SIAM Review, Volume 66, Issue 1, Page 89-89, February 2024. As modeling, simulation, and data-driven capabilities continue to advance and be adopted for an ever expanding set of applications and downstream tasks, there has been an increased need for quantifying the uncertainty in the resulting predictions. In “Easy Uncertainty Quantification (EasyUQ): Generating Predictive Distributions from Single-Valued Model Output,” authors Eva-Maria Walz, Alexander Henzi, Johanna Ziegel, and Tilmann Gneiting provide a methodology for moving beyond deterministic scalar-valued predictions to obtain particular statistical distributions for these predictions. The approach relies on training data of model output-observation pairs of scalars, and hence does not require access to higher-dimensional inputs or latent variables. The authors use numerical weather prediction as a particular example, where one can obtain repeated forecasts, and corresponding observations, of temperatures at a specific location. Given a predicted temperature, the EasyUQ approach provides a nonparametric distribution of temperatures around this value. EasyUQ uses the training data to effectively minimize an empirical score subject to a stochastic monotonicity constraint, which ensures that the predictive distribution values become larger as the model output value grows. In doing so, the approach inherits the theoretical properties of optimality and consistency enjoyed by so-called isotonic distributional regression methods. The authors emphasize that the basic version of EasyUQ does not require elaborate hyperparameter tuning. They also introduce a more sophisticated version that relies on kernel smoothing to yield predictive probability densities while preserving key properties of the basic version. The paper demonstrates how EasyUQ compares with the standard technique of applying a Gaussian error distribution to a deterministic forecast as well as how EasyUQ can be used to obtain uncertainty estimates for artificial neural network outputs. The approach will be especially of interest for settings when inputs or other latent variables are unreliable or unavailable since it offers a straightforward yet statistically principled and computationally efficient way for working only with outputs and observations.
{"title":"Research Spotlights","authors":"Stefan M. Wild","doi":"10.1137/24n975839","DOIUrl":"https://doi.org/10.1137/24n975839","url":null,"abstract":"SIAM Review, Volume 66, Issue 1, Page 89-89, February 2024. <br/> As modeling, simulation, and data-driven capabilities continue to advance and be adopted for an ever expanding set of applications and downstream tasks, there has been an increased need for quantifying the uncertainty in the resulting predictions. In “Easy Uncertainty Quantification (EasyUQ): Generating Predictive Distributions from Single-Valued Model Output,” authors Eva-Maria Walz, Alexander Henzi, Johanna Ziegel, and Tilmann Gneiting provide a methodology for moving beyond deterministic scalar-valued predictions to obtain particular statistical distributions for these predictions. The approach relies on training data of model output-observation pairs of scalars, and hence does not require access to higher-dimensional inputs or latent variables. The authors use numerical weather prediction as a particular example, where one can obtain repeated forecasts, and corresponding observations, of temperatures at a specific location. Given a predicted temperature, the EasyUQ approach provides a nonparametric distribution of temperatures around this value. EasyUQ uses the training data to effectively minimize an empirical score subject to a stochastic monotonicity constraint, which ensures that the predictive distribution values become larger as the model output value grows. In doing so, the approach inherits the theoretical properties of optimality and consistency enjoyed by so-called isotonic distributional regression methods. The authors emphasize that the basic version of EasyUQ does not require elaborate hyperparameter tuning. They also introduce a more sophisticated version that relies on kernel smoothing to yield predictive probability densities while preserving key properties of the basic version. The paper demonstrates how EasyUQ compares with the standard technique of applying a Gaussian error distribution to a deterministic forecast as well as how EasyUQ can be used to obtain uncertainty estimates for artificial neural network outputs. The approach will be especially of interest for settings when inputs or other latent variables are unreliable or unavailable since it offers a straightforward yet statistically principled and computationally efficient way for working only with outputs and observations.","PeriodicalId":49525,"journal":{"name":"SIAM Review","volume":null,"pages":null},"PeriodicalIF":10.2,"publicationDate":"2024-02-08","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"139704929","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 66, Issue 1, Page 125-146, February 2024. We analyze the spectrum of the operator $Delta^{-1} [nabla cdot (Knabla u)]$ subject to homogeneous Dirichlet or Neumann boundary conditions, where $Delta$ denotes the Laplacian and $K=K(x,y)$ is a symmetric tensor. Our main result shows that this spectrum can be derived from the spectral decomposition $K=Q Lambda Q^T$, where $Q=Q(x,y)$ is an orthogonal matrix and $Lambda=Lambda(x,y)$ is a diagonal matrix. More precisely, provided that $K$ is continuous, the spectrum equals the convex hull of the ranges of the diagonal function entries of $Lambda$. The domain involved is assumed to be bounded and Lipschitz. In addition to studying operators defined on infinite-dimensional Sobolev spaces, we also report on recent results concerning their discretized finite-dimensional counterparts. More specifically, even though $Delta^{-1} [nabla cdot (Knabla u)]$ is not compact, it turns out that every point in the spectrum of this operator can, to an arbitrary accuracy, be approximated by eigenvalues of the associated generalized algebraic eigenvalue problems arising from discretizations. Our theoretical investigations are illuminated by numerical experiments. The results presented in this paper extend previous analyses which have addressed elliptic differential operators with scalar coefficient functions. Our investigation is motivated by both preconditioning issues (efficient numerical computations) and the need to further develop the spectral theory of second order PDEs (core analysis).
{"title":"A Simple Formula for the Generalized Spectrum of Second Order Self-Adjoint Differential Operators","authors":"Bjørn Fredrik Nielsen, Zdeněk Strakoš","doi":"10.1137/23m1600992","DOIUrl":"https://doi.org/10.1137/23m1600992","url":null,"abstract":"SIAM Review, Volume 66, Issue 1, Page 125-146, February 2024. <br/> We analyze the spectrum of the operator $Delta^{-1} [nabla cdot (Knabla u)]$ subject to homogeneous Dirichlet or Neumann boundary conditions, where $Delta$ denotes the Laplacian and $K=K(x,y)$ is a symmetric tensor. Our main result shows that this spectrum can be derived from the spectral decomposition $K=Q Lambda Q^T$, where $Q=Q(x,y)$ is an orthogonal matrix and $Lambda=Lambda(x,y)$ is a diagonal matrix. More precisely, provided that $K$ is continuous, the spectrum equals the convex hull of the ranges of the diagonal function entries of $Lambda$. The domain involved is assumed to be bounded and Lipschitz. In addition to studying operators defined on infinite-dimensional Sobolev spaces, we also report on recent results concerning their discretized finite-dimensional counterparts. More specifically, even though $Delta^{-1} [nabla cdot (Knabla u)]$ is not compact, it turns out that every point in the spectrum of this operator can, to an arbitrary accuracy, be approximated by eigenvalues of the associated generalized algebraic eigenvalue problems arising from discretizations. Our theoretical investigations are illuminated by numerical experiments. The results presented in this paper extend previous analyses which have addressed elliptic differential operators with scalar coefficient functions. Our investigation is motivated by both preconditioning issues (efficient numerical computations) and the need to further develop the spectral theory of second order PDEs (core analysis).","PeriodicalId":49525,"journal":{"name":"SIAM Review","volume":null,"pages":null},"PeriodicalIF":10.2,"publicationDate":"2024-02-08","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"139705096","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 66, Issue 1, Page 149-160, February 2024. In a recent article in this journal, Gouveia and Stone [``Generating Resonant and Repeated Root Solutions to Ordinary Differential Equations Using Perturbation Methods,” SIAM Rev., 64 (2022), pp. 485--499] described a method for finding exact solutions to resonantly forced linear ordinary differential equations, and for finding the general solution of repeated root linear systems. It is shown here that applying their mathematical justification directly yields a method that is faster and algebraically simpler than the method they described. This method seems to be unknown in the undergraduate textbook literature, although it certainly should be present there as it is elegant and simple to apply, generally giving solutions with much less work than variation of parameters.
{"title":"Resonantly Forced ODEs and Repeated Roots","authors":"Allan R. Willms","doi":"10.1137/23m1545148","DOIUrl":"https://doi.org/10.1137/23m1545148","url":null,"abstract":"SIAM Review, Volume 66, Issue 1, Page 149-160, February 2024. <br/> In a recent article in this journal, Gouveia and Stone [``Generating Resonant and Repeated Root Solutions to Ordinary Differential Equations Using Perturbation Methods,” SIAM Rev., 64 (2022), pp. 485--499] described a method for finding exact solutions to resonantly forced linear ordinary differential equations, and for finding the general solution of repeated root linear systems. It is shown here that applying their mathematical justification directly yields a method that is faster and algebraically simpler than the method they described. This method seems to be unknown in the undergraduate textbook literature, although it certainly should be present there as it is elegant and simple to apply, generally giving solutions with much less work than variation of parameters.","PeriodicalId":49525,"journal":{"name":"SIAM Review","volume":null,"pages":null},"PeriodicalIF":10.2,"publicationDate":"2024-02-08","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"139704932","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}
Eva-Maria Walz, Alexander Henzi, Johanna Ziegel, Tilmann Gneiting
SIAM Review, Volume 66, Issue 1, Page 91-122, February 2024. How can we quantify uncertainty if our favorite computational tool---be it a numerical, statistical, or machine learning approach, or just any computer model---provides single-valued output only? In this article, we introduce the Easy Uncertainty Quantification (EasyUQ) technique, which transforms real-valued model output into calibrated statistical distributions, based solely on training data of model output--outcome pairs, without any need to access model input. In its basic form, EasyUQ is a special case of the recently introduced isotonic distributional regression (IDR) technique that leverages the pool-adjacent-violators algorithm for nonparametric isotonic regression. EasyUQ yields discrete predictive distributions that are calibrated and optimal in finite samples, subject to stochastic monotonicity. The workflow is fully automated, without any need for tuning. The Smooth EasyUQ approach supplements IDR with kernel smoothing, to yield continuous predictive distributions that preserve key properties of the basic form, including both stochastic monotonicity with respect to the original model output and asymptotic consistency. For the selection of kernel parameters, we introduce multiple one-fit grid search, a computationally much less demanding approximation to leave-one-out cross-validation. We use simulation examples and forecast data from weather prediction to illustrate the techniques. In a study of benchmark problems from machine learning, we show how EasyUQ and Smooth EasyUQ can be integrated into the workflow of neural network learning and hyperparameter tuning, and we find EasyUQ to be competitive with conformal prediction as well as more elaborate input-based approaches.
{"title":"Easy Uncertainty Quantification (EasyUQ): Generating Predictive Distributions from Single-Valued Model Output","authors":"Eva-Maria Walz, Alexander Henzi, Johanna Ziegel, Tilmann Gneiting","doi":"10.1137/22m1541915","DOIUrl":"https://doi.org/10.1137/22m1541915","url":null,"abstract":"SIAM Review, Volume 66, Issue 1, Page 91-122, February 2024. <br/> How can we quantify uncertainty if our favorite computational tool---be it a numerical, statistical, or machine learning approach, or just any computer model---provides single-valued output only? In this article, we introduce the Easy Uncertainty Quantification (EasyUQ) technique, which transforms real-valued model output into calibrated statistical distributions, based solely on training data of model output--outcome pairs, without any need to access model input. In its basic form, EasyUQ is a special case of the recently introduced isotonic distributional regression (IDR) technique that leverages the pool-adjacent-violators algorithm for nonparametric isotonic regression. EasyUQ yields discrete predictive distributions that are calibrated and optimal in finite samples, subject to stochastic monotonicity. The workflow is fully automated, without any need for tuning. The Smooth EasyUQ approach supplements IDR with kernel smoothing, to yield continuous predictive distributions that preserve key properties of the basic form, including both stochastic monotonicity with respect to the original model output and asymptotic consistency. For the selection of kernel parameters, we introduce multiple one-fit grid search, a computationally much less demanding approximation to leave-one-out cross-validation. We use simulation examples and forecast data from weather prediction to illustrate the techniques. In a study of benchmark problems from machine learning, we show how EasyUQ and Smooth EasyUQ can be integrated into the workflow of neural network learning and hyperparameter tuning, and we find EasyUQ to be competitive with conformal prediction as well as more elaborate input-based approaches.","PeriodicalId":49525,"journal":{"name":"SIAM Review","volume":null,"pages":null},"PeriodicalIF":10.2,"publicationDate":"2024-02-08","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"139704937","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 66, Issue 1, Page 123-123, February 2024. The SIGEST article in this issue is “A Simple Formula for the Generalized Spectrum of Second Order Self-Adjoint Differential Operators,” by Bjørn Fredrik Nielsen and Zdeněk Strakoš. This paper studies the eigenvalues of second-order self-adjoint differential operators in the continuum and discrete settings. In particular, they investigate second-order diffusion with a diffusion tensor preconditioned by the inverse Laplacian. They prove that there is a one-to-one correspondence between the spectrum of the preconditioned system and the eigenvalues of the diffusion tensor. Moreover, they investigate the relationship between the spectrum of the preconditioned operator and the generalized eigenvalue problem for its discretized counterpart and show that the latter asymptotically approximates the former. The results presented in the paper are fundamental to anyone wanting to solve elliptic PDEs. Understanding the distribution of eigenvalues is crucial for solving associated linear systems via, e.g., conjugate gradient descent whose convergence rate depends on the spread of the spectrum of the system matrix. The approach of operator preconditioning as done here with the inverse Laplacian turns the unbounded spectrum of a second-order diffusion operator into one that is completely characterized by the diffusion tensor itself. This carries over to the discrete setting, where the support of the spectrum without preconditioning is increasing as one over the squared mesh size, while in the operator preconditioned case mesh independent bounds for the eigenvalues, completely determined by the diffusion tensor, can be obtained. The original version of this article appeared in the SIAM Journal on Numerical Analysis in 2020 and has been recognized as an outstanding and well-presented result in the community. In preparing this SIGEST version, the authors have added new material to sections 1 and 2 in order to increase accessibility, added clarifications to sections 6 and 7, and added the new section 8, which contains a description of more recent results concerning the numerical approximation of the continuous spectrum. It also comments on the related differences between the (generalized) PDE eigenvalue problems for compact and noncompact operators and provides several new references.
SIAM 评论》，第 66 卷第 1 期，第 123-123 页，2024 年 2 月。 本期的 SIGEST 文章是 Bjørn Fredrik Nielsen 和 Zdeněk Strakoš 撰写的 "A Simple Formula for the Generalized Spectrum of Second Order Self-Adjoint Differential Operators"。这篇论文研究了连续和离散环境中二阶自交微分算子的特征值。他们特别研究了以反拉普拉奇为前提条件的二阶扩散张量。他们证明，预处理系统的谱与扩散张量的特征值之间存在一一对应关系。此外，他们还研究了预处理算子的频谱与其离散对应的广义特征值问题之间的关系，并证明后者近似于前者。论文中提出的结果对于任何想要求解椭圆 PDE 的人来说都是至关重要的。了解特征值的分布对于通过共轭梯度下降等方法求解相关线性系统至关重要，而共轭梯度下降的收敛速度取决于系统矩阵频谱的分布。这里使用的逆拉普拉斯算子预处理方法，将二阶扩散算子的无界频谱转化为完全由扩散张量本身表征的频谱。这一点延续到离散设置中，在没有预处理的情况下，频谱的支持率随网格大小的平方递增，而在算子预处理的情况下，可以得到完全由扩散张量决定的特征值的网格无关边界。这篇文章的原始版本于 2020 年发表在《SIAM 数值分析期刊》上，并被公认为是一项杰出的、出色的成果。在编写此 SIGEST 版本时，作者在第 1 节和第 2 节中添加了新材料，以增加可读性；在第 6 节和第 7 节中添加了说明；并添加了新的第 8 节，其中包含对有关连续谱数值逼近的最新结果的描述。这一节还评论了紧凑和非紧凑算子的（广义）PDE 特征值问题之间的相关差异，并提供了一些新的参考文献。
{"title":"SIGEST","authors":"The Editors","doi":"10.1137/24n975840","DOIUrl":"https://doi.org/10.1137/24n975840","url":null,"abstract":"SIAM Review, Volume 66, Issue 1, Page 123-123, February 2024. <br/> The SIGEST article in this issue is “A Simple Formula for the Generalized Spectrum of Second Order Self-Adjoint Differential Operators,” by Bjørn Fredrik Nielsen and Zdeněk Strakoš. This paper studies the eigenvalues of second-order self-adjoint differential operators in the continuum and discrete settings. In particular, they investigate second-order diffusion with a diffusion tensor preconditioned by the inverse Laplacian. They prove that there is a one-to-one correspondence between the spectrum of the preconditioned system and the eigenvalues of the diffusion tensor. Moreover, they investigate the relationship between the spectrum of the preconditioned operator and the generalized eigenvalue problem for its discretized counterpart and show that the latter asymptotically approximates the former. The results presented in the paper are fundamental to anyone wanting to solve elliptic PDEs. Understanding the distribution of eigenvalues is crucial for solving associated linear systems via, e.g., conjugate gradient descent whose convergence rate depends on the spread of the spectrum of the system matrix. The approach of operator preconditioning as done here with the inverse Laplacian turns the unbounded spectrum of a second-order diffusion operator into one that is completely characterized by the diffusion tensor itself. This carries over to the discrete setting, where the support of the spectrum without preconditioning is increasing as one over the squared mesh size, while in the operator preconditioned case mesh independent bounds for the eigenvalues, completely determined by the diffusion tensor, can be obtained. The original version of this article appeared in the SIAM Journal on Numerical Analysis in 2020 and has been recognized as an outstanding and well-presented result in the community. In preparing this SIGEST version, the authors have added new material to sections 1 and 2 in order to increase accessibility, added clarifications to sections 6 and 7, and added the new section 8, which contains a description of more recent results concerning the numerical approximation of the continuous spectrum. It also comments on the related differences between the (generalized) PDE eigenvalue problems for compact and noncompact operators and provides several new references.","PeriodicalId":49525,"journal":{"name":"SIAM Review","volume":null,"pages":null},"PeriodicalIF":10.2,"publicationDate":"2024-02-08","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"139705043","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}
Gabriel R. Barrenechea, Volker John, Petr Knobloch
SIAM Review, Volume 66, Issue 1, Page 3-88, February 2024. Convection-diffusion-reaction equations model the conservation of scalar quantities. From the analytic point of view, solutions of these equations satisfy, under certain conditions, maximum principles, which represent physical bounds of the solution. That the same bounds are respected by numerical approximations of the solution is often of utmost importance in practice. The mathematical formulation of this property, which contributes to the physical consistency of a method, is called the discrete maximum principle (DMP). In many applications, convection dominates diffusion by several orders of magnitude. It is well known that standard discretizations typically do not satisfy the DMP in this convection-dominated regime. In fact, in this case it turns out to be a challenging problem to construct discretizations that, on the one hand, respect the DMP and, on the other hand, compute accurate solutions. This paper presents a survey on finite element methods, with the main focus on the convection-dominated regime, that satisfy a local or a global DMP. The concepts of the underlying numerical analysis are discussed. The survey reveals that for the steady-state problem there are only a few discretizations, all of them nonlinear, that at the same time both satisfy the DMP and compute reasonably accurate solutions, e.g., algebraically stabilized schemes. Moreover, most of these discretizations have been developed in recent years, showing the enormous progress that has been achieved lately. Similarly, methods based on algebraic stabilization, both nonlinear and linear, are currently the only finite element methods that combine the satisfaction of the global DMP and accurate numerical results for the evolutionary equations in the convection-dominated scenario.
{"title":"Finite Element Methods Respecting the Discrete Maximum Principle for Convection-Diffusion Equations","authors":"Gabriel R. Barrenechea, Volker John, Petr Knobloch","doi":"10.1137/22m1488934","DOIUrl":"https://doi.org/10.1137/22m1488934","url":null,"abstract":"SIAM Review, Volume 66, Issue 1, Page 3-88, February 2024. <br/> Convection-diffusion-reaction equations model the conservation of scalar quantities. From the analytic point of view, solutions of these equations satisfy, under certain conditions, maximum principles, which represent physical bounds of the solution. That the same bounds are respected by numerical approximations of the solution is often of utmost importance in practice. The mathematical formulation of this property, which contributes to the physical consistency of a method, is called the discrete maximum principle (DMP). In many applications, convection dominates diffusion by several orders of magnitude. It is well known that standard discretizations typically do not satisfy the DMP in this convection-dominated regime. In fact, in this case it turns out to be a challenging problem to construct discretizations that, on the one hand, respect the DMP and, on the other hand, compute accurate solutions. This paper presents a survey on finite element methods, with the main focus on the convection-dominated regime, that satisfy a local or a global DMP. The concepts of the underlying numerical analysis are discussed. The survey reveals that for the steady-state problem there are only a few discretizations, all of them nonlinear, that at the same time both satisfy the DMP and compute reasonably accurate solutions, e.g., algebraically stabilized schemes. Moreover, most of these discretizations have been developed in recent years, showing the enormous progress that has been achieved lately. Similarly, methods based on algebraic stabilization, both nonlinear and linear, are currently the only finite element methods that combine the satisfaction of the global DMP and accurate numerical results for the evolutionary equations in the convection-dominated scenario.","PeriodicalId":49525,"journal":{"name":"SIAM Review","volume":null,"pages":null},"PeriodicalIF":10.2,"publicationDate":"2024-02-08","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"139705086","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 66, Issue 1, Page 1-1, February 2024. Numerical methods for partial differential equations can only be successful if their numerical solutions reflect fundamental properties of the physical solution of the respective PDE. For convection-diffusion equations, the conservation of some specific scalar quantities is crucial. When physical solutions satisfy maximum principles representing physical bounds, then the numerical solutions should respect the same bounds. In a mathematical setting, this requirement is known as the discrete maximum principle (DMP). Discretizations which fail to fulfill the DMP are prone to numerical solutions with unphysical values, e.g., spurious oscillations. However, when convection largely dominates diffusion, many discretization methods do not satisfy a DMP. In the only article of the Survey and Review section of this issue, “Finite Element Methods Respecting the Discrete Maximum Principle for Convection-Diffusion Equations,” Gabriel R. Barrenechea, Volker John, and Petr Knobloch study and analyze finite element methods that succeed in complying with DMP while providing accurate numerical solutions at the same time. This is a nontrivial task and, thus, even for the steady-state problem there are only a few such discretizations, all of them nonlinear. Most of these methods have been developed quite recently, so that the presentation highlights the state of the art and spotlights the huge progress accomplished in recent years. The goal of the paper consists in providing a survey on finite element methods that satisfy local or global DMPs for linear elliptic or parabolic problems. It is worth reading for a large audience.
SIAM Review》，第 66 卷，第 1 期，第 1-1 页，2024 年 2 月。 偏微分方程的数值方法只有在其数值解反映了相应偏微分方程物理解的基本特性时才能取得成功。对于对流扩散方程，某些特定标量的守恒性至关重要。当物理解满足代表物理边界的最大原则时，数值解也应遵守同样的边界。在数学环境中，这一要求被称为离散最大值原理（DMP）。不符合 DMP 的离散化容易导致数值解出现非物理值，例如虚假振荡。然而，当对流在很大程度上主导扩散时，许多离散化方法都不满足 DMP。Gabriel R. Barrenechea、Volker John 和 Petr Knobloch 在本期 "调查与评论 "部分的唯一一篇文章 "尊重对流-扩散方程离散最大原则的有限元方法 "中，研究并分析了成功符合 DMP 并同时提供精确数值解的有限元方法。这是一项非同小可的任务，因此，即使对于稳态问题，也只有少数几种这样的离散方法，而且都是非线性的。这些方法中的大多数都是最近才开发出来的，因此本文重点介绍了相关技术的发展状况，并突出强调了近年来所取得的巨大进步。本文的目的是对满足线性椭圆或抛物问题局部或全局 DMP 的有限元方法进行研究。它值得广大读者阅读。
{"title":"Survey and Review","authors":"Marlis Hochbruck","doi":"10.1137/24n975827","DOIUrl":"https://doi.org/10.1137/24n975827","url":null,"abstract":"SIAM Review, Volume 66, Issue 1, Page 1-1, February 2024. <br/> Numerical methods for partial differential equations can only be successful if their numerical solutions reflect fundamental properties of the physical solution of the respective PDE. For convection-diffusion equations, the conservation of some specific scalar quantities is crucial. When physical solutions satisfy maximum principles representing physical bounds, then the numerical solutions should respect the same bounds. In a mathematical setting, this requirement is known as the discrete maximum principle (DMP). Discretizations which fail to fulfill the DMP are prone to numerical solutions with unphysical values, e.g., spurious oscillations. However, when convection largely dominates diffusion, many discretization methods do not satisfy a DMP. In the only article of the Survey and Review section of this issue, “Finite Element Methods Respecting the Discrete Maximum Principle for Convection-Diffusion Equations,” Gabriel R. Barrenechea, Volker John, and Petr Knobloch study and analyze finite element methods that succeed in complying with DMP while providing accurate numerical solutions at the same time. This is a nontrivial task and, thus, even for the steady-state problem there are only a few such discretizations, all of them nonlinear. Most of these methods have been developed quite recently, so that the presentation highlights the state of the art and spotlights the huge progress accomplished in recent years. The goal of the paper consists in providing a survey on finite element methods that satisfy local or global DMPs for linear elliptic or parabolic problems. It is worth reading for a large audience.","PeriodicalId":49525,"journal":{"name":"SIAM Review","volume":null,"pages":null},"PeriodicalIF":10.2,"publicationDate":"2024-02-08","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"139705103","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 917-917, November 2023. The metric dimension $beta(G)$ of a graph $G = (V,E)$ is the smallest cardinality of a subset $S$ of vertices such that all other vertices are uniquely determined by their distances to the vertices in the resolving set $S$. Finding the metric dimension of a graph is an NP-hard problem. Determining whether the metric dimension is less than a given value is NP-complete. In the first article in the Survey and Review section of this issue, “Getting the Lay of the Land in Discrete Space: A Survey of Metric Dimension and Its Applications,” Richard C. Tillquist, Rafael M. Frongillo, and Manuel E. Lladser provide an exhaustive introduction to metric dimension. The overview of its vital results includes applications in game theory, source localization in transmission processes, and preprocessing in the computational analysis of biological sequence data. The paper is worth reading for a broad audience. The second Survey and Review article, by Ludovic Chamoin and Frédéric Legoll, is “An Introductory Review on A Posteriori Error Estimation in Finite Element Computations.” It is devoted to basic concepts and tools for verification methods that provide computable and mathematically certified error bounds and also addresses the question on the localization of errors in the spatial domain. The focus of this review is on a particular method and problem, namely, a conforming finite element method for linear elliptic diffusion problems. The tools of dual analysis and the concept of equilibrium enable a unified perspective on different a posteriori error estimation methods, e.g., flux recovery methods, residual methods, and duality-based constitutive relation error methods. Other topics considered are goal-oriented error estimation, computational costs, and extensions to other finite element schemes and other mathematical problems. While the presentation is self-contained, it is assumed that the reader is familiar with finite element methods. The text is written in an interdisciplinary style and aims to be useful for applied mathematicians and engineers.
{"title":"Survey and Review","authors":"Marlis Hochbruck","doi":"10.1137/23n975776","DOIUrl":"https://doi.org/10.1137/23n975776","url":null,"abstract":"SIAM Review, Volume 65, Issue 4, Page 917-917, November 2023. <br/> The metric dimension $beta(G)$ of a graph $G = (V,E)$ is the smallest cardinality of a subset $S$ of vertices such that all other vertices are uniquely determined by their distances to the vertices in the resolving set $S$. Finding the metric dimension of a graph is an NP-hard problem. Determining whether the metric dimension is less than a given value is NP-complete. In the first article in the Survey and Review section of this issue, “Getting the Lay of the Land in Discrete Space: A Survey of Metric Dimension and Its Applications,” Richard C. Tillquist, Rafael M. Frongillo, and Manuel E. Lladser provide an exhaustive introduction to metric dimension. The overview of its vital results includes applications in game theory, source localization in transmission processes, and preprocessing in the computational analysis of biological sequence data. The paper is worth reading for a broad audience. The second Survey and Review article, by Ludovic Chamoin and Frédéric Legoll, is “An Introductory Review on A Posteriori Error Estimation in Finite Element Computations.” It is devoted to basic concepts and tools for verification methods that provide computable and mathematically certified error bounds and also addresses the question on the localization of errors in the spatial domain. The focus of this review is on a particular method and problem, namely, a conforming finite element method for linear elliptic diffusion problems. The tools of dual analysis and the concept of equilibrium enable a unified perspective on different a posteriori error estimation methods, e.g., flux recovery methods, residual methods, and duality-based constitutive relation error methods. Other topics considered are goal-oriented error estimation, computational costs, and extensions to other finite element schemes and other mathematical problems. While the presentation is self-contained, it is assumed that the reader is familiar with finite element methods. The text is written in an interdisciplinary style and aims to be useful for applied mathematicians and engineers.","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":"71473800","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 1074-1105, November 2023. Bayesian methods have been widely used in the last two decades to infer statistical properties of spatially variable coefficients in partial differential equations from measurements of the solutions of these equations. Yet, in many cases the number of variables used to parameterize these coefficients is large, and oobtaining meaningful statistics of their probability distributions is difficult using simple sampling methods such as the basic Metropolis--Hastings algorithm---in particular, if the inverse problem is ill-conditioned or ill-posed. As a consequence, many advanced sampling methods have been described in the literature that converge faster than Metropolis--Hastings, for example, by exploiting hierarchies of statistical models or hierarchies of discretizations of the underlying differential equation. At the same time, it remains difficult for the reader of the literature to quantify the advantages of these algorithms because there is no commonly used benchmark. This paper presents a benchmark Bayesian inverse problem---namely, the determination of a spatially variable coefficient, discretized by 64 values, in a Poisson equation, based on point measurements of the solution---that fills the gap between widely used simple test cases (such as superpositions of Gaussians) and real applications that are difficult to replicate for developers of sampling algorithms. We provide a complete description of the test case and provide an open-source implementation that can serve as the basis for further experiments. We have also computed $2times 10^{11}$ samples, at a cost of some 30 CPU years, of the posterior probability distribution from which we have generated detailed and accurate statistics against which other sampling algorithms can be tested.
{"title":"A Benchmark for the Bayesian Inversion of Coefficients in Partial Differential Equations","authors":"David Aristoff, Wolfgang Bangerth","doi":"10.1137/21m1399464","DOIUrl":"https://doi.org/10.1137/21m1399464","url":null,"abstract":"SIAM Review, Volume 65, Issue 4, Page 1074-1105, November 2023. <br/> Bayesian methods have been widely used in the last two decades to infer statistical properties of spatially variable coefficients in partial differential equations from measurements of the solutions of these equations. Yet, in many cases the number of variables used to parameterize these coefficients is large, and oobtaining meaningful statistics of their probability distributions is difficult using simple sampling methods such as the basic Metropolis--Hastings algorithm---in particular, if the inverse problem is ill-conditioned or ill-posed. As a consequence, many advanced sampling methods have been described in the literature that converge faster than Metropolis--Hastings, for example, by exploiting hierarchies of statistical models or hierarchies of discretizations of the underlying differential equation. At the same time, it remains difficult for the reader of the literature to quantify the advantages of these algorithms because there is no commonly used benchmark. This paper presents a benchmark Bayesian inverse problem---namely, the determination of a spatially variable coefficient, discretized by 64 values, in a Poisson equation, based on point measurements of the solution---that fills the gap between widely used simple test cases (such as superpositions of Gaussians) and real applications that are difficult to replicate for developers of sampling algorithms. We provide a complete description of the test case and provide an open-source implementation that can serve as the basis for further experiments. We have also computed $2times 10^{11}$ samples, at a cost of some 30 CPU years, of the posterior probability distribution from which we have generated detailed and accurate statistics against which other sampling algorithms can be tested.","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":"71473801","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 1135-1135, November 2023. In this issue the Education section presents three contributions. The first paper “The Reflection Method for the Numerical Solution of Linear Systems,” by Margherita Guida and Carlo Sbordone, discusses the celebrated Gianfranco Cimmino reflection algorithm for the numerical solution of linear systems $Ax=b$, where $A$ is a nonsingular $n times n$ sparse matrix, $b in mathbb{R}^n$, and $n$ may be large. This innovative iterative algorithm proposed in 1938 uses the geometric reading of each equation of the system as a hyperplane to compute an average of all the symmetric reflections of an initial point $x^0$ with respect to hyperplanes. This leads to a new point $x^1$ which is closer to the solution. The iterative method constructs a sequence $x^k in mathbb{R}^n$ converging to the unique intersection of hyperplanes. To overcome the algorithm's efficiency issues, in 1965 Cimmino upgraded his method by introducing probabilistic arguments also discussed in this article. The method is different from widely used direct methods. Since the early 1980s, there has been increasing interest in Cimmino's method that has shown to work well in parallel computing, in particular for applications in the area of image reconstruction via X-ray tomography. Cimmino's algorithm could be an interesting subject to be deepened by students in a course on scientific computing. The second paper, “Incorporating Computational Challenges into a Multidisciplinary Course on Stochastic Processes,” is presented by Mark Jayson Cortez, Alan Eric Akil, Krešimir Josić, and Alexander J. Stewart. The authors describe their graduate-level introductory stochastic modeling course in biology for a mixed audience of mathematicians and biologists whose goal was teaching students to formulate, implement, and assess nontrivial biomathematical models and to develop research skills. This problem-based learning was addressed by proposing several computational modeling challenges based on real life applied problems; by assigning tasks to groups formed by four students, where necessarily participants had different levels of knowledge in programming, mathematics, and biology; and by creating retrospective discussion sessions. In this way the stochastic modeling was introduced using a variety of examples involving, for instance, biochemical reaction networks, gene regulatory systems, neuronal networks, models of epidemics, stochastic games, and agent-based models. As supplementary material, a detailed syllabus, homework, and the text of all computational challenges, along with code for the discussed examples, are provided. The third paper, “Hysteresis and Stability,” by Amenda N. Chow, Kirsten A. Morris, and Gina F. Rabbah, describes the phenomenon of hysteresis in some ordinary differential equations motivated by applications in a way that can be integrated into an introductory course of dynamical systems for undergraduate students.
{"title":"Education","authors":"Hèléne Frankowska","doi":"10.1137/23n975806","DOIUrl":"https://doi.org/10.1137/23n975806","url":null,"abstract":"SIAM Review, Volume 65, Issue 4, Page 1135-1135, November 2023. <br/> In this issue the Education section presents three contributions. The first paper “The Reflection Method for the Numerical Solution of Linear Systems,” by Margherita Guida and Carlo Sbordone, discusses the celebrated Gianfranco Cimmino reflection algorithm for the numerical solution of linear systems $Ax=b$, where $A$ is a nonsingular $n times n$ sparse matrix, $b in mathbb{R}^n$, and $n$ may be large. This innovative iterative algorithm proposed in 1938 uses the geometric reading of each equation of the system as a hyperplane to compute an average of all the symmetric reflections of an initial point $x^0$ with respect to hyperplanes. This leads to a new point $x^1$ which is closer to the solution. The iterative method constructs a sequence $x^k in mathbb{R}^n$ converging to the unique intersection of hyperplanes. To overcome the algorithm's efficiency issues, in 1965 Cimmino upgraded his method by introducing probabilistic arguments also discussed in this article. The method is different from widely used direct methods. Since the early 1980s, there has been increasing interest in Cimmino's method that has shown to work well in parallel computing, in particular for applications in the area of image reconstruction via X-ray tomography. Cimmino's algorithm could be an interesting subject to be deepened by students in a course on scientific computing. The second paper, “Incorporating Computational Challenges into a Multidisciplinary Course on Stochastic Processes,” is presented by Mark Jayson Cortez, Alan Eric Akil, Krešimir Josić, and Alexander J. Stewart. The authors describe their graduate-level introductory stochastic modeling course in biology for a mixed audience of mathematicians and biologists whose goal was teaching students to formulate, implement, and assess nontrivial biomathematical models and to develop research skills. This problem-based learning was addressed by proposing several computational modeling challenges based on real life applied problems; by assigning tasks to groups formed by four students, where necessarily participants had different levels of knowledge in programming, mathematics, and biology; and by creating retrospective discussion sessions. In this way the stochastic modeling was introduced using a variety of examples involving, for instance, biochemical reaction networks, gene regulatory systems, neuronal networks, models of epidemics, stochastic games, and agent-based models. As supplementary material, a detailed syllabus, homework, and the text of all computational challenges, along with code for the discussed examples, are provided. The third paper, “Hysteresis and Stability,” by Amenda N. Chow, Kirsten A. Morris, and Gina F. Rabbah, describes the phenomenon of hysteresis in some ordinary differential equations motivated by applications in a way that can be integrated into an introductory course of dynamical systems for undergraduate students. ","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":"71474874","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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