Rodrigo M. Pereira, Natacha Nguyen van yen, Kai Schneider, Marie Farge
SIAM Review, Volume 65, Issue 4, Page 1109-1134, November 2023. Adaptive Galerkin numerical schemes integrate time-dependent partial differential equations with a finite number of basis functions, and a subset of them is selected at each time step. This subset changes over time discontinuously according to the evolution of the solution; therefore the corresponding projection operator is time-dependent and nondifferentiable, and we propose using an integral formulation in time. We analyze the existence and uniqueness of this weak form of adaptive Galerkin schemes and prove that nonsmooth projection operators can introduce energy dissipation, which is a crucial result for adaptive Galerkin schemes. To illustrate this, we study an adaptive Galerkin wavelet scheme which computes the time evolution of the inviscid Burgers equation in one dimension and of the incompressible Euler equations in two and three dimensions with a pseudospectral scheme, together with coherent vorticity simulation which uses wavelet denoising. With the help of the continuous wavelet representation we analyze the time evolution of the solution of the 1D inviscid Burgers equation: We first observe that numerical resonances appear when energy reaches the smallest resolved scale, then they spread in both space and scale until they reach energy equipartition between all basis functions, as thermal noise does. Finally we show how adaptive wavelet schemes denoise and regularize the solution of the Galerkin truncated inviscid equations, and for the inviscid Burgers case wavelet denoising even yields convergence towards the exact dissipative solution, also called entropy solution. These results motivate in particular adaptive wavelet Galerkin schemes for nonlinear hyperbolic conservation laws. This SIGEST article is a revised and extended version of the article [R. M. Pereira, N. Nguyen van yen, K. Schneider, and M. Farge, Multiscale Model. Simul., 20 (2022), pp. 1147--1166].
SIAM评论，第65卷，第4期，第1109-1134页，2023年11月。自适应Galerkin数值格式集成了具有有限个基函数的含时偏微分方程，并在每个时间步长选择其中的一个子集。该子集随着时间的推移根据解决方案的演变而不连续地变化；因此，相应的投影算子是时间相关的和不可微的，我们建议使用时间积分公式。我们分析了这种弱形式的自适应Galerkin格式的存在性和唯一性，并证明了非光滑投影算子可以引入能量耗散，这是自适应Galerkn格式的一个关键结果。为了说明这一点，我们研究了一种自适应Galerkin小波格式，该格式使用伪谱格式计算一维无粘性Burgers方程和二维和三维不可压缩Euler方程的时间演化，以及使用小波去噪的相干涡度模拟。在连续小波表示的帮助下，我们分析了一维无粘Burgers方程解的时间演化：我们首先观察到，当能量达到最小的分辨尺度时，会出现数值共振，然后它们在空间和尺度上传播，直到它们达到所有基函数之间的能量均分，就像热噪声一样。最后，我们展示了自适应小波方案如何对Galerkin截断无粘方程的解进行去噪和正则化，并且对于无粘Burgers情况，小波去噪甚至产生向精确耗散解（也称为熵解）的收敛。这些结果特别激励了非线性双曲守恒律的自适应小波Galerkin格式。SIGEST的这篇文章是该文章的修订和扩展版本[R.M.Pereira，N.Nguyen van yen，K.Schneider和M.Farge，Multiscale Model.Simul.，20（2022），pp.1147-11166]。
{"title":"Are Adaptive Galerkin Schemes Dissipative?","authors":"Rodrigo M. Pereira, Natacha Nguyen van yen, Kai Schneider, Marie Farge","doi":"10.1137/23m1588627","DOIUrl":"https://doi.org/10.1137/23m1588627","url":null,"abstract":"SIAM Review, Volume 65, Issue 4, Page 1109-1134, November 2023. <br/> Adaptive Galerkin numerical schemes integrate time-dependent partial differential equations with a finite number of basis functions, and a subset of them is selected at each time step. This subset changes over time discontinuously according to the evolution of the solution; therefore the corresponding projection operator is time-dependent and nondifferentiable, and we propose using an integral formulation in time. We analyze the existence and uniqueness of this weak form of adaptive Galerkin schemes and prove that nonsmooth projection operators can introduce energy dissipation, which is a crucial result for adaptive Galerkin schemes. To illustrate this, we study an adaptive Galerkin wavelet scheme which computes the time evolution of the inviscid Burgers equation in one dimension and of the incompressible Euler equations in two and three dimensions with a pseudospectral scheme, together with coherent vorticity simulation which uses wavelet denoising. With the help of the continuous wavelet representation we analyze the time evolution of the solution of the 1D inviscid Burgers equation: We first observe that numerical resonances appear when energy reaches the smallest resolved scale, then they spread in both space and scale until they reach energy equipartition between all basis functions, as thermal noise does. Finally we show how adaptive wavelet schemes denoise and regularize the solution of the Galerkin truncated inviscid equations, and for the inviscid Burgers case wavelet denoising even yields convergence towards the exact dissipative solution, also called entropy solution. These results motivate in particular adaptive wavelet Galerkin schemes for nonlinear hyperbolic conservation laws. This SIGEST article is a revised and extended version of the article [R. M. Pereira, N. Nguyen van yen, K. Schneider, and M. Farge, Multiscale Model. Simul., 20 (2022), pp. 1147--1166].","PeriodicalId":49525,"journal":{"name":"SIAM Review","volume":null,"pages":null},"PeriodicalIF":10.2,"publicationDate":"2023-11-07","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"71473805","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":1,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Richard C. Tillquist, Rafael M. Frongillo, Manuel E. Lladser
SIAM Review, Volume 65, Issue 4, Page 919-962, November 2023. The metric dimension of a graph is the smallest number of nodes required to identify all other nodes uniquely based on shortest path distances. Applications of metric dimension include discovering the source of a spread in a network, canonically labeling graphs, and embedding symbolic data in low-dimensional Euclidean spaces. This survey gives a self-contained introduction to metric dimension and an overview of the quintessential results and applications. We discuss methods for approximating the metric dimension of general graphs, and specific bounds and asymptotic behavior for deterministic and random families of graphs. We conclude with related concepts and directions for future work.
{"title":"Getting the Lay of the Land in Discrete Space: A Survey of Metric Dimension and Its Applications","authors":"Richard C. Tillquist, Rafael M. Frongillo, Manuel E. Lladser","doi":"10.1137/21m1409512","DOIUrl":"https://doi.org/10.1137/21m1409512","url":null,"abstract":"SIAM Review, Volume 65, Issue 4, Page 919-962, November 2023. <br/> The metric dimension of a graph is the smallest number of nodes required to identify all other nodes uniquely based on shortest path distances. Applications of metric dimension include discovering the source of a spread in a network, canonically labeling graphs, and embedding symbolic data in low-dimensional Euclidean spaces. This survey gives a self-contained introduction to metric dimension and an overview of the quintessential results and applications. We discuss methods for approximating the metric dimension of general graphs, and specific bounds and asymptotic behavior for deterministic and random families of graphs. We conclude with related concepts and directions for future work.","PeriodicalId":49525,"journal":{"name":"SIAM Review","volume":null,"pages":null},"PeriodicalIF":10.2,"publicationDate":"2023-11-07","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"71474870","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":1,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Mark Jayson Cortez, Alan Eric Akil, Krešimir Josić, Alexander J. Stewart
SIAM Review, Volume 65, Issue 4, Page 1152-1170, November 2023. Quantitative methods and mathematical modeling are playing an increasingly important role across disciplines. As a result, interdisciplinary mathematics courses are increasing in popularity. However, teaching such courses at an advanced level can be challenging. Students often arrive with different mathematical backgrounds, different interests, and divergent reasons for wanting to learn the material. Here we describe a course on stochastic processes in biology delivered between September and December 2020 to a mixed audience of mathematicians and biologists. In addition to traditional lectures and homework, we incorporated a series of weekly computational challenges into the course. These challenges served to familiarize students with the main modeling concepts and provide them with an introduction on how to implement the concepts in a research-like setting. In order to account for the different academic backgrounds of the students, they worked on the challenges in small groups and presented their results and code in a dedicated discussion class each week. We discuss our experience designing and implementing an element of problem-based learning in an applied mathematics course through computational challenges. We also discuss feedback from students and describe the content of the challenges presented in the course. We provide all materials, along with example code for a number of challenges.
{"title":"Incorporating Computational Challenges into a Multidisciplinary Course on Stochastic Processes","authors":"Mark Jayson Cortez, Alan Eric Akil, Krešimir Josić, Alexander J. Stewart","doi":"10.1137/21m1445545","DOIUrl":"https://doi.org/10.1137/21m1445545","url":null,"abstract":"SIAM Review, Volume 65, Issue 4, Page 1152-1170, November 2023. <br/> Quantitative methods and mathematical modeling are playing an increasingly important role across disciplines. As a result, interdisciplinary mathematics courses are increasing in popularity. However, teaching such courses at an advanced level can be challenging. Students often arrive with different mathematical backgrounds, different interests, and divergent reasons for wanting to learn the material. Here we describe a course on stochastic processes in biology delivered between September and December 2020 to a mixed audience of mathematicians and biologists. In addition to traditional lectures and homework, we incorporated a series of weekly computational challenges into the course. These challenges served to familiarize students with the main modeling concepts and provide them with an introduction on how to implement the concepts in a research-like setting. In order to account for the different academic backgrounds of the students, they worked on the challenges in small groups and presented their results and code in a dedicated discussion class each week. We discuss our experience designing and implementing an element of problem-based learning in an applied mathematics course through computational challenges. We also discuss feedback from students and describe the content of the challenges presented in the course. We provide all materials, along with example code for a number of challenges.","PeriodicalId":49525,"journal":{"name":"SIAM Review","volume":null,"pages":null},"PeriodicalIF":10.2,"publicationDate":"2023-11-07","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"71473804","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":1,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
SIAM Review, Volume 65, Issue 4, Page 1137-1151, November 2023. We present Cimmino's reflection algorithm for the numerical solution of linear systems, which starts with an arbitrary point in $mathbb{R}^n$ that gets reflected with respect to the system's hyperplanes. The centroid of the ensuing collection of points becomes the starting point of the next iteration. We provide error estimates for the convergence at each step. A probabilistic argument is also devised to improve this elegant geometrical algorithm. This subject is an opportunity to show students how linear algebra can interact fruitfully not only with algebra, geometry, and numerical analysis, but also with probability theory and methods.
{"title":"The Reflection Method for the Numerical Solution of Linear Systems","authors":"Margherita Guida, Carlo Sbordone","doi":"10.1137/22m1470463","DOIUrl":"https://doi.org/10.1137/22m1470463","url":null,"abstract":"SIAM Review, Volume 65, Issue 4, Page 1137-1151, November 2023. <br/> We present Cimmino's reflection algorithm for the numerical solution of linear systems, which starts with an arbitrary point in $mathbb{R}^n$ that gets reflected with respect to the system's hyperplanes. The centroid of the ensuing collection of points becomes the starting point of the next iteration. We provide error estimates for the convergence at each step. A probabilistic argument is also devised to improve this elegant geometrical algorithm. This subject is an opportunity to show students how linear algebra can interact fruitfully not only with algebra, geometry, and numerical analysis, but also with probability theory and methods.","PeriodicalId":49525,"journal":{"name":"SIAM Review","volume":null,"pages":null},"PeriodicalIF":10.2,"publicationDate":"2023-11-07","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"71473803","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":1,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
SIAM Review, Volume 65, Issue 4, Page 1171-1184, November 2023. A common definition of hysteresis is that the graph of the state of the system displays looping behavior as the input of the system varies. Alternatively, a dynamical systems perspective can be used to define hysteresis as a phenomenon arising from multiple equilibrium points. Consequently, hysteresis is a topic that can be used to illustrate and extend concepts in a dynamical systems course. The concept is illustrated in this paper through examples of ordinary differential equations, most motivated by applications. Simulations are presented to complement the analysis. The examples can be used to construct student exercises, and specific additional questions are listed in an appendix. The paper concludes with a discussion of possible extensions, including hysteresis in partial differential equations.
{"title":"Hysteresis and Stability","authors":"Amenda N. Chow, Kirsten A. Morris, Gina F. Rabbah","doi":"10.1137/21m1420733","DOIUrl":"https://doi.org/10.1137/21m1420733","url":null,"abstract":"SIAM Review, Volume 65, Issue 4, Page 1171-1184, November 2023. <br/> A common definition of hysteresis is that the graph of the state of the system displays looping behavior as the input of the system varies. Alternatively, a dynamical systems perspective can be used to define hysteresis as a phenomenon arising from multiple equilibrium points. Consequently, hysteresis is a topic that can be used to illustrate and extend concepts in a dynamical systems course. The concept is illustrated in this paper through examples of ordinary differential equations, most motivated by applications. Simulations are presented to complement the analysis. The examples can be used to construct student exercises, and specific additional questions are listed in an appendix. The paper concludes with a discussion of possible extensions, including hysteresis in partial differential equations.","PeriodicalId":49525,"journal":{"name":"SIAM Review","volume":null,"pages":null},"PeriodicalIF":10.2,"publicationDate":"2023-11-07","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"71473806","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":1,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
SIAM Review, Volume 65, Issue 4, Page 963-1028, November 2023. This article is a review of basic concepts and tools devoted to a posteriori error estimation for problems solved with the finite element method. For the sake of simplicity and clarity, we mostly focus on linear elliptic diffusion problems, approximated by a conforming numerical discretization. The main goal of this review is to present in a unified manner a large set of powerful verification methods, centered around the concept of equilibrium. Methods based on that concept provide error bounds that are fully computable and mathematically certified. We discuss recovery methods, residual methods, and duality-based methods for the estimation of the whole solution error (i.e., the error in energy norm), as well as goal-oriented error estimation (to assess the error on specific quantities of interest). We briefly survey the possible extensions to nonconforming numerical methods, as well as more complex (e.g., nonlinear or time-dependent) problems. We also provide some illustrating numerical examples on a linear elasticity problem in three dimensions.
{"title":"An Introductory Review on A Posteriori Error Estimation in Finite Element Computations","authors":"Ludovic Chamoin, Frédéric Legoll","doi":"10.1137/21m1464841","DOIUrl":"https://doi.org/10.1137/21m1464841","url":null,"abstract":"SIAM Review, Volume 65, Issue 4, Page 963-1028, November 2023. <br/> This article is a review of basic concepts and tools devoted to a posteriori error estimation for problems solved with the finite element method. For the sake of simplicity and clarity, we mostly focus on linear elliptic diffusion problems, approximated by a conforming numerical discretization. The main goal of this review is to present in a unified manner a large set of powerful verification methods, centered around the concept of equilibrium. Methods based on that concept provide error bounds that are fully computable and mathematically certified. We discuss recovery methods, residual methods, and duality-based methods for the estimation of the whole solution error (i.e., the error in energy norm), as well as goal-oriented error estimation (to assess the error on specific quantities of interest). We briefly survey the possible extensions to nonconforming numerical methods, as well as more complex (e.g., nonlinear or time-dependent) problems. We also provide some illustrating numerical examples on a linear elasticity problem in three dimensions.","PeriodicalId":49525,"journal":{"name":"SIAM Review","volume":null,"pages":null},"PeriodicalIF":10.2,"publicationDate":"2023-11-07","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"71474871","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":1,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
SIAM Review, Volume 65, Issue 4, Page 1031-1073, November 2023. Solutions to many partial differential equations satisfy certain bounds or constraints. For example, the density and pressure are positive for equations of fluid dynamics, and in the relativistic case the fluid velocity is upper bounded by the speed of light, etc. As widely realized, it is crucial to develop bound-preserving numerical methods that preserve such intrinsic constraints. Exploring provably bound-preserving schemes has attracted much attention and has been actively studied in recent years. This is, however, still a challenging task for many systems, especially those involving nonlinear constraints. Based on some key insights from geometry, we systematically propose an innovative and general framework, referred to as geometric quasilinearization (GQL), which paves a new effective way for studying bound-preserving problems with nonlinear constraints. The essential idea of GQL is to equivalently transform all nonlinear constraints to linear ones, by properly introducing some free auxiliary variables. We establish the fundamental principle and general theory of GQL via the geometric properties of convex regions and propose three simple effective methods for constructing GQL. We apply the GQL approach to a variety of partial differential equations and demonstrate its effectiveness and remarkable advantages for studying bound-preserving schemes, using diverse challenging examples and applications which cannot be easily handled by direct or traditional approaches.
{"title":"Geometric Quasilinearization Framework for Analysis and Design of Bound-Preserving Schemes","authors":"Kailiang Wu, Chi-Wang Shu","doi":"10.1137/21m1458247","DOIUrl":"https://doi.org/10.1137/21m1458247","url":null,"abstract":"SIAM Review, Volume 65, Issue 4, Page 1031-1073, November 2023. <br/> Solutions to many partial differential equations satisfy certain bounds or constraints. For example, the density and pressure are positive for equations of fluid dynamics, and in the relativistic case the fluid velocity is upper bounded by the speed of light, etc. As widely realized, it is crucial to develop bound-preserving numerical methods that preserve such intrinsic constraints. Exploring provably bound-preserving schemes has attracted much attention and has been actively studied in recent years. This is, however, still a challenging task for many systems, especially those involving nonlinear constraints. Based on some key insights from geometry, we systematically propose an innovative and general framework, referred to as geometric quasilinearization (GQL), which paves a new effective way for studying bound-preserving problems with nonlinear constraints. The essential idea of GQL is to equivalently transform all nonlinear constraints to linear ones, by properly introducing some free auxiliary variables. We establish the fundamental principle and general theory of GQL via the geometric properties of convex regions and propose three simple effective methods for constructing GQL. We apply the GQL approach to a variety of partial differential equations and demonstrate its effectiveness and remarkable advantages for studying bound-preserving schemes, using diverse challenging examples and applications which cannot be easily handled by direct or traditional approaches.","PeriodicalId":49525,"journal":{"name":"SIAM Review","volume":null,"pages":null},"PeriodicalIF":10.2,"publicationDate":"2023-11-07","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"71474873","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":1,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
SIAM Review, Volume 65, Issue 4, Page 1107-1107, November 2023. The SIGEST article in this issue is “Are Adaptive Galerkin Schemes Dissipative?” by Rodrigo M. Pereira, Natacha Nguyen van yen, Kai Schneider, and Marie Farge. “Although this may seem a paradox, all exact science is dominated by the idea of approximation.” With this quote from Bertrand Russell from 1931 commences this issue's SIGEST article. Indeed, approximation is at the core of mathematics associated to studying partial differential equations (PDEs) with the idea of approximating the solution to the continuous equation with a finite number of modes. The finite element method for PDEs is a prime exemplar of such an approximation, and much research has been dedicated to getting this approximation as accurate and computationally efficient as possible. In this context, adaptive finite element methods and especially Galerkin methods are often the method of choice. Here, typically, when used for solving evolutionary PDEs the number of modes in the Galerkin scheme is fixed over time. In this article, the authors consider adaptive Galerkin schemes in which the number of modes can change over time, and they introduce a mathematical framework for studying evolutionary PDEs discretized with these schemes. In particular, they show that the associated projection operators, i.e., the operators that project the continuous solution onto the finite-dimensional finite element spaces, are discontinuous and introduce energy dissipation. That this is a significant result is demonstrated by studying adaptive Galerkin schemes for the time evolution of the inviscid Burgers equation in 1D and the incompressible Euler equations in 2D and 3D. They show that adaptive wavelet schemes regularize the solution of the Galerkin truncated equations and yield convergence towards the exact dissipative solution for the inviscid Burgers equation. Also for the Euler equations this regularizing effect can be numerically observed though no exact reference solutions are available in this case. This motivates, in particular, adaptive wavelet Galerkin schemes for nonlinear hyperbolic conservation laws and leave their systematic study for this class of PDEs for an interesting future work. For the SIGEST article the authors have expanded their original Multiscale Modeling & Simulation article by providing a more comprehensive discussion on adaptive Galerkin methods fit for a general mathematical audience. They have also added a new section on continuous wavelet analysis of the inviscid Burgers equation, analyzing its time evolution, and added an illustration for the development of thermal resonances in wavelet space. Overall, adaptive Galerkin methods and their mathematical properties will be of interest to a wide range of applied mathematicians who study PDE models, and also to applied analysts and numerical analysts who wish to simulate PDEs numerically.
SIAM评论，第65卷第4期，第1107-1107页，2023年11月。本期SIGEST的文章是Rodrigo M.Pereira、Natacha Nguyen van yen、Kai Schneider和Marie Farge的《自适应伽辽金方案是耗散的吗？》。“尽管这似乎是一个悖论，但所有精确科学都被近似的思想所支配。”引用伯特兰·罗素1931年的这句话，开始了本期SIGEST的文章。事实上，近似是研究偏微分方程（PDE）的数学核心，其思想是用有限个模式近似连续方程的解。偏微分方程的有限元方法是这种近似的一个主要例子，许多研究都致力于使这种近似尽可能准确和高效。在这种情况下，自适应有限元方法，尤其是伽辽金方法通常是首选方法。这里，通常，当用于求解进化偏微分方程时，Galerkin格式中的模式数随时间固定。在本文中，作者考虑了模式数量可以随时间变化的自适应Galerkin格式，并介绍了一个研究用这些格式离散的进化偏微分方程的数学框架。特别地，他们表明，相关的投影算子，即将连续解投影到有限维有限元空间上的算子，是不连续的，并引入能量耗散。通过研究一维无粘性Burgers方程和二维和三维不可压缩Euler方程的自适应Galerkin格式，证明了这是一个重要的结果。他们证明了自适应小波格式正则化了Galerkin截断方程的解，并使无粘Burgers方程的精确耗散解收敛。同样，对于欧拉方程，这种正则化效应可以在数值上观察到，尽管在这种情况下没有精确的参考解。这尤其激发了非线性双曲守恒律的自适应小波Galerkin格式，并为这类偏微分方程的系统研究留下了有趣的未来工作。对于SIGEST的文章，作者扩展了他们最初的多尺度建模&；仿真文章通过提供一个更全面的讨论自适应伽辽金方法适合一般数学受众。他们还增加了一个关于无粘性Burgers方程的连续小波分析的新章节，分析了其时间演化，并增加了小波空间中热共振发展的说明。总的来说，自适应伽辽金方法及其数学性质将引起广泛研究偏微分方程模型的应用数学家的兴趣，也会引起希望数值模拟偏微分方程的应用分析师和数值分析师的兴趣。
{"title":"SIGEST","authors":"The Editors","doi":"10.1137/23n97579x","DOIUrl":"https://doi.org/10.1137/23n97579x","url":null,"abstract":"SIAM Review, Volume 65, Issue 4, Page 1107-1107, November 2023. <br/> The SIGEST article in this issue is “Are Adaptive Galerkin Schemes Dissipative?” by Rodrigo M. Pereira, Natacha Nguyen van yen, Kai Schneider, and Marie Farge. “Although this may seem a paradox, all exact science is dominated by the idea of approximation.” With this quote from Bertrand Russell from 1931 commences this issue's SIGEST article. Indeed, approximation is at the core of mathematics associated to studying partial differential equations (PDEs) with the idea of approximating the solution to the continuous equation with a finite number of modes. The finite element method for PDEs is a prime exemplar of such an approximation, and much research has been dedicated to getting this approximation as accurate and computationally efficient as possible. In this context, adaptive finite element methods and especially Galerkin methods are often the method of choice. Here, typically, when used for solving evolutionary PDEs the number of modes in the Galerkin scheme is fixed over time. In this article, the authors consider adaptive Galerkin schemes in which the number of modes can change over time, and they introduce a mathematical framework for studying evolutionary PDEs discretized with these schemes. In particular, they show that the associated projection operators, i.e., the operators that project the continuous solution onto the finite-dimensional finite element spaces, are discontinuous and introduce energy dissipation. That this is a significant result is demonstrated by studying adaptive Galerkin schemes for the time evolution of the inviscid Burgers equation in 1D and the incompressible Euler equations in 2D and 3D. They show that adaptive wavelet schemes regularize the solution of the Galerkin truncated equations and yield convergence towards the exact dissipative solution for the inviscid Burgers equation. Also for the Euler equations this regularizing effect can be numerically observed though no exact reference solutions are available in this case. This motivates, in particular, adaptive wavelet Galerkin schemes for nonlinear hyperbolic conservation laws and leave their systematic study for this class of PDEs for an interesting future work. For the SIGEST article the authors have expanded their original Multiscale Modeling & Simulation article by providing a more comprehensive discussion on adaptive Galerkin methods fit for a general mathematical audience. They have also added a new section on continuous wavelet analysis of the inviscid Burgers equation, analyzing its time evolution, and added an illustration for the development of thermal resonances in wavelet space. Overall, adaptive Galerkin methods and their mathematical properties will be of interest to a wide range of applied mathematicians who study PDE models, and also to applied analysts and numerical analysts who wish to simulate PDEs numerically.","PeriodicalId":49525,"journal":{"name":"SIAM Review","volume":null,"pages":null},"PeriodicalIF":10.2,"publicationDate":"2023-11-07","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"71473802","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":1,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
SIAM Review, Volume 65, Issue 4, Page 1029-1029, November 2023. This issue's two Research Spotlights highlight techniques for obtaining ever more realistic solutions to challenging systems of partial differential equations (PDEs). Although borne from different fields of applied mathematics, both papers aim to leverage prior information to improve the fidelity and practical solution of PDEs. How predictive is a model if it violates constraints known to be satisfied by the underlying physical phenomena or otherwise imposed by numerical stability requirements? Fundamentally, one desires to avoid nonlinear instabilities, nonphysical solutions, and numerical method divergence whenever these constraints are known a priori, but this pursuit is often easier said than done. In this issue's first Research Spotlight, “Geometric Quasilinearization Framework for Analysis and Design of Bound-Preserving Schemes,” authors Kailiang Wu and Chi-Wang Shu extend the range of systems of PDEs for which bound constraints can be imposed on solutions. For example, solutions of the special relativistic magnetohydrodynamic equations have fluid velocities upper bounded by the speed of light. Such constraint equations, and many others illustrated by the authors, are nonlinear and hence challenging to enforce. The authors lift such problems into a higher-dimensional space with the benefit of representing the original nonlinear constraints with higher-dimensional linear constraints based on the geometric properties of the underlying convex regions. The authors illuminate when such lifting results in an equivalent representation---a geometric quasilinearization (GQL)---and derive three techniques for constructing GQL-based bound-preserving methods in practice. The applicability of the resulting framework is based on the form of the nonlinear constraint, in this case based on convex feasible regions, but provides a potential path forward for satisfying even more general constraints. The second Research Spotlight addresses the estimation of unknown, spatially varying PDE system parameters from data. Of particular interest to authors David Aristoff and Wolfgang Bangerth are Bayesian formulations for such inverse problems since these formulations yield predictive distributions on the unknown parameters. Obtaining such a distribution can be highly beneficial for uncertainty quantification and other downstream uses, but Bayesian inversion quickly becomes computationally impractical as the dimension of the unknown parameters grows. More difficult still is validating the obtained distributions. In “A Benchmark for the Bayesian Inversion of Coefficients in Partial Differential Equations” the authors seek to advance the field and understanding of the state of the art through a comprehensive specification of a 64-dimensional benchmark problem. The authors provide a complete description of the underlying physical problem, data-generating process, likelihood, and prior, as well as open
{"title":"Research Spotlights","authors":"Stefan M. Wild","doi":"10.1137/23n975788","DOIUrl":"https://doi.org/10.1137/23n975788","url":null,"abstract":"SIAM Review, Volume 65, Issue 4, Page 1029-1029, November 2023. <br/> This issue's two Research Spotlights highlight techniques for obtaining ever more realistic solutions to challenging systems of partial differential equations (PDEs). Although borne from different fields of applied mathematics, both papers aim to leverage prior information to improve the fidelity and practical solution of PDEs. How predictive is a model if it violates constraints known to be satisfied by the underlying physical phenomena or otherwise imposed by numerical stability requirements? Fundamentally, one desires to avoid nonlinear instabilities, nonphysical solutions, and numerical method divergence whenever these constraints are known a priori, but this pursuit is often easier said than done. In this issue's first Research Spotlight, “Geometric Quasilinearization Framework for Analysis and Design of Bound-Preserving Schemes,” authors Kailiang Wu and Chi-Wang Shu extend the range of systems of PDEs for which bound constraints can be imposed on solutions. For example, solutions of the special relativistic magnetohydrodynamic equations have fluid velocities upper bounded by the speed of light. Such constraint equations, and many others illustrated by the authors, are nonlinear and hence challenging to enforce. The authors lift such problems into a higher-dimensional space with the benefit of representing the original nonlinear constraints with higher-dimensional linear constraints based on the geometric properties of the underlying convex regions. The authors illuminate when such lifting results in an equivalent representation---a geometric quasilinearization (GQL)---and derive three techniques for constructing GQL-based bound-preserving methods in practice. The applicability of the resulting framework is based on the form of the nonlinear constraint, in this case based on convex feasible regions, but provides a potential path forward for satisfying even more general constraints. The second Research Spotlight addresses the estimation of unknown, spatially varying PDE system parameters from data. Of particular interest to authors David Aristoff and Wolfgang Bangerth are Bayesian formulations for such inverse problems since these formulations yield predictive distributions on the unknown parameters. Obtaining such a distribution can be highly beneficial for uncertainty quantification and other downstream uses, but Bayesian inversion quickly becomes computationally impractical as the dimension of the unknown parameters grows. More difficult still is validating the obtained distributions. In “A Benchmark for the Bayesian Inversion of Coefficients in Partial Differential Equations” the authors seek to advance the field and understanding of the state of the art through a comprehensive specification of a 64-dimensional benchmark problem. The authors provide a complete description of the underlying physical problem, data-generating process, likelihood, and prior, as well as open","PeriodicalId":49525,"journal":{"name":"SIAM Review","volume":null,"pages":null},"PeriodicalIF":10.2,"publicationDate":"2023-11-07","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"71474872","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":1,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
SIAM Review, Volume 65, Issue 3, Page 806-828, August 2023. In $d$ dimensions, accurately approximating an arbitrary function oscillating with frequency $lesssim k$ requires $sim$$k^d$ degrees of freedom. A numerical method for solving the Helmholtz equation (with wavenumber $k$ and in $d$ dimensions) suffers from the pollution effect if, as $k→∞$, the total number of degrees of freedom needed to maintain accuracy grows faster than this natural threshold (i.e., faster than $k^d$ for domain-based formulations, such as finite element methods, and $k^{d-1}$ for boundary-based formulations, such as boundary element methods). It is well known that the $h$-version of the finite element method (FEM) (where accuracy is increased by decreasing the meshwidth $h$ and keeping the polynomial degree $p$ fixed) suffers from the pollution effect, and research over the last $sim$30 years has resulted in a near-complete rigorous understanding of how quickly the number of degrees of freedom must grow with $k$ to maintain accuracy (and how this depends on both $p$ and properties of the scatterer). In contrast to the $h$-FEM, at least empirically, the $h$-version of the boundary element method (BEM) does not suffer from the pollution effect (recall that in the boundary element method the scattering problem is reformulated as an integral equation on the boundary of the scatterer, with this integral equation then solved numerically using a finite element--type approximation space). However, the current best results in the literature on how quickly the number of degrees of freedom for the $h$-BEM must grow with $k$ to maintain accuracy fall short of proving this. In this paper, we prove that the $h$-version of the Galerkin method applied to the standard second-kind boundary integral equations for solving the Helmholtz exterior Dirichlet problem does not suffer from the pollution effect when the obstacle is nontrapping (i.e., does not trap geometric-optic rays). While the proof of this result relies on information about the large-$k$ behavior of Helmholtz solution operators, we show in an appendix how the result can be proved using only Fourier series and asymptotics of Hankel and Bessel functions when the obstacle is a 2-d ball.
{"title":"Does the Helmholtz Boundary Element Method Suffer from the Pollution Effect?","authors":"J. Galkowski, E. A. Spence","doi":"10.1137/22m1474199","DOIUrl":"https://doi.org/10.1137/22m1474199","url":null,"abstract":"SIAM Review, Volume 65, Issue 3, Page 806-828, August 2023. <br/> In $d$ dimensions, accurately approximating an arbitrary function oscillating with frequency $lesssim k$ requires $sim$$k^d$ degrees of freedom. A numerical method for solving the Helmholtz equation (with wavenumber $k$ and in $d$ dimensions) suffers from the pollution effect if, as $k→∞$, the total number of degrees of freedom needed to maintain accuracy grows faster than this natural threshold (i.e., faster than $k^d$ for domain-based formulations, such as finite element methods, and $k^{d-1}$ for boundary-based formulations, such as boundary element methods). It is well known that the $h$-version of the finite element method (FEM) (where accuracy is increased by decreasing the meshwidth $h$ and keeping the polynomial degree $p$ fixed) suffers from the pollution effect, and research over the last $sim$30 years has resulted in a near-complete rigorous understanding of how quickly the number of degrees of freedom must grow with $k$ to maintain accuracy (and how this depends on both $p$ and properties of the scatterer). In contrast to the $h$-FEM, at least empirically, the $h$-version of the boundary element method (BEM) does not suffer from the pollution effect (recall that in the boundary element method the scattering problem is reformulated as an integral equation on the boundary of the scatterer, with this integral equation then solved numerically using a finite element--type approximation space). However, the current best results in the literature on how quickly the number of degrees of freedom for the $h$-BEM must grow with $k$ to maintain accuracy fall short of proving this. In this paper, we prove that the $h$-version of the Galerkin method applied to the standard second-kind boundary integral equations for solving the Helmholtz exterior Dirichlet problem does not suffer from the pollution effect when the obstacle is nontrapping (i.e., does not trap geometric-optic rays). While the proof of this result relies on information about the large-$k$ behavior of Helmholtz solution operators, we show in an appendix how the result can be proved using only Fourier series and asymptotics of Hankel and Bessel functions when the obstacle is a 2-d ball.","PeriodicalId":49525,"journal":{"name":"SIAM Review","volume":null,"pages":null},"PeriodicalIF":10.2,"publicationDate":"2023-08-08","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"71509718","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":1,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
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