x−θfα(x|t)
+(64)
+so that fα,θ(x|t) is defined for θ/α + 1 > 0, or θ > −α. We then consider a two-parameter
+16
+
+function hα,θ(x|λ) defined by:
+α hα,θ(x|λ) = x
+� ∞
+0
+fα,θ(x|t) t−1e−λt dt
+(65)
+=
+Γ(θ + 1)
+Γ(θ/α + 1) x1−θ
+� ∞
+0
+fα(x|t) tθ/α−1 e−λt dt
+u = x−αt :
+hα,θ(x|λ) =
+� ∞
+0
+e−λxαu dPα,θ(u)
+(66)
+where
+Pα,θ(t) =
+Γ(θ + 1)
+Γ(θ/α + 1)
+1
+α
+� t
+0
+fα(u−1/α) u(θ−1)/α−1 du
+(67)
+or
+dPα,θ(t) =
+Γ(θ + 1)
+Γ(θ/α + 1) tθ/α dPα(t)
+(68)
+It is clear from (66) that hα,θ(x|λ) may be written as hα,θ(λxα). It follows that:
+1. hα,θ(x) is completely monotone
+2. θ = 0: Pα,0(t) = Pα(t) =⇒ hα,0(x) = Eα(−x), as directly apparent from comparing (32)
+and (65).
+3. hα,θ(η) is completely monotone where η is a Bernstein function as discussed in Section 7.
+In particular, hα,θ(λxα) is completely monotone and thus expressible as the Laplace trans-
+form of a corresponding distribution Qα,θ(t|λ) (distinct from Qα,β(t|λ) discussed in Sec-
+tion 7).
+We are not aware of a representation of hα,θ other than that generated by Pα,θ in (66). By
+comparison, the two-parameter Mittag-Leffler function Eα,β has a well-established infinite se-
+ries representation (37), in addition to the representation (40) generated by the two-parameter
+Pollard distribution Pα,β.
+9
+Discussion
+The integral representation (49) of Eγ
+α,β(−λxα) in Theorem 3, arising from the limit (48), con-
+tains the L´evy measure t−1e−λtdt of the infinitely divisible gamma distribution. There is indeed
+an intimate relationship between completely monotone functions and the theory of infinitely di-
+visible distributions on the nonnegative half-line R+ = [0, ∞) (Feller [7] (XIII.4, XIII.7), Steutel
+and van Harn [21] (III)). Sato [19] considers infinitely divisible distributions on Rd, but the de-
+liberate restriction to R+ makes for simpler discussion and relates directly to the core concept of
+complete monotonicity that is of interest here. There is also an intimate link to the generalised
+gamma convolutions studied by Bondesson [4].
+The limit (48) of Theorem 3 is an instance of a limit rule to generate the L´evy measure of
+an infinitely divisible distribution given in Steutel and van Harn [21] (III(4.7)) and Sato [19]
+(Corollary 8.9 restricted to R+ rather than Rd). Barndorff-Nielsen and Hubalek [2] also cite
+Sato’s Corollary.
+Further exploration using the probabilistic machinery of this paper possibly includes the ex-
+plicit determination of the three-parameter distribution Qγ
+α,β(t|λ), whose Laplace transform is
+Eγ
+α,β(−λxα), as per (63).
+17
+
+10
+Conclusion
+We have presented a probabilistic derivation of the complete monotonicity of the three-parameter
+Mittag-Leffler function (also known as the Prabhakar function) by expressing it as the Laplace
+transform of a distribution that we referred to as the three-parameter Pollard distribution. This
+is a generalisation of a result due to Pollard for the one-parameter case.
+References
+[1] Julyan Arbel, Pierpaolo De Blasi, and Igor Pr¨unster. Stochastic Approximations to the
+Pitman–Yor Process. Bayesian Analysis, 14(4):1201 – 1219, 2019.
+[2] Ole E. Barndorff-Nielsen and Friedrich Hubalek. Probability measures, L´evy measures and
+analyticity in time. Bernoulli, 14(3):764 – 790, 2008.
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+The Annals of Statistics, 1(2):353–355, 1973.
+[4] Lennart Bondesson. Generalized Gamma Convolutions and Related Classes of Distributions
+and Densities. Lecture Notes in Statistics, 76. Springer-Verlag, New York, 1992.
+[5] E. Capelas de Oliveira, F. Mainardi, and J. Vaz. Models based on Mittag-Leffler functions
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+19
+

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+A STREAMLINE UPWIND PETROV-GALERKIN REDUCED ORDER
+METHOD FOR ADVECTION-DOMINATED PARTIAL DIFFERENTIAL
+EQUATIONS UNDER OPTIMAL CONTROL
+FABIO ZOCCOLAN1, MARIA STRAZZULLO2, AND GIANLUIGI ROZZA3
+Abstract. In this paper we will consider distributed Linear-Quadratic Optimal Control Problems
+dealing with Advection-Diffusion PDEs for high values of the P´eclet number. In this situation,
+computational instabilities occur, both for steady and unsteady cases.
+A Streamline Upwind
+Petrov–Galerkin technique is used in the optimality system to overcome these unpleasant effects.
+We will apply a finite element method discretization in a optimize-then-discretize approach. For
+the parabolic case, a space-time framework will be considered and stabilization will also occur in
+the bilinear forms involving time derivatives. Then we will build Reduced Order Models on this
+discretization procedure and two possible settings can be analyzed: whether or not stabilization is
+needed in the online phase, too. In order to build the reduced bases for state, control, and adjoint
+variables we will consider a Proper Orthogonal Decomposition algorithm in a partitioned approach.
+The discussion is supported by computational experiments, where relative errors between the FEM
+and ROM solutions are studied together with the respective computational times.
+1. Introduction
+The main goal of Optimal Control theory is to modify a physical or engineering system through an
+input, called control, to obtain a desired output. From a theoretical point of view, one can describe
+the state problem through partial differential equations (PDEs), following the approach of J.L. Lions
+[30, 31]. Applying an optimal control means to solve a constrained optimization problem, where
+a cost functional has to be minimized. This process translates into an optimality system, which
+will be discretized for numerical simulations, that, in this framework, are more and more needed.
+Thus, effective and fast numerical techniques are required to exploit optimal control in scientific and
+industrial applications.
+In this work, we will consider Advection-Diffusion equations [42] for large P´eclet numbers. These
+equations are widespread in many engineering contexts since they can model transfer of particles,
+of energy, of heat and so on. In the case of high values of the P´eclet number, numerical instabilities
+occur during discretization: this can happen for related optimal control problems, too. Thus, it
+becomes necessary to introduce some stabilization techniques to overcome this undesired behaviour.
+We exploit a Streamline Upwind Petrov–Galerkin (SUPG) technique over a finite element method
+(FEM) [11, 26, 38] in a optimize-than-discretize approach, as done in [14], to provide strongly-
+consistency to the discretization. When we deal with unsteady problems, a space-time discretization
+[21, 46, 50, 51, 52, 57] will be used together with the SUPG stabilization for bilinear forms related
+to the derivative over time.
+The discretization procedure can easily request a huge amount of
+computational resources, especially for parametric time-dependent problems. The parameters can
+represent physical or geometrical features of the system at hand. In this scenario, we decide to exploit
+the parameter dependence of the equations to build Reduced Order Models (ROMs) [22, 40, 39, 43]
+by means of Proper Orthogonal Decomposition (POD) algorithm in a partitioned approach. Namely,
+1 Section de Math´ematiques, ´Ecole Polytechnique F´ed´erale de Lausanne, 1015 Lausanne, Switzerland,
+email: fabio.zoccolan@epfl.ch
+2 DISMA, Politecnico di Torino, Corso Duca degli Abruzzi 24, Turin, Italy.
+email: maria.strazzullo@polito.it
+3 mathLab, Mathematics Area, SISSA, via Bonomea 265, I-34136 Trieste, Italy.
+email: gianluigi.rozza@sissa.it
+1
+arXiv:2301.01973v1  [math.NA]  5 Jan 2023
+
+2
+SUPG ROMS FOR ADVECTION-DOMINATED PDES UNDER OPTIMAL CONTROL
+the discretization process is divided in two phases: an offline stage where a low-dimensional space is
+built through FEM solutions computed in properly chosen parameters, and an online stage, where
+the system is solved for a new parametric instance in the new low-dimensional framework. Thus,
+we consider two possible strategies: the former is to stabilize the system only in the offline phase;
+the latter uses SUPG in the online one, too. This setting was considered for problems without
+optimal control in [37, 55]. To the best of our knowledge, it is the first time that SUPG stabilization
+for time-dependent Advection-Dominated problems under distributed control is analyzed in a ROM
+setting.
+This work is organized as follows. The first section will illustrate some theoretical aspects about
+Optimal Control Theory for PDEs. Section 3 shows the FEM discretization that will be used for
+numerical experiments, an introduction to Advection-Dominated problems, and SUPG technique
+in an optimize-than-discretize approach. Instead, in Section 4, we will focus on the ROM setting
+and Section 5 refers to the related numerical simulations. Firstly, we will introduce two specific
+examples of Advection-Diffusion problems: the Graetz-Poiseuille and the Propagating Front in a
+Square Problems. The former was studied in various forms without optimal control in [18, 37, 44, 55]
+and with optimal control but without stabilization in [34, 50]. The latter is studied without optimal
+control in a similar version in [37, 55]. Here, both the problems will be analyzed under a distributed
+optimal control for high values of the P´eclet number, both in the steady and unsteady cases. Relative
+errors between FEM and ROM solutions will be shown, as well as an analysis on the computational
+times.
+2. Problem Formulation
+In this Section we will illustrate the fundamentals of Linear-Quadratic Optimal Control Problem
+(OCP) for steady and unsteady PDEs.
+The aim of Optimal Control is to achieve a prescribed
+optimality condition by minimizing a suitable cost functional under the constraint of satisfying the
+PDE Problem. The proposed framework follows the J.L. Lions theory [30, 31].
+2.1. Parametric Optimal Control Problems governed by PDEs. The main features of an
+OCP are:
+(1) a controlled system, i.e. an input-output process given by a system of PDEs;
+(2) the output of the system, or an observation of it, when the output cannot be measured
+directly. In our case, we will consider the solution of the system as the output;
+(3) a control, which constitutes the input of the system. It influences the output which can be
+expressed as a function of it. In this work we will only consider distributed control;
+(4) an objective condition to be fulfilled, which can be represented by a real functional.
+Therefore, from a mathematical perspective, we can state that an OCP is characterized by:
+• e, the state equation function, which expresses the relationship between the output and the
+control within the system in terms of a PDE problem or PDEs in a weak formulation. A
+pair (y, u) ∈ X := Y × U is said to be physical or feasible if it is a solution of the state
+equation e; y is called the state variable, the output, and u is the control variable, the input.
+Xad is the set of all the feasible pairs (y, u);
+• z(y) = Oy, a direct observation of the output. Here, a linear operator O is applied to the
+state to describe the observation: we will denote the space of observation as Z. We will only
+deal with state variables that can be measured on a portion of the domain;
+• J, the objective functional, which describes the objective to achieve.
+• suitable spaces Y and U, as the state space and control space respectively.
+Domains of
+definition for control and/or state can be taken smaller due to possible restrictions; hence
+we have to introduce Yad ⊆ Y and Uad ⊆ U as the admissible state space and admissible
+control space respectively. However, we will always consider unconstrained problems, i.e.
+Xad = X. The theory of well-posedness can make use of the Lagrangian approach as in
+[12, 34] or it can be consider as a particular case of the general Adjoint approach when we
+can deal with Xad ⊂ Yad × Uad [24, 30, 38].
+
+SUPG ROMS FOR ADVECTION-DOMINATED PDES UNDER OPTIMAL CONTROL
+3
+Let us consider Ω ⊂ Rn, an open and bounded regular domain, and the time interval (0, T) ⊂ R+:
+for us it will always be the case of n = 2. Let us denote with ΓD and ΓN the portions of the boundary
+of ∂Ω where Dirichlet and Neumann boundary conditions are specified, respectively. We define the
+observation domain Ωobs ⊆ Ω as the portion of the domain where we want that the state variable
+assumes a desired value. P ⊆ Rp, for natural number p, is the parameter space and µ ∈ P is a
+p-vector which can represent physical or geometrical parameter of interest. In this work we deal
+with Parametric Optimal Control Problems (OCP(µ)s), i.e. systems where there is a dependency on
+the parameter µ.
+Problem 2.1.1 (Parametric Optimal Control Problem). Given Y, U real Banach spaces, consider
+the state equation e : Y × U → Q, with Q a Banach space, which fulfills a set of boundary and/or
+initial conditions, and the objective functional J : Y ×U → R. Given µ ∈ P, then find
+�
+y(µ), u(µ)
+�
+∈
+X such that the cost functional J(y(µ), u(µ); µ) is minimized subject to e(y(µ), u(µ); µ) = 0.
+2.2. Lagrangian Approach. We refer to the Lagrangian approach to state the well-posedness of
+OCP(µ)s in full admissibility setting, i.e. when Xad = Y × U. We want to solve:
+min
+(y(µ),u(µ))∈Y ×U J(y(µ), u(µ); µ) s.t. e(y(µ), u(µ); µ) = 0,
+thus we define the Lagrangian operator L : Y × U × Q∗ → R as:
+(1)
+L(y(µ), u(µ), p(µ); µ) = J(y(µ), u(µ); µ) + ⟨p(µ), e(y(µ), u(µ); µ)⟩Q∗Q,
+where p(µ) is a Lagrange multiplier belonging to Q∗, the dual space of Q. For the sake of notation,
+we write y := y(µ), u := u(µ) and p := p(µ): we will explicit the parameter dependence only when
+necessary. The discussion inherent to the Lagrangian approach is based on [12], the same reference
+presents a comparison between this approach and the adjoint one. For the sake of simplicity, we
+make some regularity assumptions [12]:
+Assumption 2.2.1. The objective functional J and the state equation e are Fr´echet differentiable,
+more precisely the differential operator related to J is continuous, i.e. J′(µ) ∈ C(Y ×U, B(Y ×U, R)),
+where B(V , ˜V ) is the space of linear bounded operators between Banach spaces V and ˜V .
+The following theorem and proposition claim that under Assumption 2.2.1 minimizers of the
+function J, subject to equality constraints e, can be critical points of (1) [53].
+Theorem 2.2.2 (Lagrange Multipliers). Let X := Y × U and V ⊆ X be an open subset such that
+J and e are Frech´et differentiable on V. Assume x = (y, u) ∈ V to be a minimizer of J subject to
+the constraint e(x; µ) = 0, and e′(x; µ) ∈ B(X, Q) to be surjective. Then, there exists a Lagrange
+multiplier p ∈ Q∗ such that (x, p) is an unconstrained stationary point of the Lagrangian L in (1).
+Therefore, in order to find a stationary point (y, u, p) of L, one has to solve the following optimality
+system [12]:
+(2)
+�
+�
+�
+�
+�
+Ly(y, u, p; µ)(¯y) = Jy(y, u; µ)(¯y) + ⟨p, ey(y, u; µ)(¯y)⟩Q∗Q = 0,
+∀¯y ∈ Y,
+Lu(y, u, p; µ)(¯u) = Ju(y, u; µ)(¯u) + ⟨p, eu(y, u; µ)(¯u)⟩Q∗Q = 0,
+∀¯u ∈ U,
+Lp(y, u, p; µ)(¯p) = ⟨¯p, e(y, u; µ)⟩Q∗Q = 0,
+∀¯p ∈ Q∗.
+In the Lagrangian formulation Q∗ is said the adjoint space. The above result easily implies the
+following useful proposition [38], where we derive another system of three equations that we will use
+in the numerical simulations.
+Proposition 2.2.3 (Optimality System). Suppose Xad = Y × U and Assumption 2.2.1 holds, then
+for some p ∈ Q∗ a minimizer x = (y, u) of 2.1.1 where e′(y, u; µ) is surjective must satisfy
+(3)
+�
+�
+�
+�
+�
+Ly(y, u, p; µ) = Jy(y, u; µ) + ey(y, u; µ)∗p = 0,
+in Y ∗,
+Lu(y, u, p; µ) = Ju(y, u; µ) + eu(y, u; µ)∗p = 0,
+in U ∗,
+Lp(y, u, p; µ) = e(y, u; µ) = 0,
+in Q.
+
+4
+SUPG ROMS FOR ADVECTION-DOMINATED PDES UNDER OPTIMAL CONTROL
+In (3), the first equation is called the adjoint equation, the second one is the gradient equation
+and, as we have already seen, the state equation is the third one. We remark that we will always
+consider Linear-Quadratic problems.
+Definition 2.2.4 (Linear-Quadratic Problem). Consider a Banach space Z and α > 0. Let the
+Observation map O : Y → Z be a linear and bounded operator. Consider an element zd(µ) ∈ Z,
+which is the so-called desired solution profile (the desired observed output). Let J be a quadratic
+objective functional of the form
+(4)
+J(y, u; µ) = 1
+2m (Oy(µ) − zd(µ), Oy(µ) − zd(µ)) + α
+2 n(u(µ), u(µ)),
+where m : Z × Z → R and n : U × U → R are symmetric and continuous bilinear forms. Let e be
+affine, i.e. there exist A(µ) ∈ B(Y, Q), B(µ) ∈ B(U, Q) and f(µ) ∈ Q such that
+(5)
+e(y, u; µ) = A(µ)y + B(µ)u − f(µ),
+∀
+�
+y(µ), u(µ)
+�
+∈ Y × U.
+Then an OCP(µ)s with the above properties is said a Linear-Quadratic Optimal Control Problem.
+For Linear-Quadratic OCP(µ)s Proposition 2.2.3 implies that a solution (y, u) to Problem 2.1.1
+must satisfy, for some p ∈ Q∗ [12],
+(6)
+�
+�
+�
+�
+�
+m(Oy, O¯y; µ) + ⟨A∗(µ)p, ¯y⟩Y ∗Y = m (O¯y, zd; µ) ,
+∀¯y ∈ Y,
+αn(u, ¯u; µ) + ⟨B∗(µ)p, ¯u⟩U ∗U = 0,
+∀¯u ∈ U,
+⟨¯p, A(µ)y + B(µ)u⟩Q∗Q = ⟨¯p, f(µ)⟩Q∗Q,
+∀¯p ∈ Q∗.
+In this context, if (y, u, p) is a saddle point of L [56], then (y, u) minimizes J over all zeroes of
+e [12]. Moreover, under some precise hypotheses existence and uniqueness of a saddle point can be
+provided using Brezzi Theorem [9, 10, 12]. Therefore, a possible strategy to prove well-posedness
+of an Linear-Quadratic OCP(µ)s can be to demonstrate that a stationary point of (6) is a saddle
+point. At this purpose, System (6) can also be recast in a saddle-point structure [7, 12, 36]. In order
+to derive this structure, assume x ∈ X := Y × U. We define M(µ) ∈ B (Z, Z∗) , N(µ) ∈ B (U, U ∗)
+as the unique operators that satisfy the following relations:
+⟨M(µ)z, ¯z⟩Z∗Z = m(z, ¯z; µ),
+⟨N(µ)u, ¯u⟩U ∗U = n(u, ¯u; µ),
+∀z, ¯z ∈ Z, ∀u, ¯u ∈ U.
+This directly implies that m(Oy, O¯y; µ) = ⟨O∗M(µ)Oy, ¯y⟩Y ∗Y . Using Proposition (2.2.3) and a
+matrix notation as follows [12]:
+(7)
+E(µ) =
+� A(µ)
+B(µ) �
+,
+D(µ) =
+� O∗M(µ)O
+0
+0
+αN(µ)
+�
+,
+E∗(µ) =
+� A∗(µ)
+B∗(µ)
+�
+,
+defining also ¯g(µ) = O∗M(µ)zd, the optimality system (6) for Linear-Quadratic OCP(µ)s can be
+written in a more compact form as
+(8)
+� D(µ)
+E∗(µ)
+E(µ)
+0
+� � x
+p
+�
+=
+� ¯g(µ)
+f(µ)
+�
+in X∗,
+in Q.
+For Linear-Quadratic Problems, a saddle point of L is a stationary point [56], so it satisfies (6).
+For Linear-Quadratic problems the solution to system (8), and hence to (6), is a saddle point of L
+when D(µ) is self-adjoint [12]. In this case Brezzi Theorem gives us well-posedness [9, 10, 12].
+Lemma 2.2.5. [12] If Y is reflexive so that D(µ) = D∗(µ), then (x, p) = (y, u, p) is a saddle point
+of L if and only if it solves the system (8).
+Assumption 2.2.6. We assume that Y, U are reflexive, A(µ) is weakly coercive, the operator B(µ)
+is not null, αN(µ) is coercive with constant α > 0 and m(z, z; µ) ≥ 0, ∀z ∈ Z.
+Considering Linear-Quadratic OCP(µ)s and Assumption 2.2.6, it follows that E(µ) is inf-sup
+stable and D(µ) is coercive over the kernel of E(µ). Consequently, the well-posedness of the system
+(8) is assured by Theorem 2.2.7.
+Theorem 2.2.7 (Brezzi). [9, 10, 12] Let X be a reflexive Banach space. Then the equivalence of
+the following statements holds:
+
+SUPG ROMS FOR ADVECTION-DOMINATED PDES UNDER OPTIMAL CONTROL
+5
+(1) D(µ) ∈ B (X, X∗) , E(µ) ∈ B(X, Q) with the following properties:
+• D(µ) is weakly coercive over the kernel of E(µ),
+• E(µ) is inf-sup stable.
+(2) The system (8) has a unique solution (x, p) ∈ X × Q∗ for all ¯g(µ) ∈ X∗, f(µ) ∈ Q, which
+satisfies for some constant C > 0 ∥x∥X + ∥p∥Q∗ ≤ C (∥¯g(µ)∥X∗ + ∥f(µ)||Q) .
+(3) The operator S(µ) :=
+�
+D(µ)
+E∗(µ)
+E(µ)
+0
+�
+is an isomorphism in X∗ × Q.
+Remark 2.2.8 (Notation). From now on, we will always involve Hilbert spaces. For the sake of
+notation, there we will denote the various bilinear forms defined by A(µ), B(µ) and their adjoints
+ones in the following unique way:
+⟨A(µ)y, p⟩QQ∗ := a(y, p; µ)
+⟨B(µ)u, p⟩QQ∗ := b(u, p; µ).
+2.3. Unsteady Problems. We briefly recall results on well-posedness for time-dependent Linear-
+Quadratic OCP(µ)s based on [50, 51]. We consider saddle-point formulation in order to prove well-
+posedness by using tools of the previous Sections in the case of null initial conditions. Differently
+from the steady case, here we will make some more technical assumptions, which will be fulfilled by
+both “Graetz-Poiseuille” and “Propagating Front in a Square” problems.
+Consider two separable Hilbert spaces Y and H satisfying Y �→ H �→ Y ∗ and, moreover, other
+two Hilbert spaces U and Z ⊇ Y , where Y and U are the usual state and control spaces, and Z is
+the space of observation. We endow them with the standard norms inherited from their respectively
+scalar products: (·, ·)Y , (·, ·)Z, (·, ·)U and (·, ·)H. We define the following Hilbert spaces:
+Y = L2(0, T; Y ),
+Y∗ = L2 (0, T; Y ∗) ,
+U = L2(0, T; U)
+Z := L2(0, T; Z) ⊇ Y.
+with respective norms, for instance in the case of Y and U given by
+(9)
+∥y∥2
+Y :=
+T
+�
+0
+∥y∥2
+Y dt,
+and
+∥u∥2
+U :=
+T
+�
+0
+∥u∥2
+Udt
+and similarly for the others. Furthermore, let us define the Hilbert space Yt with its scalar product
+(·, ·)Yt:
+Yt :=
+�
+y ∈ Y
+s.t.
+∂y
+∂t ∈ Y∗
+�
+,
+(y, z)Yt :=
+T
+�
+0
+(y, z)Y dt +
+T
+�
+0
+�∂y
+∂t , ∂z
+∂t
+�
+Y ∗dt.
+Our aim is to solve the following unconstrained Linear-Quadratic Parametric Parabolic OCP(µ):
+Problem 2.3.1 (Parametric Parabolic OCP(µ)). For a given µ ∈ P find the pair (y(µ), u(µ)) ∈
+Yt × U that satisfies
+(10)
+�
+�
+�
+�
+�
+�
+�
+�
+�
+�
+�
+�
+�
+�
+�
+∂y(µ)
+∂t
++ A(µ)y(µ) + B(µ)u(µ) − f(µ) = 0,
+in Ω × (0, T),
+∂y(µ)
+∂n
+= 0,
+on ΓN × (0, T),
+y(µ) = l,
+on ΓD × (0, T),
+y(µ)(0) = y0,
+in Ω,
+and minimizes
+min
+(y(µ),u(µ))∈Yt×U J(y, u; µ) = 1
+2m (Oy(µ) − zd(µ), Oy(µ) − zd(µ)) + α
+2 n(u(µ), u(µ)),
+where m : Yt × Yt → R and n : U × U → R are symmetric and continuous bilinear forms, zd(µ) ∈ Z
+is the observed desired solution profile and α > 0 is the fixed penalization parameter. In our test
+case we will always take y0 ≡ 0.
+
+6
+SUPG ROMS FOR ADVECTION-DOMINATED PDES UNDER OPTIMAL CONTROL
+Also in this case, we denote y := y(µ) and u := u(µ) omitting the parameter dependence. We
+can state the weak formulation of (10) as
+�
+�
+�
+�
+�
+�
+�
+T
+�
+0
+�∂y
+∂t , q
+�
+Y∗Y
+dt +
+T
+�
+0
+⟨A(µ)y, q⟩Y∗Ydt +
+T
+�
+0
+⟨B(µ)u, q⟩Y∗Ydt −
+T
+�
+0
+⟨f(µ), q⟩Y ∗Y dt = 0,
+∀q ∈ Yt,
+y(0) = y0,
+in Ω,
+where f(µ) ∈ Y∗ gathers all forcing, boundary and, eventually, lifting terms of the state equation.
+Nevertheless, for the sake of notation, we will consider a : Yt × Yt → R and b : U × Yt → R the
+bilinear forms defined as a(y, q; µ) = ⟨A(µ)y, q⟩Y∗Y and b(u, q; µ) = ⟨B(µ)u, q⟩Y∗Y, respectively.
+For a proper definition of the adjoint variable, it is opportune to take q ∈ Yt rather than q ∈ Y [50].
+Let us define the state-control product space X = Yt × U. Then we define the operators E,D and ¯g
+similarly as made in the steady case in order to make the formulation more compact [50]:
+(11)
+D(µ) : X × X → R,
+D(x, ¯x, µ) =m(Oy, O¯y; µ) + αn(u, ¯u; µ);
+E(µ) : X × Yt → R,
+E(x, q, µ) =
+T
+�
+0
+�∂y
+∂t , q
+�
+Y∗Y
+dt +
+T
+�
+0
+a(y, q, µ)dt +
+T
+�
+0
+b(u, q, µ)dt;
+¯g(µ) ∈ X ∗,
+T
+�
+0
+⟨¯g(µ), ¯x⟩dt =m (O¯y, zd(µ)) .
+Denoting p := p(µ) and considering Q∗ = Yt [50], the Lagrangian and objective functionals are,
+respectively:
+(12) L(x, p; µ) = J(x; µ)+E(x, p; µ)−
+T
+�
+0
+⟨f(µ), p⟩Y ∗Y dt,
+J(x, µ) = 1
+2D(x, x; µ)−
+T
+�
+0
+⟨¯g(µ), x⟩dt.
+As made in the steady case, the minimization of Problem 2.3.1 means to seek the solution of the
+following system: given µ ∈ D, find (y, u, p) = (x, p) ∈ X × Yt which solve
+(13)
+�
+�
+�
+�
+�
+Ly(y, u, p; µ)[¯y] = 0,
+∀¯y ∈ Yt,
+Lu(y, u, p; µ)[¯u] = 0,
+∀¯u ∈ U,
+Lp(y, u, p; µ)[¯p] = 0,
+∀¯p ∈ Yt,
+and satisfy boundary and initial conditions in Problem 2.3.1 with p(T) = 0 [30]. The saddle-point
+structure of steady Linear-Quadratic OCP(µ)s (8) can be derived in the parabolic case, too (here
+expressed in the weak formulation) [50]:
+(14)
+�
+�
+�
+�
+�
+�
+�
+�
+�
+�
+�
+�
+�
+�
+�
+�
+�
+D(x, w; µ) + E(w, p; µ) =
+T
+�
+0
+⟨¯g(µ), w⟩dt,
+∀w ∈ X,
+E(x, q; µ) =
+T
+�
+0
+⟨f(µ), q⟩Y ∗Y dt,
+∀q ∈ Yt.
+The equivalence between the optimality system and saddle-point formulation for Linear-Quadratic
+Parabolic OCP(µ)s is straighforward. For well-posedness the following assumption is needed [50].
+Assumption 2.3.2. The bilinear forms n(·, · ; µ), m(·, · ; µ), b(·, · ; µ), and a(·, · ; µ) satisfy the
+following features:
+(1) m(·, · ; µ) is positive definite, continuous, and symmetric.
+(2) n(·, · ; µ) is coercive, continuous, and symmetric;
+(3) there exists Ca > 0 s.t. a(w, w; µ) ≥ Ca(µ)∥w∥2
+Y ,
+∀w ∈ Yt;
+(4) there exists ca > 0 s.t. |a(w, p; µ)| ≤ ca(µ)∥w∥Y ∥p∥Y ,
+∀w, p ∈ Yt;
+
+SUPG ROMS FOR ADVECTION-DOMINATED PDES UNDER OPTIMAL CONTROL
+7
+(5) there exists cb > 0 s.t. |b(v, p; µ)| ≤ cb(µ)∥v∥U∥p∥Y ,
+∀v ∈ U and ∀p ∈ Yt;
+Finally, one can prove the well-posedness of Problem 2.3.1 (for more details, we refer to [50]).
+Theorem 2.3.3 (Well-posedness of Parabolic OCP(µ)s). [50] Under Assumption 2.3.2 the saddle-
+point formulation (14) satisfies the hypothesis (1) of Theorem 2.2.7, hence the solution is unique.
+Assumption 2.3.4. For both steady and unsteady problems, we will consider the Identity operator
+restricted to our observation domain Ωobs as the Observation function O. Therefore, Z = Y is
+assumed and our desired state will be denoted by yd.
+3. Truth Discretization
+In this Section we firstly pursue a numerical method for the solution of an OCP: a discretization
+of the optimality sistem (6) will be given following an one shot or all-at-once approach [23, 46, 47].
+Secondly, we will consider SUPG stabilization for Advection-Dominated equations in case of high
+P´eclet number. An optimize-then-discretize approach is followed, i.e. at first we derive optimality
+conditions as system (6) and then we discretize it. Therefore, we obtain a discretized system:
+(15)
+�
+�
+�
+�
+�
+LyN (yN , uN , pN ) = JyN (yN , uN ) + eyN (yN , uN )∗pN = 0
+in
+�
+Y N �∗
+LuN (yN , uN , pN ) = JuN (yN , uN ) + euN (yN , uN )∗pN = 0
+in
+�
+U N �∗
+LpN (yN , uN , pN ) = e(yN , uN ) = 0
+in QN ,
+where LyN , LuN , LpN are the discretizations of partial derivatives of L and Y N , U N , QN are the
+approximation of Y, U, Q, respectively.
+Let us start our discussion from the steady case. From now on we will always assume to work with
+Y, U, Q Hilbert spaces. We employ a FEM discretization, which will be named as the high-fidelity
+or truth approximation. We consider Ωh as a quasi-uniform mesh on the domain Ω, for which the
+parameter h indicates the mesh size, i.e. maximum diameter of an element of the grid. Th is a
+regular triangularization on Ω and
+Ωh := int
+� �
+K∈Th
+K
+�
+,
+where K is a triangle of Th. We define the FEM spaces Y N = Y ∩ XN ,r, U N = U ∩ XN ,r and
+�
+QN �∗ = Q∗ ∩ XN ,r, where
+XN ,r =
+�
+vN ∈ C0(¯Ω) : vN
+|K ∈ Pr(K), ∀K ∈ Th
+�
+and Pr(K) represents the space of polynomials of degree at most equal to r defined on a triangle K.
+As we will remark later, we will always use the same triangulation Th and a P1-FEM approximation
+for state, control and adjoint variables. The dimensions of Y N , U N , QN are all equal to N. The
+overall dimension of the discrete problem is Ntot = 3 · N. For the sake of simplicity, we assume
+Q∗
+h = Y N . Moreover, we indicate with XN = Y N × U N ⊂ X. From now on we will refer to the
+same symbol yd to also indicate the FEM discretization version of the desired state.
+The discretization of a Linear-Quadratic OCP of Problem 2.1.1 reads as
+min
+(yN ,uN )∈Y N ×U N J
+�
+yN , uN �
+= 1
+2m
+�
+yN − yd, yN − yd
+�
++ α
+2 n(uN , uN ) s.t. e
+�
+yN , uN �
+= 0.
+Moreover, the operators m and n will be the L2 product on Ωobs and on Ω, respectively.
+For the saddle-point system, we define the operators ¯gN : XN → R, f N : Y N → R, DN : XN →
+�
+XN �∗ , and EN : XN →
+�
+QN �∗ as just the usual restrictions
+(16)
+�
+¯gN , ¯xN �
+(XN )∗XN =
+�
+¯g, ¯xN �
+X∗X ,
+�
+DN xN , ¯xN �
+(XN)∗XN =
+�
+DxN , ¯xN �
+X∗X ,
+⟨f N , ¯pN ⟩(Y N )∗Y N =
+�
+f, ¯pN �
+Q∗∗Q∗ ,
+�
+EN xN , ¯pN �
+(Y N )∗Y N =
+�
+ExN , ¯pN �
+Q∗∗Q∗ ,
+for all xN ∈ XN , ¯pN ∈ Y N .
+
+8
+SUPG ROMS FOR ADVECTION-DOMINATED PDES UNDER OPTIMAL CONTROL
+We highlight the algebraic structure of the discretize optimality system. We define the basis of
+the finite spaces XN and Y N as below:
+(17)
+�
+ϕj ∈ XN �2N
+j=1 ,
+�
+ψk ∈ Y N �N
+k=1 .
+As a result, we can rewrite a pair
+�
+xN , pN �
+∈ XN × Y N in the following way:
+�
+�xN =
+2N
+�
+j=1
+xjϕj,
+pN =
+N
+�
+k=1
+pkψk
+�
+� .
+Therefore, we can define D ∈ R2N ·2N , E ∈ RN ·2N , ¯g ∈ R2N and f ∈ RN as follows:
+(18)
+Dij = ⟨DN ϕi, ϕj⟩(XN )∗XN ,
+Elm = ⟨EN ϕl, ψm⟩(Y N )∗Y N ,
+¯gk =
+�
+¯gN , ϕk
+�
+(XN )∗XN ,
+fn =
+�
+f N , ψn
+�
+(Y N )∗Y N .
+Finally, we can build the following saddle point system, with a block structure:
+(19)
+�
+D
+ET
+E
+0
+� � x
+p
+�
+=
+� ¯g
+f
+�
+,
+where (x)i = xi, i = 1, · · · 2N and (p)k = pk, k = 1, · · · N. For this purpose, let us denote with y,
+u and p the vectors of coefficients of yN , uN and pN , expressed in terms of the nodal basis (17)
+by splitting components of XN in those of Y N and U N . We express with yd the vector with the
+components of the discretized desired state, i.e. the Galerkin projection of yd on Y N . Moreover,
+let us indicate the stiffness matrix derived from the bilinear form a(·, ·) with K, KT is the stiffness
+matrix related to a∗ and the mass matrix is denoted with M. In addition, we call B, BT is the mass
+matrix related to the forms b and b∗. We have that:
+D =
+� M
+0
+0
+αM
+�
+,
+E =
+� K
+B �
+,
+x =
+� y
+u
+�
+,
+¯g =
+� Myd
+0
+�
+.
+and the optimality system shows this block structure:
+(20)
+�
+�
+M
+0
+KT
+0
+αM
+BT
+K
+B
+0
+�
+�
+�
+�
+y
+u
+p
+�
+� =
+�
+�
+Myd
+0
+f
+�
+� .
+3.1. SUPG stabilization for Advection-Dominated OCP(µ)s. In this Section we illustrate
+Advection-Dominated OCP(µ)s and the SUPG technique applied to an optimize-then-discretize
+approach. From now, we recall the dependence on parameters of our operators. Let us start from
+our definition of an Advection-Diffusion equation.
+Definition 3.1.1 (Advection-Diffusion Equations). Let us consider the following problem:
+(21)
+L(µ)y := −ε(µ)∆y + b(µ) · ∇y = f(µ) in Ω ⊂ R2,
+with suitable boundary conditions on ∂Ω. Let us suppose that:
+• the diffusion coefficient ε : Ω → R belongs to L∞(Ω) and depends on the parameter µ. We
+assume there exists a constant ¯ε > 0 such that ε(x) ≥ ¯ε, ∀x ∈ Ω;
+• the advection field b : Ω → R2 belongs to (L∞(Ω))2 and depends on the parameter µ. We
+suppose that 0 ≥ div b(x) ≥ −˜k, holds for all x ∈ Ω, with ˜k ∈ R+
+0 ;
+• f(µ) : Ω → R is an L2(Ω)-function that can depend on the parameter µ.
+In this case, (21) is an Advection-Diffusion problem and the operator L(µ)y := −ε(µ)∆y +b(µ)·
+∇y is said the Advection-Diffusion operator.
+From (21), we can easily derive the weak formulation of an Advection-Diffusion problem:
+(22)
+find y ∈ Y s.t. a (y, q; µ) = F (q; µ)
+∀q ∈ Q∗,
+
+SUPG ROMS FOR ADVECTION-DOMINATED PDES UNDER OPTIMAL CONTROL
+9
+where
+(23)
+a (y, q; µ) :=
+�
+Ω
+ε(µ)∇y∇q + b(µ) · ∇yq dx,
+F(q; µ) :=
+�
+Ω
+f(µ)q dx,
+y ∈ Y, q ∈ Q∗.
+From a numerical point of view, when the advection term b(µ) · ∇u “dominates” the diffusive
+one −ε(µ)∆u, i.e. when |b(µ)| ≫ ε(µ), the approximated solution can show instability phenomena
+along the direction of the advection field [42]. In order to give an indicator of the instability, let us
+consider the regular triangulation Th related to FEM discretization. For any element K ∈ Th, we
+can then define the local P´eclet number as [42, 38]:
+(24)
+PeK(x) := |b(x)|hK
+2ε(x)
+,
+∀x ∈ K,
+where hK is the diameter of K.
+Definition 3.1.2 (Advection-Dominated problem). Considering Definition 3.1.1 we are dealing
+with an Advection-Dominated problem if PeK(x) > 1, ∀x ∈ K, ∀K ∈ Th.
+To solve the issue of the instability, we will exploit the SUPG method [11, 25, 26, 42], which is a
+strongly consistent stabilization technique; i.e. is consistent for weak PDEs and its order of accuracy
+can be greater than one. Let us now consider the Advection-Diffusion operator (21): for the sake of
+simplicity, we define it on H1
+0(Ω) and we do not indicate the parameter dependence. The operator
+L can be split into its symmetric and skew-symmetric parts [42], defined as:
+(25)
+symmetric part: LSy = −ε∆y − 1
+2(div b)y,
+∀y ∈ H1
+0(Ω),
+skew-symmetric part: LSSy = b · ∇y + 1
+2(div b)y,
+∀y ∈ H1
+0(Ω),
+i.e. L = LS +LSS. Symmetric and skew-symmetric parts can be directly derived using the formulae:
+(26)
+LS = L + L∗
+2
+,
+LSS = L − L∗
+2
+,
+where L∗ is the adjoint operator related to L.
+Now, let us analyze our OCP problem (6): we follow the optimize-then-discretize approach in
+[14]. The discretized state equation is described as follows, where the control is distributed, i.e. it
+acts on the whole domain Ω:
+(27)
+as
+�
+yN , qN �
++ bs
+�
+uN , qN �
+= Fs(qN ),
+∀qN ∈
+�
+QN �∗ ,
+with
+(28)
+as
+�
+yN , qN �
+:= a
+�
+yN , qN �
++
+�
+K∈Th
+δK
+�
+LyN , hK
+|b| LSSqN
+�
+K
+,
+(29)
+bs
+�
+uN , qN �
+:= −
+�
+Ω
+uN qN −
+�
+K∈Th
+δK
+�
+uN , hK
+|b| LSSqN
+�
+K
+,
+and
+(30)
+Fs(qN ) := F
+�
+qN �
++
+�
+K∈Th
+δK
+�
+f, hK
+|b| LSSqN
+�
+K
+,
+where
+�
+·, ·
+�
+K indicates the usual L2(K)-product, f collects all the forcing and lifting terms, and δK
+denotes a positive dimensionless stabilization parameter related to an element K ∈ Th. In principle,
+since δK is local, it can be different for each K. Considering the adjoint equation, we can see that
+it is also an Advection-Dominated equation, but with an advective term with opposite sign with
+respect to the state one. As a matter of fact, from (26) we obtain that L∗ = LS − LSS. The SUPG
+method leads to the discretized adjoint equation
+(31)
+a∗
+s
+�
+zN , pN �
++
+�
+yN − yd, zN �
+s = 0,
+∀zN ∈ Y N ,
+
+10
+SUPG ROMS FOR ADVECTION-DOMINATED PDES UNDER OPTIMAL CONTROL
+with
+(32)
+a∗
+s
+�
+zN , pN �
+:= a∗ �
+zN , pN �
++
+�
+K∈Th
+δa
+K
+�
+(LS − LSS)pN , hK
+|b| (−LSS) zN
+�
+K
+,
+�
+yN − yd, zN �
+s :=
+�
+Ωobs
+(yN − yd)zN dx +
+�
+K∈Th|Ωobs
+δa
+K
+�
+yN − yd, hK
+|b| (−LSS) zN
+�
+K
+,
+where a∗ is the adjoint form of a and δa
+K is the parameter related to the stabilized adjoint bilinear
+forms. As in this work we consider δK = δa
+K in numerical simulations, from now on we will always
+denote both stabilization parameter with δK. Instead, the discretized gradient equation is not affected
+by the SUPG and it remains untouched:
+(33)
+b∗�
+vN , pN �
++ αn
+�
+uN , vN �
+= 0,
+∀vN ∈ U N .
+With this setting it follows a nonsymmetric system for the computation of the numerical solution,
+but we gain the strongly-consistency of the method for the optimality system if y, u, p are regular
+[14]. To summarize, the SUPG optimality system for a steady OCP is the following:
+(34)
+discretized adjoint equation:
+a∗
+s
+�
+zN , pN �
++
+�
+yN − yd, zN �
+s = 0,
+∀zN ∈ Y N ,
+discretized gradient equation:
+b∗�
+vN , pN �
++ αn
+�
+uN , vN �
+= 0,
+∀vN ∈ U N ,
+discretized state equation:
+as
+�
+yN , qN �
++ bs
+�
+uN , qN �
+= Fs(qN ),
+∀qN ∈
+�
+QN �∗ ,
+and, referring to (20), the discretized algebraic system reads as:
+(35)
+�
+�
+Ms
+0
+KT
+s
+0
+αM
+BT
+Ks
+Bs
+0
+�
+�
+�
+�
+y
+u
+p
+�
+� =
+�
+�
+Msyd
+0
+fs
+�
+� ,
+where Ms is the stabilized mass matrix related to m, M is the not-stabilized mass matrix related
+to n, Ks and KT
+s are the stiffness matrices related to as and a∗
+s, respectively, Bs is the stabilized
+mass matrix related to bs, BT is the block linked to b∗ and fs is the vector whose components are
+the coefficients of the stabilized force term. Every block is derived as in (18).
+We indicate with |∥ · ∥| the energy norm related to the bilinear form a belonging to Advection-
+Diffusion equations (3.1.1), i.e.
+(36)
+|∥w∥|2 := ε∥∇w∥2
+L2(Ω) + 1
+2
+���(div b)
+1
+2 w
+���
+2
+L2(Ω) ,
+∀w ∈ H1
+0(Ω).
+Therefore, we define the SUPG norm on H1
+0(Ω) as
+(37)
+∥w∥2
+SUP G := |∥w∥|2 +
+�
+K∈Th
+δK
+�
+LSSw, hK
+|b| LSSw
+�
+K
+,
+∀w ∈ H1
+0(Ω).
+Considering that (38) holds true, it is immediate to see that the SUPG bilinear form (28) is coercive
+with respect to the SUPG norm [42]. Finally, we can illustrate an estimate of the error for the
+adjoint and the state variables of the solution of an OCP [14].
+Theorem 3.1.3 (Error for state and adjoint variables). Let m, r ≥ 1 and (y, u, p) be the solution
+of (6) with y ∈ Hm+1(Ω), p ∈ Hr+1(Ω). Furthermore, let yN , uN , pN be the numerical solution of
+(34). If δK satisfies
+(38)
+0 < δK ≤ hK
+εη2
+inv
+and
+δK =
+�
+�
+�
+δ1
+hK
+ε ,
+PeK(x) ≤ 1,
+δ2,
+PeK(x) > 1,
+where δ1, δ2 > 0 are chosen constant, and ηinv is defined as the following inverse constant
+|yN |1,K ≤ ηinvh−1
+K ∥yN ∥L2(K)
+and
+∥∆yN ∥L2(K) ≤ ηinvh−1
+K ∥∇yN ∥L2(K)
+∀yN ∈ Y N ,
+
+SUPG ROMS FOR ADVECTION-DOMINATED PDES UNDER OPTIMAL CONTROL
+11
+with | · |1,K, ∥ · ∥K seminorm and L2-norm on K, respectively, then there exists C > 0 such that
+(39)
+��y − yN ��
+SUP G ≤ C
+�
+hm �
+ε1/2 + h1/2�
+|y|Hm+1(Ω) +
+��uN − u
+��
+L2(Ω)
+�
+,
+∀h,
+��p − pN ��
+SUP G ≤ C
+�
+hr �
+ε1/2 + h1/2�
+|p|Hr+1(Ω) +
+��yN − y
+��
+L2(Ω)
+�
+,
+∀h.
+3.2. SUPG for Time-Dependent Advection-Dominated OCP(µ)s. We briefly discuss the
+SUPG technique employed with time-dependent problems. Referring to (13), the main challenge
+comes from the fact that the time derivative should also enter into stabilization framework to ensure
+consistency [27]. However, other approaches have been proposed: in [45], for instance, the time-
+derivative is not stabilized. Nevertheless, our discussion follows works inherent to Graetz-Poiseuille
+and Propagating Front in a Square problems without optimal control [37, 54], where stabilization
+is used for time derivative, too. This adds nonsymmetric terms to the discretized state and adjoint
+equations for time derivatives.
+To the best of our knowledge SUPG for Parabolic OCPs in an
+optimize-then-discretized approach is still a novelty element in literature from a theoretical point of
+view. However, we refer to [17, 20, 27] for SUPG applied to general Parabolic equations.
+We firstly discretize the equation in time, considering each discrete time as a steady-state Advection-
+Diffusion equation, in a space-time approach, and then stabilized it with the SUPG. The time interval
+(0, T) is divided in Nt sub-intervals of equal length ∆t := ti − ti−1, i ∈ {1, . . . , Nt}. On the other
+hand, all terms involving time-derivative go through a time discretization equivalent to a classical
+implicit Euler approach [3, 23, 46, 50, 51, 52]. The backward Euler method is used to discretize the
+state equation forward in time, instead the adjoint equation is discretized backward in time using
+the forward Euler method, which is equivalent to the backward Euler with respect to time T − t, for
+t ∈ (0, T) [16, 50]. The global dimension of the discrete spaces is Ntot = 3 · N · Nt. We recall that
+Y, U, Q are Hilbert Spaces and that Y N ≡ (QN )∗.
+For the state equation, the stabilized term added to the form related to the time derivative of the
+state ∂y
+∂t and the bilinear form a is the following [27, 37, 54]:
+s
+�
+yN (t), qN �
+=
+�
+K∈Th
+δK
+�∂yN (t)
+∂t
++ (LS + LSS) yN (t), hK
+|b| LSSqN
+�
+K
+,
+where yN (t) ∈ Y N for each t ∈ (0, T) and qN ∈ Y N . Instead, the stabilized term added to the form
+related to the time derivative of the adjoint ∂p
+∂t and the bilinear form a∗ is:
+s∗ �
+zN , pN (t)
+�
+=
+�
+K∈Th
+δK
+�
+−∂pN (t)
+∂t
++ (LS − LSS) pN (t), −hK
+|b| LSSzN
+�
+K
+.
+We can write the discretized state formulation using a backward Euler approach as follows:
+(40)
+for each i ∈ {1, 2, · · · , Nt}, find yN
+i
+∈ Y N s.t. ∀qN ∈ Y N ,
+1
+∆tms
+�
+yN
+i (µ) − yN
+i−1(µ), qN ; µ
+�
++ as
+�
+yN
+i (µ), qN ; µ
+�
++ bs
+�
+uN
+i , qN ; µ
+�
+= Fs
+�
+qN ; µ
+�
+,
+given the initial condition yN
+0 which satisfies
+(41)
+�
+yN
+0 , qN �
+L2(Ω) =
+�
+y0, qN �
+L2(Ω) ,
+∀qN ∈ Y N .
+The stabilized term ms above is defined as:
+(42)
+ms
+�
+yN , qN ; µ
+�
+=
+�
+yN , qN �
+L2(Ω) +
+�
+K∈Th
+δK
+�
+yN , hK
+|b| LSSqN
+�
+K
+and it is related to the time discretization; instead, as and Fs are defined as in the steady case.
+Similarly we can derive the same for the adjoint forms applying a forward Euler method:
+(43)
+for each i ∈ {Nt − 1, Nt − 2, ..., 1}, find pN
+i
+∈ Y N s.t.
+1
+∆tm∗
+s
+�
+pN
+i (µ) − pN
+i+1(µ), zN ; µ
+�
++ a∗
+s
+�
+zN , pN
+i (µ); µ
+�
+= −
+�
+yN
+i − ydi, zN �
+s
+∀zN ∈ Y N .
+
+12
+SUPG ROMS FOR ADVECTION-DOMINATED PDES UNDER OPTIMAL CONTROL
+The stabilized term m∗
+s above is defined as:
+(44)
+m∗
+s
+�
+pN , zN ; µ
+�
+=
+�
+pN , zN �
+L2(Ω) −
+�
+K∈Th
+δK
+�
+pN , hK
+|b| LSSzN
+�
+K
+.
+Now we give a look at the discretization scheme. As in the steady case, yi ∈ Y N , ui ∈ U N and
+pi ∈ Y N , for 1 ≤ i ≤ Nt, represent the column vectors including the coefficients of the FEM dis-
+cretization for state, control and adjoint, respectively. Therefore, we define y =
+�
+yT
+1 , . . . , yT
+Nt
+�T ,
+u =
+�
+uT
+1 , . . . , uT
+Nt
+�T and p =
+�
+pT
+1 , . . . , pT
+Nt
+�T . The vector f s =
+�
+f T
+s1, . . . , f T
+sNt
+�T
+indicates the com-
+ponents of the stabilized forcing term, yd =
+�
+yT
+d1, . . . , yT
+dNt
+�T
+is the vector made of discrete time
+components of our desired state solution; instead, y0 =
+�
+yT
+0 , 0T , . . . , 0T �T indicates the vector of ini-
+tial condition for the state, where 0 is the zero vector in RN . The block matrix system is described
+as follows.
+• State equation.
+We recall that Ks and Bs are the matrices associated to the stabilized
+bilinear forms as and bs. Using the backward Euler along time, one has to solve
+(45)
+Msyi + ∆tKsyi + ∆tBsui = Msyi−1 + fsi∆t
+for i ∈ {1, 2, . . . , Nt} ,
+where Ms is the stabilized mass matrix relative to the FEM discretization of ms. Therefore
+the related block matrix subsystem is
+�
+����
+Ms + ∆tKs
+0
+−Ms
+Ms + ∆tKs
+0
+...
+...
+0
+0
+−Ms
+Ms + ∆tKs
+���
+����
+�
+��
+�
+As
+y+∆t
+�
+��
+Bs
+0
+0
+...
+0
+0
+Bs
+�
+��
+�
+��
+�
+Cs
+u = Msy0 + ∆tf s,
+where Ms is a block diagonal matrix in RN ·Nt ×RN ·Nt whose element on the main diagonal
+are [Ms, . . . , Ms]. Then everything can be recast in a more compact form as
+(46)
+Asy+∆tCsu = Msy0 + ∆tf s.
+• Gradient equation. We recall that BT indicates the mass matrix related to the b∗ form and
+hence at every time step we have to solve the equation
+(47)
+α∆tMui+∆tBT pi = 0,
+∀i ∈ {1, 2, . . . , Nt} ,
+which translates into the following block system:
+∆t · α
+�
+����
+M
+M
+...
+...
+M
+�
+����
+�
+��
+�
+M
+�
+����
+u1
+u2
+...
+uNt
+�
+���� +∆t
+�
+����
+BT
+0
+· · ·
+BT
+...
+BT
+�
+����
+�
+��
+�
+CT
+�
+����
+p1
+p2
+...
+pNt
+�
+���� =
+�
+����
+0
+0
+...
+0
+�
+���� .
+In a vector notation we have
+(48)
+α∆tMu+∆tCT p = 0.
+• Adjoint equation: we have to solve the first equation of the optimality system (6) at each
+time step as follows, considering M T
+s the matrix formulation of m∗
+s:
+M T
+s pi = M T
+s pi+1 + ∆t
+�
+−M T
+s yi − KT
+s pi + M T
+s ydi
+�
+for i ∈ {Nt − 1, Nt − 2, . . . , 1} .
+
+SUPG ROMS FOR ADVECTION-DOMINATED PDES UNDER OPTIMAL CONTROL
+13
+As did in previous steps, we derive the following block system:
+�
+����
+M T
+s + ∆tKT
+s
+−M T
+s
+...
+...
+M T
+s + ∆tKT
+s
+−M T
+s
+M T
+s + ∆tKT
+s
+�
+����
+�
+��
+�
+AT
+s
+p +
+�
+�����
+∆tM T
+s y1
+...
+...
+∆tM T
+s yNt
+�
+�����
+=
+�
+�����
+∆tM T
+s yd1
+...
+...
+∆tM T
+s ydNt
+�
+�����
+.
+Then, defining MT
+s as the diagonal matrix in RN ·Nt × RN ·Nt which diagonal entries are
+[M T
+s , . . . , M T
+s ], the adjoint system to be solved is:
+∆tMT
+s y + AT
+s p = ∆tMT
+s yd.
+In the end, we solve system (49) via an one shot approach:
+(49)
+�
+�
+∆tMT
+s
+0
+AT
+s
+0
+α∆tM
+∆tCT
+As
+∆tCs
+0
+�
+�
+�
+�
+y
+u
+p
+�
+� =
+�
+�
+∆tMT
+s yd
+0
+Msy0 + ∆tf s
+�
+� .
+4. ROMs for advection-dominated OCP(µ)s
+FEM simulations can be expensive in terms of computational time and memory storage: this issue
+is obviously more evident in case of high-dimensional discrete spaces. Moreover, when we talk about
+parametrized PDEs, one can require to repeat the simulations for several values of the parameter µ.
+To overcome these difficulties, we will use ROMs approach. The basic idea of ROMs is to create a
+low-dimensional space, called the reduced space, exploiting the parameter dependence of the problem
+at hand, such that it is a good approximation of the discrete initial space [8, 22, 41, 40, 39]. Let us
+consider a generic Parametrized OCPs described by the optimality conditions (6). We can define
+the set of the parametric solutions of the optimality system with respect to the functional space
+W = Y × U × Q∗ for steady OCP(µ)s and W = Yt × U × Yt for the unsteady ones as
+(50)
+M := {(y(µ), u(µ), p(µ)) solution of (6) | µ ∈ P}.
+The extension to space-time formulation for time-dependent problem is straightforward [6, 50] and
+requires small modifications, thus, we will exclusively refer to the steady framework.
+Assumption 4.0.1 (Smoothness of the solution manifold). The continuous solution manifold M is
+smooth with respect to the parameter µ ∈ P.
+Let WN ⊂ W be our FEM approximation of the continuous space W, we call WN := Y N ×
+U N ×
+�
+QN �∗ the high-fidelity space. Then, for stabilized problems we define the discrete parametric
+solution manifold as
+(51)
+MN :=
+��
+yN (µ), uN (µ), pN (µ)
+�
+FEM solution of the (35) | µ ∈ P
+�
+.
+Starting from MN , ROM techniques create a reduced space of low dimension N denoted with
+WN, via a linear combination of snapshots, i.e. high-fidelity evaluations of the optimal solution
+�
+yN (µ), uN (µ), pN (µ)
+�
+computed in properly chosen parameters values µ. Obviously we have that
+WN ⊂ WN and we denote WN = Y N × U N ×
+�
+QN�∗. Here, Y N, U N and (QN)∗ are the reduced
+spaces for the state, the control and the adjoint variables, respectively. The snapshots are collected by
+a POD algorithm using a partitioned approach. This strategy is followed due to good results shown
+in literature [28, 35, 49]. After having built these reduced function spaces, a standard Galerkin
+projection is performed onto these ones [5, 38, 42].
+
+14
+SUPG ROMS FOR ADVECTION-DOMINATED PDES UNDER OPTIMAL CONTROL
+4.1. Offline-Online Procedure for ROMs. ROM procedure is divided in two stages:
+• offline phase: here the snapshots are collected by solving the high-fidelity system (35).
+Secondly, the low-dimensional bases are created and hence all reduced spaces Y N, U N and
+(QN)∗ are built and stored, too. Moreover, all the µ-independent quantities are assembled
+and stored. It is potentially an expensive phase, which depends on N.
+• online phase: here a parameter µ is chosen and all the previous store µ-independent quan-
+tities are combined with the just-computed µ-dependent ones to build the reduced block
+matrix system based on a Galerkin projection.
+To be convenient, this phase should be
+N-independent. Whereas in the offline phase stabilization is present due to stabilized snap-
+shots, for the online phase this cannot be necessary. Therefore, we have two possibilities: if
+stabilization is performed also here, we talk about Online-Offline stabilization, otherwise we
+denote the setting as Only-Offline stabilization.
+As already said, the online phase should be performed in a number of operations independent
+of N. A sufficient condition is to admit the separation of the variables depending on µ and the
+solution (y, u, p) in the affine decomposition [22].
+Assumption 4.1.1. We require that all the forms in (35) are affine in µ ∈ P.
+In Section 4.2 we describe the POD algorithm used in the offline phase. Now, we illustrate the
+explicit expression of the reduced solutions. Let us make clear the structure of the three reduced
+spaces in terms of their bases. Therefore, we define
+(52)
+Y N = span {ηy
+n, n = 1, . . . , N},
+U N = span {ηu
+n, n = 1, . . . , N} ,
+(QN)∗ = span {ηp
+n, n = 1, . . . , N} ,
+the reduced state, the reduced control and the reduced adjoint space, respectively. After having
+built them, we consider an enriched space for state and adjoint variables. Therefore, let us denote
+with {τn}2N
+n=1 = {ηy
+n}N
+n=1 ∪ {ηp
+n}N
+n=1 the basis functions for the space ZN, with ZN ≡ Y N ≡ (QN)∗,
+then we have ZN = span {τn, n = 1, . . . , 2N} [15, 19, 28, 29, 36, 35].
+4.2. Proper Orthogonal Decomposition. In this Section we briefly describe the Proper Orthog-
+onal Decomposition (POD) Galerkin algorithm [6, 22, 49, 50] for the construction of a discrete
+solution manifold and the relative reduced spaces. Since in the unsteady case we use a space-time
+structure, this procedure can be described making no distinction between time-dependency and
+steadiness. Firstly, we make a sampling of P by choosing Ntrain of its elements. Therefore, let us
+define the set of the train samples as PNtrain: we have that obviously PNtrain ⊂ P and the cardinality
+is |PNtrain| = Ntrain. The set PNtrain is denoted as the training set. We should pursue that Ntrain
+is large enough so as to ensure that PNtrain is a good “approximation” of the parameter space P.
+PNtrain is built through a Monte-Carlo sampling method with respect to a uniform density with
+support equal to P.
+Starting from the sampling, the POD algorithm manipulates Ntrain snapshots for the state, the
+adjoint and the control variables:
+(53)
+��
+yN (µj), uN (µj), pN (µj)
+��Ntrain
+j=1
+with µj ∈ PNtrain.
+After this step, a compressing stage is performed: from (53) we build N basis functions by only
+considering the most important parametric information and throwing away the redundant ones, with
+N ≤ Ntrain. A partitioned approach is used, which means that, after the deterministic sampling,
+
+SUPG ROMS FOR ADVECTION-DOMINATED PDES UNDER OPTIMAL CONTROL
+15
+we perform the POD algorithm separately for all the three variables. Namely, we find three N-
+dimensional reduced spaces Y N, U N and (QN)∗ that minimizes the following three quantities:
+�
+�
+�
+�
+1
+Ntrain
+�
+µj∈PNtrain
+min
+¯y∈Y N
+��yN �
+µj
+�
+− ¯y
+��2
+Y ,
+�
+�
+�
+�
+1
+Ntrain
+�
+µj∈PNtrain
+min
+¯u∈U N
+��uN �
+µj
+�
+− ¯u
+��2
+U,
+�
+�
+�
+�
+1
+Ntrain
+�
+µj∈PNtrain
+min
+¯p∈(QN)∗
+��pN �
+µj
+�
+− ¯p
+��2
+Q∗,
+where obviously Y N ⊂ Y N , U N ⊂ U N and (QN)∗ ⊂ (QN )∗.
+Let us discuss the data compression procedure of the POD for the state variable y(µ) [6, 22, 49,
+50]. As we are following a partitioned approach, the control and the adjoint variables follow the
+below discussion with usual modifications, as well. Firstly we collect a set of ordered parameters
+µ1, . . . , µNtrain ∈ PNtrain, which the ordered snapshots yN (µ1) , . . . , yN �
+µNtrain
+�
+are linked to. Let
+us define Cy ∈ RNtrain×Ntrain as the correlation matrix of the snapshots for the state variable as
+follows:
+(54)
+Cy
+ij :=
+1
+Ntrain
+�
+yN (µi) , yN �
+µj
+��
+Y ,
+1 ≤ i, j ≤ Ntrain.
+The next step is to find the pair eigenvalue-eigenvector (λy
+n, ey
+n), where ey
+n has norm equal to one, of
+the following problem:
+Cyey
+n = λy
+ney
+n,
+1 ≤ n ≤ Ntrain.
+For the sake of simplicity, we organise the eigenvalues λy
+1, . . . , λy
+Ntrain in decreasing order. Consider
+the first N ones, specifically λy
+1, . . . , λy
+N together with the related eigenvectors ey
+1, . . . , ey
+N. We refer
+to the k-th component of the state eigenvector ey
+n ∈ RNtrain with the notation (ey
+n)k. After having
+finished the computation of the pair eigenvalue-eigenvector, the basis functions ηy
+n for the state
+equation are built through the following formula:
+(55)
+ηy
+n =
+1
+√
+λy
+n
+Ntrain
+�
+k=1
+(ey
+n)k yN (µk) ,
+1 ≤ n ≤ N.
+Therefore, our reduced spaces are built as (52) and, then, aggregated space technique is applied.
+As both N and Ntrain can be chosen by us, we should find sharp criteria in order to decide them.
+A possibility can be to set them in based on a study of the eigenvalues, using the estimate [22, 39, 58]:
+(56)
+�
+�
+�
+�
+1
+Ntrain
+Ntrain
+�
+k=1
+∥yN (µk) − ΠN (yN (µk))∥2
+Y =
+�
+�
+�
+�
+Ntrain
+�
+k=N+1
+λy
+k,
+where ΠN : Y → Y N is a Galerkin projector of functions from Y onto Y N. (56) holds for the
+control and the adjoint in a partitioned approach, too. The second member of equation (56) can be
+a measure of how well the FEM space is approximated by N reduced basis over the chosen training
+set of cardinality Ntrain. We summarise the whole POD procedure in the below Algorithm 1.
+Remark 4.2.1 (Time-dependent problems). When we are dealing with time-dependent OCPs, the
+time instances are not separated in the POD procedure. Therefore, the space-time problem is studied
+as a steady one and each snapshot carries all the time instances.
+
+16
+SUPG ROMS FOR ADVECTION-DOMINATED PDES UNDER OPTIMAL CONTROL
+Algorithm 1 POD algorithm for OCP problems in a partitioned approach
+Input: parameter domain P, FEM spaces Y N , U N and (QN )∗ and Ntrain.
+Output: reduced spaces Y N, U N and (QN)∗.
+Starting from the high-fidelity spaces Y N , U N and (QN )∗:
+1: Sample Ptrain ⊂ P;
+2: for all µ ∈ Ptrain do
+3:
+Solve the high-fidelity OCP system (34) (in this case a stabilized one);
+4: end for
+5: Assemble the matrix Cy
+ij :=
+1
+Ntrain
+�
+yN (µi) , yN �
+µj
+��
+Y , 1 ≤ i, j ≤ Ntrain. Do the same for u
+and p variables;
+6: Compute its eigenvalues λy
+1, . . . , λy
+Ntrain and the corresponding orthonormalised eigenvectors
+ey
+1, . . . , ey
+Ntrain. Do the same for u and p variables;
+7: After having chosen N according to a certain criterion, define Y N = span {ηy
+n, n = 1, . . . , N},
+where ηy
+n =
+1
+√
+λy
+n
+�Ntrain
+k=1
+(ey
+n)k yN (µk). Do the same for u and p variables.
+8: Define the aggregated space ZN = span
+�
+{ηy
+n}N
+n=1 ∪ {ηp
+n}N
+n=1
+�
+and impose ZN ≡ Y N ≡ (QN)∗.
+5. Numerical Results
+In this Section we propose simulations regarding the Graetz-Poiseuille and the Propagating Front
+in a Square problems. Regarding the steady case, the numerical experiments are coded through
+the RBniCS library [2]; instead, the unsteady ones are implemented employing both RBniCS and
+multiphenics [1] libraries. They are python-based libraries, built on FEniCS [32].
+When we will perform the Online-Offline stabilization procedure, we will always use the same
+stabilization parameter δK of the high-fidelity approximation also at the reduced level, both in
+steady and unsteady cases.
+We will illustrate an analysis over relative errors between the FEM and the reduced solutions for
+all three variables, defined as
+(57)
+ey,N(µ) :=
+��yN (µ) − yN(µ)
+��
+Y
+∥yN (µ)∥Y
+, eu,N(µ) :=
+��uN (µ) − uN(µ)
+��
+U
+∥uN (µ)∥U
+, ep,N(µ) :=
+��pN (µ) − pN(µ)
+��
+Q∗
+∥pN (µ)∥Q∗
+,
+for the state, the control and the adjoint, respectively. As we are dealing with parametrized OCPs,
+we will evaluate a simple average of (57) for µ uniformly distributed in a testing set Ptest ⊆ P of size
+Ntest for every dimension N = 1, . . . , Nmax of the reduced space obtained by our POD procedure.
+More precisely, we will plot the base-10 logarithm of the average of (57). For parabolic problems we
+will consider the sum of the errors with respect to each discretized instant of time t.
+Regarding the efficiency of ROMs, we use the speedup-index to compare the computational cost
+of the FEM solution with that of the reduced one. This quantity is defined as:
+(58)
+speedup-index = computational time of the high-fidelity solution
+computational time of the reduced solution
+,
+which will be computed for each µ in the testing set with respect to the dimension N of the reduced
+spaces. As made with the relative error, we will consider the sample average of this quantity with
+respect to N; however, for the sake of completeness, we will add its minumum and maximum value
+computed through the testing set. For each test case, we will use the same Ptest to compute relative
+errors and the speedup-index. The steady results are obtained with 16GB of RAM and Intel Core
+i7-7500U Dual Core, 2.7GHz for the CPU; instead, the FEM and ROM parabolic simulations are
+run with 16GB of RAM and Intel Core i7 − 7700 Quad Core, 3.60GHz for the CPU.
+5.1. Numerical Experiments for the Graetz-Poiseuille Problem. The Graetz-Poiseuille prob-
+lem concerns the heat conduction in a straight duct, whose walls can be characterized by heat
+
+SUPG ROMS FOR ADVECTION-DOMINATED PDES UNDER OPTIMAL CONTROL
+17
+exchange or maintained at a certain fixed temperature. This example is very well-known in the nu-
+merical Advection-Dominated literature [18, 37, 44, 55]. We start by presenting the stationary case.
+We apply a distributed control in the whole domain and the parameter µ = µ1 > 0 is a physical
+component and characterizes the diffusion term. The spatial coordinates of the system are denoted
+Ωobs
+Ωobs
+Ω
+Γ1
+Γ2
+Γ3
+Γ4
+Γ5
+Γ6
+(0,0)
+(1,0)
+(2,0)
+(2,0.2)
+(2,0.8)
+(2,1)
+(1,1)
+(0,1)
+Figure 1. Geometry of the Graetz-Poiseuille Problem.
+with (x0, x1). The boundary of Ω is Γ. We consider Dirichlet boundary conditions (BC) on sides
+Γ1 := [0, 1] × {0}, Γ5 := [0, 1] × {1}, Γ6 := {0} × [0, 1] by imposing y = 0 and Γ2 := [1, 2] × {0} and
+Γ4 := [1, 2] × {1} by imposing y = 1, referring to Figure 1. We deal with homogeneous Neumann
+conditions on Γ3 := {2} × [0, 1]. The classic formulation of the problem is:
+(59)
+�
+�
+�
+�
+�
+�
+�
+�
+�
+�
+�
+�
+�
+�
+�
+− 1
+µ1
+∆y(µ) + 4x1(1 − x1)∂x0y(µ) = u,
+in Ω,
+y(µ) = 0,
+on Γ1 ∪ Γ5 ∪ Γ6,
+y(µ) = 1,
+on Γ2 ∪ Γ4,
+∂y(µ)
+∂ν
+= 0,
+on Γ3.
+Now we want to derive the optimality system. Ωobs := [1, 2]×[0.8, 1]∪[1, 2]×[0, 0.2] as illustrated
+in Figure 1. In this case, the state belongs to the space:
+˜Y :=
+�
+v ∈ H1�
+Ω
+�
+s.t. it satisfies the BC in (59)
+�
+.
+For the sake of practice, it is better to introduce a lifting function Ry ∈ H1(Ω), such that it fulfills
+the BC in (59). Therefore we define the variable ¯y := y − Ry, with ¯y ∈ Y , where
+Y :=
+�
+v ∈ H1
+0
+�
+Ω
+�
+s.t. ∂¯y
+∂ν = 0, on Γ3 and ¯y = 0 on Γ \ Γ3
+�
+.
+Nevertheless, without loss of generality, we will denote the new variable ¯y with y and we settle
+U := L2(Ω) and Q := Y ∗.
+Therefore, the adjoint variable p is null on Γ \ Γ3. The mathematical
+formulation is described as follows (we omitted the dependence from µ). Fixed α > 0, find the pair
+(y, u) ∈ Y × U that realizes
+(60)
+min
+(y,u)∈Y ×U J(y, u) = 1
+2
+�
+Ωobs
+�
+y − yd
+�2 dx + α
+2
+�
+Ω
+u2 dx
+such that e (y, u, p; µ) = 0,
+where e (y, u, p; µ) := a (y, p; µ)+b (u, p; µ)−⟨p, f(µ)⟩Y ∗Y . As explained in Sections 2 and 3, we follow
+a Lagrangian approach and we use SUPG stabilization in a optimize-then-discretize framework. We
+exploit P1-FEM approximation for the state, control and adjoint spaces. Here the stabilized forms
+as and a∗
+s are, respectively:
+as
+�
+yN , qN ; µ
+�
+:= a
+�
+yN , qN ; µ
+�
++
+�
+K∈Th
+δK
+�
+K
+�
+4x1(1 − x1)∂x0yN � �
+hK∂x0qN �
+,
+yN , qN ∈ Y N ,
+a∗
+s
+�
+zN , pN ; µ
+�
+:= a∗ �
+zN , pN ; µ
+�
++
+�
+K∈Th
+δK
+�
+K
+�
+4x1(1 − x1)∂x0pN � �
+hK∂x0zN �
+,
+zN , pN ∈ Y N .
+
+18
+SUPG ROMS FOR ADVECTION-DOMINATED PDES UNDER OPTIMAL CONTROL
+We consider a parameter space P :=
+�
+104, 106�
+and a quite coarse mesh of size h = 0.029 for the
+FEM spaces. The training set Ptrain has cardinality Ntrain = 100. We choose δK = 1.0 for all
+K ∈ Th and the penalization term is α = 0.01. We pursue the convergence in the L2-norm of the
+state to the desired solution profile yd(x) = 1.0, function defined on Ωobs of Figure 1. We perform
+the POD algorithm for Nmax = 20 in a partitioned approach. We illustrate the reduced solution for
+the state and adjoint variables in the best relative error scenario in Figure 2. Namely, we plot the
+Only-Offline and Online-Offline Stabilized solutions for N = 1 and N = 6. The values of N can be
+deduced by Figure 3. From the gradient equation (34), we expect the distributed control u to be
+equal to the adjoint p up to the multiplicative constant α.
+Figure 2. (Top) Only-Offline stabilized state (left) and adjoint (right) for N = 1;
+(Bottom) Online-Offline stabilized state (left) and adjoint (right) for N = 6; for the
+Graetz-Poiseuille Problem; P =
+�
+104, 106�
+, µ1 = 105, h = 0.029, δK = 1.0, α = 0.01.
+We consider the relative errors between the FEM and the reduced solution in Figure 3.
+We
+use a testing set Ptest of 100 elements in P. As previously cited, at N = 6 we reach the minima
+for all the three errors for the Online-Offline stabilization; more precisely for the state we touch
+ey,6 = 9.65 · 10−9, for the adjoint ep,6 = 1.98 · 10−8 and the control eu,6 = 6.00 · 10−9. In contrast
+with this situation, Only-Offline stabilization never falls under 10−2. This implies that the best
+choice is to pursue the Online-Offline stabilization procedure for this problem. However, after N = 6
+the errors begin slightly to increase. Our interpretation to this fact relies on P. Despite the fact
+that this parameter space might be too large, however the coefficient which multiplies the diffusion
+operator is still absolutely low in value for every µ1 ∈ P, nearly 10−4 and 10−5. Therefore, also
+thanks to SUPG stabilization and the distributed control action, the majority of snapshots can be
+very similar referring to the solution for µ1 = 105: this translates in very few bases to reach a good
+relative error. As a matter of fact, the eigenvalues λy
+7, λu
+7 are ≈ 10−15 and λp
+7 ≈ 10−16; by their
+decreasing order, all the subsequent eigenvalues are very close to zero machine. Thus, recalling (55)
+it follows that all basis components with N ≥ 7 are affected by some rounding errors due to the
+orthonormalization procedure of the POD (for details see [39]).
+Finally, we take a look at the speedup-index in Table 1. All the average values are better for
+the Only-Offline stabilized ROM procedure due to the fact that the stabilized forms are not taken
+into account in the online phase. However, the Online-Offline stabilized reduced solution shows very
+good behaviour, for instance, we have an average equal to 284.3 for N = 6. Generally, in this case
+speedup-index takes average value around 2 · 102 order of magnitude for the first 20 basis elements.
+
+-3.9e-02
+0.2
+0.4
+0.6
+0.8
+1 1.1e+00-2.1e-03
+0
+0.0020.004.0.0060.0080.010.012
+1.5e-020.0e+00
+0.2
+0.3
+0.4
+0.5
+0.6
+0.7
+0.8
+0.91.0e+002.2e-03
+0.002 0.0040.0060.0080.01 0.012
+1.6e-02SUPG ROMS FOR ADVECTION-DOMINATED PDES UNDER OPTIMAL CONTROL
+19
+Figure 3. Relative errors between FEM and reduced solution for state (left), control
+(center) and adjoint (right), for Online-Offline and Only-Offline stabilization, α = 0.01,
+Ntest = 100, h = 0.029, P =
+�
+104, 106�
+. Graetz-Poiseuille Problem.
+Only-Offline Stabilization
+Offline-Online Stabilization
+N
+min
+average
+max
+min
+average
+max
+1
+162.1
+296.6
+338.1
+170.8
+261.7
+285.9
+2
+172.2
+342.1
+391.3
+178.4
+298.5
+327.1
+3
+168.5
+336.2
+383.7
+192.0
+298.9
+325.3
+4
+165.1
+336.1
+385.6
+256, 7
+298.0
+322.6
+5
+164.8
+331.6
+376.6
+220.1
+287.6
+307.7
+6
+198.3
+321.0
+366.4
+192.3
+284.3
+305.7
+7
+186.1
+318.4
+348.6
+228.6
+282.6
+306.9
+Table 1. Speedup-index of the Graetz-Poiseuille Problem for Online-Offline and Only-
+Offline stabilizations with Ptest sampled from P =
+�
+104, 106], Ntest = 100, α = 0.01,
+h = 0.029, δK = 1.0, α = 0.01.
+Let us give a look to the unsteady version of Problem (59) with null initial condition.
+The
+unsteady Graetz-Poiseuille problem without control has been presented in [37, 55], instead the OCP
+Graetz Problem under boundary control without Advection-dominancy is studied in [50, 48].
+Recalling Figure 1, for a fixed T > 0 we state the parabolic Graetz-Poiseuille Problem as follows:
+(61)
+�
+�
+�
+�
+�
+�
+�
+�
+�
+�
+�
+�
+�
+�
+�
+�
+�
+�
+�
+∂y(µ)
+∂t
+− 1
+µ1
+∆y(µ) + 4x1(1 − x1)∂x0y(µ) = u,
+in Ω × (0, T),
+y(µ) = 0,
+on Γ1 ∪ Γ5 ∪ Γ6 × (0, T),
+y(µ) = 1,
+on Γ2 ∪ Γ4 × (0, T),
+∂y(µ)
+∂ν
+= 0,
+on Γ3 × (0, T),
+y(µ)(0) = 0,
+in Ω.
+We do simulations in a space-time framework as discussed in Section 3.2 for a prearranged number of
+time-steps Nt using a P1-FEM approximation for the high-fidelity solutions. The relative stabilized
+forms in (49) for derivatives along time for state and adjoint are, respectively:
+(62)
+ms
+�
+yN , qN ; µ
+�
+=
+�
+yN , qN �
+L2(Ω) +
+�
+K∈Th
+δKhK
+�
+yN , ∂x0qN �
+K ,
+yN , qN ∈ Y N ,
+m∗
+s
+�
+pN , zN ; µ
+�
+=
+�
+pN , zN �
+L2(Ω) −
+�
+K∈Th
+δKhK
+�
+pN , ∂x0zN �
+K ,
+pN , zN ∈ Y N .
+We consider a final time of T = 3.0 and a time step of ∆t = 0.1, hence we have Nt = 30. We choose
+a quite coarse mesh of h = 0.038 and the overall high-fidelity dimension is Ntot = 314820. This
+means that a single FEM space for a fixed t has a dimension of N = 3498. We consider a initial
+condition of y0(x) = 0 for all x ∈ Ω referring to Figure 1. We want the state solution to converge
+
+FEM vs ROM averaged relative error - y (state)
+Online stab
+101
+Online not stab.
+Log-Error
+10-1
+10-
+Relative L
+10-5
+10-7
+1
+2
+3
+4
+5
+6
+7
+8
+NFEM vs ROM averaged relative error - u (control)
+101
+Online stab
+Online not stab
+10-1
+10-5
+10-7
+1
+2
+3
+4
+5
+6
+7
+8
+NFEM vs ROM averaged relative error - p (adjoint)
+Online stab.
+Online not stab.
+101
+10
+10-3
+10-5
+10-7
+1
+2
+3
+4
+5
+6
+7
+8
+N20
+SUPG ROMS FOR ADVECTION-DOMINATED PDES UNDER OPTIMAL CONTROL
+in the L2-norm to a desired solution profile yd(x, t) = 1.0, function defined for all t ∈ [0, 3.0] and
+for all x in Ωobs. Here the SUPG stabilization is implemented with parameters δK = 1.0 for all
+K ∈ Th. P :=
+�
+104, 106�
+and we choose a training set Ptrain of cardinality Ntrain = 100. Then, we
+performed the POD algorithm with Nmax = 20. The penalization parameter is α = 0.01.
+Figure 4. (Top) SUPG FEM solution for the state and (Bottom) for the adjoint at
+t = 0.1, t = 1.5, t = 3.0.
+Unsteady Graetz-Poiseuille Problem, µ1 = 105, Nt = 30,
+h = 0.038, δK = 1.0, α = 0.01.
+As we will see in Figure 5 the performance of the Only-Offline stabilized reduced solutions are not
+so good in terms of accuracy, unlike the Online-Offline stabilized ones. We consider a testing set of
+100 elements in P. As succeeded in the steady case in Section 5.1, after nearly N = 6 Online-Offline
+stabilized errors begin to fluctuate due to the nature of the eigenvalues of the correlation matrix
+(54) that are closed to zero machine. For this reason we present the trend of error from 1 to 10.
+However, errors stays close to 10−7 for the state and the adjoint and 10−6 to the control. For N = 6
+we have ey,6 = 4.20 · 10−7, eu,6 = 1.10 · 10−6 and ep,6 = 3.18 · 10−7, instead for N = 20 we have
+ey,20 = 1.93 · 10−7, eu,20 = 3.25 · 10−7 and ep,20 = 1.21 · 10−7 for the Online-Offline stabilization
+ROM.
+Figure 5. Relative errors between the FEM and Only-Offline and Online-Offline stabi-
+lized solutions for the state (left), control (center) and adjoint (right), Unsteady Graetz-
+Poiseuille problem, Nt = 30, Ntest = 100, P =
+�
+104, 106�
+, h = 0.038.
+Finally, we can see the speedup-index for some value of N in Table 2. In both situation we can
+compute a huge number of reduced solutions in the time of a high-fidelity one: for the Offline-Online
+stabilization we have an average speedup-index of nearly 26000 for N = 6. On the whole, average
+speedup-index has an order of magnitude of 2 · 104 for N ≤ 20.
+5.2. Numerical Experiments for Propagating Front in a Square Problem. In this Section,
+we consider a problem studied in the Advection-Dominated form in [37, 55] from a numerical point
+of view and we will add a distributed control to it. Let Ω be the unit square in R2. We consider
+the representation in Figure 6. Also in this case, (x0, x1) are the coordinates of the square domain.
+
+1.0e+00
+0.8
+0.6
+0.4
+0.2
+-7.2e-021.2e-02
+0.01
+0.008
+0.006
+0.004
+0.002
+0
+-1.5e-03FEM vs ROM averaged relative error - y (state)
+Online stab.
+Online not stab.
+100
+Relative Log-Error
+10-2
+10-6
+1
+2
+3
+4
+5
+6
+7
+8
+NFEM vs ROM averaged relative error - u (control)
+Online stab
+Online not stab.
+101
+Relative Log-Error
+10-1
+10-3
+10-5
+1
+2
+3
+4
+5
+6
+7
+8
+NFEM vs ROM averaged relative error - p (adjoint)
+Online stab.
+Online not stab.
+100
+Relative Log-Error
+10-2
+10-4
+10-6
+1
+2
+3
+4
+5
+6
+7
+8
+NSUPG ROMS FOR ADVECTION-DOMINATED PDES UNDER OPTIMAL CONTROL
+21
+Only-Offline Stabilization
+Offline-Online Stabilization
+N
+min
+average
+max
+min
+average
+max
+1
+21588.3
+26588.8
+30971.5
+18968.4
+23588.0
+27062.7
+2
+23821.3
+29723.4
+34817.2
+20757.2
+26018.9
+29929.1
+3
+23571.0
+29468.6
+34349.5
+20547.9
+25698.2
+29662.5
+4
+23062.2
+28880.6
+33702.7
+21385.2
+25380.9
+28883.3
+5
+25762.9
+28767.9
+33488.8
+23021.5
+25882.4
+29388.9
+6
+27003.2
+29707.7
+34544.7
+23236.5
+26054.7
+29677.5
+7
+26658.5
+29481.1
+34277.3
+23206.6
+25879.5
+29505.4
+Table 2. Speedup-index of the Unsteady Graetz Problem for Online-Offline and Only-
+Offline stabilization with P =
+�
+104, 106], α = 0.01, Nt = 30, Ntest = 100, h = 0.038.
+Γ1
+Γ2
+Γ3
+Γ4
+Γ5
+Ω
+Ωobs
+(0,0.25)
+(0,1)
+(1,0.75)
+(1,1)
+(0.25,1)
+(1,0)
+(0,0)
+Figure 6. Geometry of the Square Problem
+Referring to Figure 6, Γ1 := {0} × [0, 0.25], Γ2 := [0, 1] × {0}, Γ3 := {1} × [0, 1], Γ4 := [0, 1] × {1},
+Γ5 := {0} × [0.25, 1]; Ωobs := [0.25, 1] × [0.75, 1]. Given µ = (µ1, µ2), the problem is formulated as
+(63)
+�
+�
+�
+�
+�
+�
+�
+− 1
+µ1
+∆y(µ) + (cos µ2, sin µ2) · ∇y(µ) = u,
+in Ω,
+y(µ) = 1,
+on Γ1 ∪ Γ2,
+y(µ) = 0,
+on Γ3 ∪ Γ4 ∪ Γ5.
+We assume that the Identity restricted to Ωobs as the Observation operator and Z := L2(Ωobs). In
+our test cases, P :=
+�
+104, 105�
+×
+�
+0, 1.57
+�
+. In this case, we have that the domain of definition of our
+state y is
+˜Y :=
+�
+v ∈ H1�
+Ω
+�
+s.t. BC in (63)
+�
+.
+Exactly as done in the previous paragraph, we define a lifting function Ry ∈ H1�
+Ω
+�
+such that
+satisfies BC in (63). We define ¯y := y − Ry, even though we denote ¯y as y again for the sake of
+notation. We consider Y := H1
+0(Ω), U = L2(Ω) and Q := Y ∗, hence the adjoint p is such that p = 0
+on ∂Ω. We define the objective functional J exactly as in (60); instead, a and b are
+a (y, p; µ) :=
+�
+Ω
+1
+µ1
+∇y · ∇p + (cos µ2, sin µ2) · ∇yp dx, and b (u, p; µ) := −
+�
+Ω
+up dx.
+and ⟨p, f(µ)⟩Y ∗Y = −a (Ry, p; µ) . Then we follow usual discussions of Sections 2 and 3.
+We exploit a P1-FEM approximation for the optimality system by using the usual SUPG stabi-
+lization technique, arriving to system (34). Here, for the sake of completeness, we remark that the
+
+22
+SUPG ROMS FOR ADVECTION-DOMINATED PDES UNDER OPTIMAL CONTROL
+stabilized forms as and a∗
+s are, respectively:
+as
+�
+yN , qN ; µ
+�
+:= a
+�
+yN , qN ; µ
+�
++
+�
+K∈Th
+δK
+�
+K
+hK (cos µ2, sin µ2) · ∇yN (cos µ2, sin µ2) · ∇qN ,
+a∗
+s
+�
+zN , pN ; µ
+�
+:= a∗ �
+zN , pN ; µ
+�
++
+�
+K∈Th
+δK
+�
+K
+hK (cos µ2, sin µ2) · ∇pN (cos µ2, sin µ2) · ∇zN ,
+for all yN , qN , zN , pN ∈ Y N .
+As previously done, we build a training set Ptrain and a testing
+set Ptest with both cardinality ntrain = 100. The mesh size h is 0.025 and therefore the overall
+dimension of the high-fidelity approximation is 12087, which implies that state, control and adjoint
+spaces have dimension equal N = 4029. The SUPG stabilization is implemented with parameters
+δK = 1.0 for all K ∈ Th. The penalization parameter is α = 0.01 and we pursue the state solution
+to be convergent in the L2-norm to a desired solution profile yd(x) = 0.5, defined for all x in Ωobs of
+Figure 6. In Figure 7 we observe state and adjoint FEM solutions for µ = (2 · 104, 1.2). Instead, in
+Figure 8 we illustrate Only-Offline and Online-Offline reduced solution for the state and the adjoint
+variable with µ = (2 · 104, 1.2) for N = 50.
+Figure 7. Numerical solution without stabilization and SUPG FEM solution with µ =
+(2 · 104, 1.2) for state (left) and adjoint (right) variables in the Propagating Front in a
+Square Problem, α = 0.01, h = 0.025, δK = 1.0.
+Figure 8. Only-Offline stabilized and Online-Offline stabilized reduced solutions with
+µ = (2 · 104, 1.2) or state (left) and adjoint (right) variables in the Propagating Front in a
+Square Problem, α = 0.01, N = 50, h = 0.025, δK = 1.0, P =
+�
+104, 105] ×
+�
+0, 1.57].
+These computational evidences and the analysis of the relative errors show that Online-Offline
+stabilization procedure is preferable in this setting. In Figure 9, the trend is the same of all three
+variables, where errors continue to decrease along all N: we have ey,50 = 1.20·10−3, eu,50 = 7.67·10−4
+and ep,50 = 3.16 · 10−3.
+Concerning the speedup-index, the performance are quite good as seen in Table 3. For the best
+approximation, we have that we can compute an average of 44 Online-Offline reduced solutions when
+we build the associated FEM one. Obviously, the Only-Offline stabilized one is slightly better. On
+the whole, speedup-index takes average value around 101, 102 order of magnitude for N ≤ 50.
+
+1.2e+00
+1.1
+0.9
+0.8
+0.7
+0.6
+0.5
+0.4
+0.3
+0.2
+0.1
+-1.8e-029.7e-03
+0.008
+- 0.006
+- 0.004
+0.002
+0
+-0.002
+-0.004
+-0.006
+8.4e-031.2e+00
+1.1
+0.9
+0.8
+0.7
+0.6
+0.5
+0.4
+0.3
+0.2
+0.1
+-1.8e-029.7e-03
+0.008
+0.006
+0.004
+0.002
+0
+-0.002
+-0.004
+-0.006
+8.4e-03SUPG ROMS FOR ADVECTION-DOMINATED PDES UNDER OPTIMAL CONTROL
+23
+Figure 9. Relative errors between FEM and reduced solutions with P =
+�
+104, 105] ×
+�
+0, 1.57] for the state (left), control (center) and adjoint (right) in the Propagating Front
+in a Square Problem, Ntest = 100, α = 0.01, h = 0.025, δK = 1.0.
+Only-Offline Stabilization
+Offline-Online Stabilization
+N
+min
+average
+max
+min
+average
+max
+1
+123.1
+198.8
+243.1
+110.0
+162.7
+181.9
+10
+132.2
+200.4
+244.2
+110.8
+158.3
+176.9
+20
+84.6
+158.3
+191.7
+60.1
+124.3
+141.3
+30
+78.8
+114.7
+137.8
+65.0
+92.5
+104.7
+40
+54.2
+78.6
+96.1
+46.9
+64.2
+72.3
+50
+33.2
+53.0
+64.8
+28.5
+44.0
+49.9
+Table 3. Speedup-index of the Propagating Front in a Square Problem for Online-Offline
+and Only-Offline stabilization with training set P =
+�
+104, 105]×
+�
+0, 1.57], α = 0.01, Ntest =
+100, h = 0.025, δK = 1.0.
+Now we study the unsteady case of the Propagating Front in a Square Problem for a fix T > 0:
+(64)
+�
+�
+�
+�
+�
+�
+�
+�
+�
+�
+�
+�
+�
+∂y(µ)
+∂t
+− 1
+µ1
+∆y(µ) + (cos µ2, sin µ2) · ∇y(µ) = u,
+in Ω × (0, T),
+y(µ) = 1,
+on Γ1 ∪ Γ2 × (0, T),
+y(µ) = 0,
+on Γ3 ∪ Γ4 ∪ Γ5 × (0, T),
+y(µ)(0) = 0,
+in Ω,
+with initial value y0(x) = 0 for all x ∈ Ω referring to the domain in Figure 6. We build a parabolic
+problem for a final time T = 3.0 and a time-step ∆t = 0.1, hence Nt = 30. We choose a quite
+coarse mesh of size h = 0.036 and the overall dimension of the space-time system is Ntot = 174780,
+which means that a single FEM space has dimension N = 1942. Again, our aim is to achieve in a
+L2-mean a desired solution profile yd(x, t) = 0.5, defined for all t ∈ [0, 3] and x in Ωobs of Figure 6.
+The penalization parameter is α = 0.01. We set δK = 1.0 for all K ∈ Th. Here we have that the
+stabilized forms in (49) for derivatives along time for state and adjoint equations are, respectively:
+ms
+�
+yN , qN ; µ
+�
+=
+�
+yN , qN �
+L2(Ω) +
+�
+K∈Th
+δKhK
+�
+yN , (cos µ2, sin µ2) · ∇qN �
+K ,
+yN , qN ∈ Y N ,
+m∗
+s
+�
+pN , zN ; µ
+�
+=
+�
+pN , zN �
+L2(Ω) −
+�
+K∈Th
+δKhK
+�
+pN , (cos µ2, sin µ2) · ∇zN �
+K ,
+pN , zN ∈ Y N .
+We consider a parameter space equal to the steady case, i.e. P :=
+�
+104, 105�
+×
+�
+0, 1.57
+�
+. Our training
+set has cardinality Ntrain = 100. In Figure 10 we show a representative stabilized FEM solution for
+µ = (2·104, 1.2) for some instants of time. We choose to perform a POD procedure with Nmax = 30.
+In Figure 11 one can see the relative errors of the three variables. As previously said, Only-
+Offline procedure has not good error behaviour. Instead, it is worth to note that in a Online-Offline
+
+FEM vs ROM averaged relative error - y (state)
+100
+10-2
+Online stab
+Online not stab.
+10-3
+10
+20
+30
+40
+50
+NFEM vs ROM averaged relative error - u (contro
+100
+Relative Log-Error
+10-2
+Online stab.
+10-3.
+Online not stab.
+10
+20
+30
+40
+50
+NFEM vs ROM averaged relative error - p (adjoint)
+Online stab.
+ Online not stab.
+101
+100
+10-1
+10-2
+10
+20
+30
+40
+50
+N24
+SUPG ROMS FOR ADVECTION-DOMINATED PDES UNDER OPTIMAL CONTROL
+Figure 10. (Top) SUPG FEM state solution and (Bottom) SUPG FEM adjoint solution
+for µ = (2·104, 1.2) for time t = 0.1 (left), t = 1.5 (center), t = 3.0 (right), in the Parabolic
+Propagating Front in a Square Problem, h = 0.036, α = 0.01, δK = 1.0.
+stabilization context, errors between the FEM and the reduced solutions decrease as N grows. The
+fact that we deal with a two-dimensional parameter space implies to require more N basis for a
+good approximation of the reduced solution. We have ey,30 = 2.17 · 10−3, eu,30 = 1.59 · 10−3 and
+ep,30 = 5.62 · 10−3. Therefore, also for this case test we can state that the SUPG stabilization is an
+efficient procedure for the ROMs.
+Figure 11. Relative errors between the FEM and the Only-Offline and Online-Offline
+stabilized reduced solution for the state (left), control (center) and adjoint (right) solutions,
+respectively with P =
+�
+104, 105�
+×
+�
+0, 1.57
+�
+, Nt = 30, Ntest = 100, δK = 1.0, h = 0.036.
+Finally, we show the results about the speedup-index in Table 4. For N = 30, not only we have
+the best accuracy for the reduced problem, but we are able to computed, averagely, 3981 reduced
+solution in the interval of a FEM simulation. Speedup-index has an average order of magnitude of
+103 overall.
+6. Conclusions and Perspectives
+In this work, we presented the numerical experiments concerning Advection-Dominated OCPs in
+a ROM context with high P´eclet number, both in the steady and the unsteady cases, under SUPG
+stabilization. Concerning ROMs, we can have two possibilities of stabilization: we can apply SUPG
+only to the offline phase or we can use it in both online and offline phases. We analyzed relative
+
+1.2e+00
+0.9
+0.8
+0.7
+0.6
+0.5
+0.4
+0.3
+0.2
+0.1
+-1.8e-029.7e-03
+0.008
+0.006
+0.004
+0.002
+0
+-0.002
+-0.004
+-0.006
+-8.4e-03FEM vs ROM averaged relative error - y (state)
+100
+10-1
+Relative L
+10-2
+Online stab
+Online not stab
+5
+10
+15
+20
+25
+30
+NFEM vs ROM averaged relative error - u (control)
+100
+Relative Log-Error
+10-2
+Online stab.
+Online not stab
+5
+10
+15
+20
+25
+30
+NFEM vs ROM averaged relative error - p (adjoint)
+100
+10-1
+10-2
+-Online stab.
+Online not stab.
+5
+10
+15
+20
+25
+30
+NSUPG ROMS FOR ADVECTION-DOMINATED PDES UNDER OPTIMAL CONTROL
+25
+Only-Offline Stabilization
+Offline-Online Stabilization
+N
+min
+average
+max
+min
+average
+max
+5
+6136.6
+8659.4
+12654.5
+4588.3
+6605.8
+9914.7
+10
+6018.7
+8278.8
+11989.5
+4282.9
+6231.7
+9353.3
+15
+5611.3
+7721.4
+11359.5
+3725.3
+5779.0
+8794.3
+20
+4820.0
+7041.2
+10143.0
+3567.7
+5318.9
+8091.1
+25
+3814.1
+5970.5
+9212.2
+2751.7
+4366.6
+6678.3
+30
+3432.0
+5420.1
+8462.8
+2424.7
+3981.3
+6147.9
+Table 4. Speedup-index of the unsteady Propagating Front in a Square Problem for
+Online-Offline and Only-Offline stabilization with training set P :=
+�
+104, 105] ×
+�
+0, 1.57],
+h = 0.036, α = 0.01, Nt = 30, Ntest = 100, δK = 1.0.
+errors between the reduced and the high fidelity solutions and of the speedup-index concerning the
+Graetz-Poiseuille and Propagating Front in a Square Problems, always under a distributed control.
+A P1-FEM approximation for the state, control and adjoint spaces is used in a optimize-then-
+discretize framework. Concerning parabolic problems, a space-time approach is followed and we
+applied in a suitable way the SUPG stabilization. For the ROM, we considered a partitioned approach
+for all three variables using the POD algorithm. In all the steady and unsteady experiments, the
+ROM technique performed excellently in a Online-Offline stabilization framework. Especially for
+parabolic problems, the speedup-index features large values thanks to the space-time formulation.
+Only-Offline stabilization technique performed very poorly in terms of errors, despite the little
+favorable speedup values. Thus, Online-Offline stabilization is preferable.
+We also performed experiments inherent a geometrical parametrization and boundary control for
+the Graetz-Poiseuille Problem that are not shown in this work. Results were quite good for Online-
+Offline stabilization: we had some little oscillations regarding relative errors due to the complexity
+of the problem. As a perspective, it might be interesting to create a strongly-consistent stabilization
+technique that flattens all the fluctuation for these two configurations, since, to the best of our
+knowledge, this topic is still a novelty in literature.
+Regarding the SUPG stabilization for parabolic OCPs in a optimize-then-discretize framework,
+it would be also worth to derive some theoretical results that gives us the accuracy of the numerical
+solution with respect to the time-step and the mesh-size.
+In conclusion, as another goal it might be interesting to study the performance of new stabilization
+techniques for the online phases, such as the Online Vanishing Viscosity and the Online Rectification
+methods [4, 13, 33]. Moreover, the extension of this setting to the uncertainty certification context
+will be the topic of future research.
+Acknowledgements
+We acknowledge the support by European Union Funding for Research and Innovation – Horizon
+2020 Program – in the framework of European Research Council Executive Agency: Consolidator
+Grant H2020 ERC CoG 2015 AROMA-CFD project 681447 “Advanced Reduced Order Methods
+with Applications in Computational Fluid Dynamics”. We also acknowledge the PRIN 2017 “Nu-
+merical Analysis for Full and Reduced Order Methods for the efficient and accurate solution of
+complex systems governed by Partial Differential Equations” (NA-FROM-PDEs) and the INDAM-
+GNCS project “Tecniche Numeriche Avanzate per Applicazioni Industriali”. The computations in
+this work have been performed with RBniCS [2] library, developed at SISSA mathLab, which is
+an implementation in FEniCS [32] of several reduced order modelling techniques; we acknowledge
+developers and contributors to both libraries.
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