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":"5 4","pages":""},"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":"5 2","pages":""},"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":"11 11","pages":""},"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. <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
{"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":"5 3","pages":""},"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 774-805, August 2023. The beauty and simplicity of compartment modeling makes it a useful approach for simulating dynamics in an amazingly wide range of applications, which are growing rapidly especially in global carbon cycling, hydrological network flows, and epidemiology and population dynamics. These contexts, however, often involve compartment-to-compartment flows that are incongruent with the conventional assumption of complete mixing that results in exponential residence times in linear models. Here we detail a general method for assigning any desired residence time distribution to a given intercompartmental flow, extending compartment modeling capability to transport operations, power-law residence times, diffusions, etc., without resorting to composite compartments, fractional calculus, or partial differential equations (PDEs) for diffusive transport. This is achieved by writing the mass exchange rate coefficients as functions of age-in-compartment as done in one of the first compartment models in 1917, at the cost of converting the usual ordinary differential equations to a system of first-order PDEs. The PDEs are readily converted to a system of integral equations for which a numerical method is devised. Example calculations demonstrate incorporation of advective lags, advective-dispersive transport, power-law residence time distributions, or diffusive domains in compartment models.
{"title":"Compartment Models with Memory","authors":"Timothy Ginn, Lynn Schreyer","doi":"10.1137/21m1437160","DOIUrl":"https://doi.org/10.1137/21m1437160","url":null,"abstract":"SIAM Review, Volume 65, Issue 3, Page 774-805, August 2023. <br/> The beauty and simplicity of compartment modeling makes it a useful approach for simulating dynamics in an amazingly wide range of applications, which are growing rapidly especially in global carbon cycling, hydrological network flows, and epidemiology and population dynamics. These contexts, however, often involve compartment-to-compartment flows that are incongruent with the conventional assumption of complete mixing that results in exponential residence times in linear models. Here we detail a general method for assigning any desired residence time distribution to a given intercompartmental flow, extending compartment modeling capability to transport operations, power-law residence times, diffusions, etc., without resorting to composite compartments, fractional calculus, or partial differential equations (PDEs) for diffusive transport. This is achieved by writing the mass exchange rate coefficients as functions of age-in-compartment as done in one of the first compartment models in 1917, at the cost of converting the usual ordinary differential equations to a system of first-order PDEs. The PDEs are readily converted to a system of integral equations for which a numerical method is devised. Example calculations demonstrate incorporation of advective lags, advective-dispersive transport, power-law residence time distributions, or diffusive domains in compartment models.","PeriodicalId":49525,"journal":{"name":"SIAM Review","volume":"32 1","pages":""},"PeriodicalIF":10.2,"publicationDate":"2023-08-08","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"71509717","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":"31 12","pages":""},"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}
SIAM Review, Volume 65, Issue 3, Page 735-773, August 2023. We analyze neural ordinary differential equations (NODEs) from a control theoretical perspective to address some of the main properties and paradigms of deep learning (DL), in particular, data classification and universal approximation. These objectives are tackled and achieved from the perspective of the simultaneous control of systems of NODEs. For instance, in the context of classification, each item to be classified corresponds to a different initial datum for the control problem of the NODE, to be classified, all of them by the same common control, to the location (a subdomain of the Euclidean space) associated to each label. Our proofs are genuinely nonlinear and constructive, allowing us to estimate the complexity of the control strategies we develop. The nonlinear nature of the activation functions governing the dynamics of NODEs under consideration plays a key role in our proofs, since it allows deforming half of the phase space while the other half remains invariant, a property that classical models in mechanics do not fulfill. This very property allows us to build elementary controls inducing specific dynamics and transformations whose concatenation, along with properly chosen hyperplanes, allows us to achieve our goals in finitely many steps. The nonlinearity of the dynamics is assumed to be Lipschitz. Therefore, our results apply also in the particular case of the ReLU activation function. We also present the counterparts in the context of the control of neural transport equations, establishing a link between optimal transport and deep neural networks.
{"title":"Neural ODE Control for Classification, Approximation, and Transport","authors":"Domènec Ruiz-Balet, Enrique Zuazua","doi":"10.1137/21m1411433","DOIUrl":"https://doi.org/10.1137/21m1411433","url":null,"abstract":"SIAM Review, Volume 65, Issue 3, Page 735-773, August 2023. <br/> We analyze neural ordinary differential equations (NODEs) from a control theoretical perspective to address some of the main properties and paradigms of deep learning (DL), in particular, data classification and universal approximation. These objectives are tackled and achieved from the perspective of the simultaneous control of systems of NODEs. For instance, in the context of classification, each item to be classified corresponds to a different initial datum for the control problem of the NODE, to be classified, all of them by the same common control, to the location (a subdomain of the Euclidean space) associated to each label. Our proofs are genuinely nonlinear and constructive, allowing us to estimate the complexity of the control strategies we develop. The nonlinear nature of the activation functions governing the dynamics of NODEs under consideration plays a key role in our proofs, since it allows deforming half of the phase space while the other half remains invariant, a property that classical models in mechanics do not fulfill. This very property allows us to build elementary controls inducing specific dynamics and transformations whose concatenation, along with properly chosen hyperplanes, allows us to achieve our goals in finitely many steps. The nonlinearity of the dynamics is assumed to be Lipschitz. Therefore, our results apply also in the particular case of the ReLU activation function. We also present the counterparts in the context of the control of neural transport equations, establishing a link between optimal transport and deep neural networks.","PeriodicalId":49525,"journal":{"name":"SIAM Review","volume":"48 1","pages":""},"PeriodicalIF":10.2,"publicationDate":"2023-08-08","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"71518431","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 599-599, August 2023. Apart from a short erratum, which concerns the correction of some coefficients in a differential equation in the original paper, this issue contains two Survey and Review articles. “On and Beyond Total Variation Regularization in Imaging: The Role of Space Variance,” authored by Monica Pragliola, Luca Calatroni, Alessandro Lanza, and Fiorella Sgallari, reviews total variation (TV)-type image reconstruction algorithms with a focus on Bayesian interpretations. The paper scientifically travels across various disciplines by considering a standard example problem to highlight extensions for the TV regularization model. A main contribution is a space-variant framework which allows one to describe the contents of an image at a local scale. Important applications of space-variant models are tomography, e.g., magnetic resonance imaging, electrical impedance tomography, positron emission tomography, and photoacoustic tomography, or noninvasive digital reconstruction, e.g., for ancient frescoes, illuminated manuscripts, surface colorization, etc. The unified view of many of the different models within the Bayesian framework enables one to design flexible and adaptive image regularization functionals which take advantage of the form of the underlying gradient distributions through statistical approaches. The paper contains theoretical results as well as sections on algorithmic optimization (based on the alternating direction methods of multipliers) and numerical tests for examples from image deblurring. Thus it should be interesting for researchers from several disciplines. “What Are Higher-Order Networks” is a question raised and answered by Christian Bick, Elizabeth Gross, Heather A. Harrington, and Michael T. Schaub. In short, higher-order networks are a refurbishment of graphs, removing/overcoming some of the limitations of pairwise relationships by enabling the modeling of polyadic relations in real-world systems, such as reactions in biochemical systems with several species or reagents, or interactions of multiple people in social networks. The main topics of discussion are the understanding of the “shape” of data (by identifying and classifying topological and geometrical properties of the data), the modeling of relational data via higher-order networks, and network dynamical systems (describing couplings between dynamical units). The focus of the presentation is on the mathematical aspects of the topics, but a multitude of applications are mentioned. The impressive list of references comprises 316 entries. We believe the paper to be interesting for a broad audience.
{"title":"Survey and Review","authors":"Marlis Hochbruck","doi":"10.1137/23n975727","DOIUrl":"https://doi.org/10.1137/23n975727","url":null,"abstract":"SIAM Review, Volume 65, Issue 3, Page 599-599, August 2023. <br/> Apart from a short erratum, which concerns the correction of some coefficients in a differential equation in the original paper, this issue contains two Survey and Review articles. “On and Beyond Total Variation Regularization in Imaging: The Role of Space Variance,” authored by Monica Pragliola, Luca Calatroni, Alessandro Lanza, and Fiorella Sgallari, reviews total variation (TV)-type image reconstruction algorithms with a focus on Bayesian interpretations. The paper scientifically travels across various disciplines by considering a standard example problem to highlight extensions for the TV regularization model. A main contribution is a space-variant framework which allows one to describe the contents of an image at a local scale. Important applications of space-variant models are tomography, e.g., magnetic resonance imaging, electrical impedance tomography, positron emission tomography, and photoacoustic tomography, or noninvasive digital reconstruction, e.g., for ancient frescoes, illuminated manuscripts, surface colorization, etc. The unified view of many of the different models within the Bayesian framework enables one to design flexible and adaptive image regularization functionals which take advantage of the form of the underlying gradient distributions through statistical approaches. The paper contains theoretical results as well as sections on algorithmic optimization (based on the alternating direction methods of multipliers) and numerical tests for examples from image deblurring. Thus it should be interesting for researchers from several disciplines. “What Are Higher-Order Networks” is a question raised and answered by Christian Bick, Elizabeth Gross, Heather A. Harrington, and Michael T. Schaub. In short, higher-order networks are a refurbishment of graphs, removing/overcoming some of the limitations of pairwise relationships by enabling the modeling of polyadic relations in real-world systems, such as reactions in biochemical systems with several species or reagents, or interactions of multiple people in social networks. The main topics of discussion are the understanding of the “shape” of data (by identifying and classifying topological and geometrical properties of the data), the modeling of relational data via higher-order networks, and network dynamical systems (describing couplings between dynamical units). The focus of the presentation is on the mathematical aspects of the topics, but a multitude of applications are mentioned. The impressive list of references comprises 316 entries. We believe the paper to be interesting for a broad audience.","PeriodicalId":49525,"journal":{"name":"SIAM Review","volume":"32 6","pages":""},"PeriodicalIF":10.2,"publicationDate":"2023-08-08","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"71509712","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 732-732, August 2023. This erratum corrects an error in the coefficients of equation (6.23) in the original paper [H. Miao, X. Xia, A. S. Perelson, and H. Wu, SIAM Rev., 53 (2011), pp. 3--39].
{"title":"Erratum: On Identifiability of Nonlinear ODE Models and Applications in Viral Dynamics","authors":"Hongyu Miao, Alan S. Perelson, Hulin Wu","doi":"10.1137/23m1568958","DOIUrl":"https://doi.org/10.1137/23m1568958","url":null,"abstract":"SIAM Review, Volume 65, Issue 3, Page 732-732, August 2023. <br/> This erratum corrects an error in the coefficients of equation (6.23) in the original paper [H. Miao, X. Xia, A. S. Perelson, and H. Wu, SIAM Rev., 53 (2011), pp. 3--39].","PeriodicalId":49525,"journal":{"name":"SIAM Review","volume":"32 3","pages":""},"PeriodicalIF":10.2,"publicationDate":"2023-08-08","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"71509715","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 887-902, August 2023. Some simple models of a child swinging on a playground swing are presented. These are analyzed using techniques from Lagrangian mechanics with a twist: the child changes the configuration of the system by sudden movements of their body at key moments in the oscillation. This can lead to jumps in the generalized coordinates describing the system and/or their velocities. Jump conditions can be determined by integrating the Euler--Lagrange equations over a short time interval and then taking the limit as this time interval goes to zero. These models give insights into strategies used by swingers, and answer such vexed questions such as whether it is possible for the swing to go through a full 360$^circ$ turn over its pivot. A model of an instability at the pivot observed by Colin Furze in a rigid swing constructed to rotate through 360$^circ$ is also described. This uses a novel double pendulum configuration in which the two components of the pendulums are constrained to move in orthogonal planes.
{"title":"Piecewise Smooth Models of Pumping a Child's Swing","authors":"Brigid Murphy, Paul Glendinning","doi":"10.1137/19m1268574","DOIUrl":"https://doi.org/10.1137/19m1268574","url":null,"abstract":"SIAM Review, Volume 65, Issue 3, Page 887-902, August 2023. <br/> Some simple models of a child swinging on a playground swing are presented. These are analyzed using techniques from Lagrangian mechanics with a twist: the child changes the configuration of the system by sudden movements of their body at key moments in the oscillation. This can lead to jumps in the generalized coordinates describing the system and/or their velocities. Jump conditions can be determined by integrating the Euler--Lagrange equations over a short time interval and then taking the limit as this time interval goes to zero. These models give insights into strategies used by swingers, and answer such vexed questions such as whether it is possible for the swing to go through a full 360$^circ$ turn over its pivot. A model of an instability at the pivot observed by Colin Furze in a rigid swing constructed to rotate through 360$^circ$ is also described. This uses a novel double pendulum configuration in which the two components of the pendulums are constrained to move in orthogonal planes.","PeriodicalId":49525,"journal":{"name":"SIAM Review","volume":"43 2","pages":""},"PeriodicalIF":10.2,"publicationDate":"2023-08-08","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"71516803","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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