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.
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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":"9 1","pages":""},"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":"12 1","pages":""},"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.
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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.
{"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":"5 1","pages":""},"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}
Rodrigo M. Pereira, Natacha Nguyen van yen, Kai Schneider, Marie Farge
SIAM Review, Volume 65, Issue 4, Page 1109-1134, November 2023. Adaptive Galerkin numerical schemes integrate time-dependent partial differential equations with a finite number of basis functions, and a subset of them is selected at each time step. This subset changes over time discontinuously according to the evolution of the solution; therefore the corresponding projection operator is time-dependent and nondifferentiable, and we propose using an integral formulation in time. We analyze the existence and uniqueness of this weak form of adaptive Galerkin schemes and prove that nonsmooth projection operators can introduce energy dissipation, which is a crucial result for adaptive Galerkin schemes. To illustrate this, we study an adaptive Galerkin wavelet scheme which computes the time evolution of the inviscid Burgers equation in one dimension and of the incompressible Euler equations in two and three dimensions with a pseudospectral scheme, together with coherent vorticity simulation which uses wavelet denoising. With the help of the continuous wavelet representation we analyze the time evolution of the solution of the 1D inviscid Burgers equation: We first observe that numerical resonances appear when energy reaches the smallest resolved scale, then they spread in both space and scale until they reach energy equipartition between all basis functions, as thermal noise does. Finally we show how adaptive wavelet schemes denoise and regularize the solution of the Galerkin truncated inviscid equations, and for the inviscid Burgers case wavelet denoising even yields convergence towards the exact dissipative solution, also called entropy solution. These results motivate in particular adaptive wavelet Galerkin schemes for nonlinear hyperbolic conservation laws. This SIGEST article is a revised and extended version of the article [R. M. Pereira, N. Nguyen van yen, K. Schneider, and M. Farge, Multiscale Model. Simul., 20 (2022), pp. 1147--1166].
SIAM评论，第65卷，第4期，第1109-1134页，2023年11月。自适应Galerkin数值格式集成了具有有限个基函数的含时偏微分方程，并在每个时间步长选择其中的一个子集。该子集随着时间的推移根据解决方案的演变而不连续地变化；因此，相应的投影算子是时间相关的和不可微的，我们建议使用时间积分公式。我们分析了这种弱形式的自适应Galerkin格式的存在性和唯一性，并证明了非光滑投影算子可以引入能量耗散，这是自适应Galerkn格式的一个关键结果。为了说明这一点，我们研究了一种自适应Galerkin小波格式，该格式使用伪谱格式计算一维无粘性Burgers方程和二维和三维不可压缩Euler方程的时间演化，以及使用小波去噪的相干涡度模拟。在连续小波表示的帮助下，我们分析了一维无粘Burgers方程解的时间演化：我们首先观察到，当能量达到最小的分辨尺度时，会出现数值共振，然后它们在空间和尺度上传播，直到它们达到所有基函数之间的能量均分，就像热噪声一样。最后，我们展示了自适应小波方案如何对Galerkin截断无粘方程的解进行去噪和正则化，并且对于无粘Burgers情况，小波去噪甚至产生向精确耗散解（也称为熵解）的收敛。这些结果特别激励了非线性双曲守恒律的自适应小波Galerkin格式。SIGEST的这篇文章是该文章的修订和扩展版本[R.M.Pereira，N.Nguyen van yen，K.Schneider和M.Farge，Multiscale Model.Simul.，20（2022），pp.1147-11166]。
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Richard C. Tillquist, Rafael M. Frongillo, Manuel E. Lladser
SIAM Review, Volume 65, Issue 4, Page 919-962, November 2023. The metric dimension of a graph is the smallest number of nodes required to identify all other nodes uniquely based on shortest path distances. Applications of metric dimension include discovering the source of a spread in a network, canonically labeling graphs, and embedding symbolic data in low-dimensional Euclidean spaces. This survey gives a self-contained introduction to metric dimension and an overview of the quintessential results and applications. We discuss methods for approximating the metric dimension of general graphs, and specific bounds and asymptotic behavior for deterministic and random families of graphs. We conclude with related concepts and directions for future work.
{"title":"Getting the Lay of the Land in Discrete Space: A Survey of Metric Dimension and Its Applications","authors":"Richard C. Tillquist, Rafael M. Frongillo, Manuel E. Lladser","doi":"10.1137/21m1409512","DOIUrl":"https://doi.org/10.1137/21m1409512","url":null,"abstract":"SIAM Review, Volume 65, Issue 4, Page 919-962, November 2023. <br/> The metric dimension of a graph is the smallest number of nodes required to identify all other nodes uniquely based on shortest path distances. Applications of metric dimension include discovering the source of a spread in a network, canonically labeling graphs, and embedding symbolic data in low-dimensional Euclidean spaces. This survey gives a self-contained introduction to metric dimension and an overview of the quintessential results and applications. We discuss methods for approximating the metric dimension of general graphs, and specific bounds and asymptotic behavior for deterministic and random families of graphs. We conclude with related concepts and directions for future work.","PeriodicalId":49525,"journal":{"name":"SIAM Review","volume":"5 5","pages":""},"PeriodicalIF":10.2,"publicationDate":"2023-11-07","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"71474870","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":1,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Mark Jayson Cortez, Alan Eric Akil, Krešimir Josić, Alexander J. Stewart
SIAM Review, Volume 65, Issue 4, Page 1152-1170, November 2023. Quantitative methods and mathematical modeling are playing an increasingly important role across disciplines. As a result, interdisciplinary mathematics courses are increasing in popularity. However, teaching such courses at an advanced level can be challenging. Students often arrive with different mathematical backgrounds, different interests, and divergent reasons for wanting to learn the material. Here we describe a course on stochastic processes in biology delivered between September and December 2020 to a mixed audience of mathematicians and biologists. In addition to traditional lectures and homework, we incorporated a series of weekly computational challenges into the course. These challenges served to familiarize students with the main modeling concepts and provide them with an introduction on how to implement the concepts in a research-like setting. In order to account for the different academic backgrounds of the students, they worked on the challenges in small groups and presented their results and code in a dedicated discussion class each week. We discuss our experience designing and implementing an element of problem-based learning in an applied mathematics course through computational challenges. We also discuss feedback from students and describe the content of the challenges presented in the course. We provide all materials, along with example code for a number of challenges.
{"title":"Incorporating Computational Challenges into a Multidisciplinary Course on Stochastic Processes","authors":"Mark Jayson Cortez, Alan Eric Akil, Krešimir Josić, Alexander J. Stewart","doi":"10.1137/21m1445545","DOIUrl":"https://doi.org/10.1137/21m1445545","url":null,"abstract":"SIAM Review, Volume 65, Issue 4, Page 1152-1170, November 2023. <br/> Quantitative methods and mathematical modeling are playing an increasingly important role across disciplines. As a result, interdisciplinary mathematics courses are increasing in popularity. However, teaching such courses at an advanced level can be challenging. Students often arrive with different mathematical backgrounds, different interests, and divergent reasons for wanting to learn the material. Here we describe a course on stochastic processes in biology delivered between September and December 2020 to a mixed audience of mathematicians and biologists. In addition to traditional lectures and homework, we incorporated a series of weekly computational challenges into the course. These challenges served to familiarize students with the main modeling concepts and provide them with an introduction on how to implement the concepts in a research-like setting. In order to account for the different academic backgrounds of the students, they worked on the challenges in small groups and presented their results and code in a dedicated discussion class each week. We discuss our experience designing and implementing an element of problem-based learning in an applied mathematics course through computational challenges. We also discuss feedback from students and describe the content of the challenges presented in the course. We provide all materials, along with example code for a number of challenges.","PeriodicalId":49525,"journal":{"name":"SIAM Review","volume":"11 9","pages":""},"PeriodicalIF":10.2,"publicationDate":"2023-11-07","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"71473804","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":1,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
SIAM Review, Volume 65, Issue 4, Page 1137-1151, November 2023. We present Cimmino's reflection algorithm for the numerical solution of linear systems, which starts with an arbitrary point in $mathbb{R}^n$ that gets reflected with respect to the system's hyperplanes. The centroid of the ensuing collection of points becomes the starting point of the next iteration. We provide error estimates for the convergence at each step. A probabilistic argument is also devised to improve this elegant geometrical algorithm. This subject is an opportunity to show students how linear algebra can interact fruitfully not only with algebra, geometry, and numerical analysis, but also with probability theory and methods.
{"title":"The Reflection Method for the Numerical Solution of Linear Systems","authors":"Margherita Guida, Carlo Sbordone","doi":"10.1137/22m1470463","DOIUrl":"https://doi.org/10.1137/22m1470463","url":null,"abstract":"SIAM Review, Volume 65, Issue 4, Page 1137-1151, November 2023. <br/> We present Cimmino's reflection algorithm for the numerical solution of linear systems, which starts with an arbitrary point in $mathbb{R}^n$ that gets reflected with respect to the system's hyperplanes. The centroid of the ensuing collection of points becomes the starting point of the next iteration. We provide error estimates for the convergence at each step. A probabilistic argument is also devised to improve this elegant geometrical algorithm. This subject is an opportunity to show students how linear algebra can interact fruitfully not only with algebra, geometry, and numerical analysis, but also with probability theory and methods.","PeriodicalId":49525,"journal":{"name":"SIAM Review","volume":"11 10","pages":""},"PeriodicalIF":10.2,"publicationDate":"2023-11-07","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"71473803","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":1,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
SIAM Review, Volume 65, Issue 4, Page 1171-1184, November 2023. A common definition of hysteresis is that the graph of the state of the system displays looping behavior as the input of the system varies. Alternatively, a dynamical systems perspective can be used to define hysteresis as a phenomenon arising from multiple equilibrium points. Consequently, hysteresis is a topic that can be used to illustrate and extend concepts in a dynamical systems course. The concept is illustrated in this paper through examples of ordinary differential equations, most motivated by applications. Simulations are presented to complement the analysis. The examples can be used to construct student exercises, and specific additional questions are listed in an appendix. The paper concludes with a discussion of possible extensions, including hysteresis in partial differential equations.
{"title":"Hysteresis and Stability","authors":"Amenda N. Chow, Kirsten A. Morris, Gina F. Rabbah","doi":"10.1137/21m1420733","DOIUrl":"https://doi.org/10.1137/21m1420733","url":null,"abstract":"SIAM Review, Volume 65, Issue 4, Page 1171-1184, November 2023. <br/> A common definition of hysteresis is that the graph of the state of the system displays looping behavior as the input of the system varies. Alternatively, a dynamical systems perspective can be used to define hysteresis as a phenomenon arising from multiple equilibrium points. Consequently, hysteresis is a topic that can be used to illustrate and extend concepts in a dynamical systems course. The concept is illustrated in this paper through examples of ordinary differential equations, most motivated by applications. Simulations are presented to complement the analysis. The examples can be used to construct student exercises, and specific additional questions are listed in an appendix. The paper concludes with a discussion of possible extensions, including hysteresis in partial differential equations.","PeriodicalId":49525,"journal":{"name":"SIAM Review","volume":"11 7","pages":""},"PeriodicalIF":10.2,"publicationDate":"2023-11-07","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"71473806","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":1,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
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