SIAM Review最新文献

{"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}

{"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":"11 12","pages":""},"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}

{"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. &lt;br/&gt; 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}

{"title":"Are Adaptive Galerkin Schemes Dissipative?","authors":"Rodrigo M. Pereira, Natacha Nguyen van yen, Kai Schneider, Marie Farge","doi":"10.1137/23m1588627","DOIUrl":"https://doi.org/10.1137/23m1588627","url":null,"abstract":"SIAM Review, Volume 65, Issue 4, Page 1109-1134, November 2023. <br/> Adaptive Galerkin numerical schemes integrate time-dependent partial differential equations with a finite number of basis functions, and a subset of them is selected at each time step. This subset changes over time discontinuously according to the evolution of the solution; therefore the corresponding projection operator is time-dependent and nondifferentiable, and we propose using an integral formulation in time. We analyze the existence and uniqueness of this weak form of adaptive Galerkin schemes and prove that nonsmooth projection operators can introduce energy dissipation, which is a crucial result for adaptive Galerkin schemes. To illustrate this, we study an adaptive Galerkin wavelet scheme which computes the time evolution of the inviscid Burgers equation in one dimension and of the incompressible Euler equations in two and three dimensions with a pseudospectral scheme, together with coherent vorticity simulation which uses wavelet denoising. With the help of the continuous wavelet representation we analyze the time evolution of the solution of the 1D inviscid Burgers equation: We first observe that numerical resonances appear when energy reaches the smallest resolved scale, then they spread in both space and scale until they reach energy equipartition between all basis functions, as thermal noise does. Finally we show how adaptive wavelet schemes denoise and regularize the solution of the Galerkin truncated inviscid equations, and for the inviscid Burgers case wavelet denoising even yields convergence towards the exact dissipative solution, also called entropy solution. These results motivate in particular adaptive wavelet Galerkin schemes for nonlinear hyperbolic conservation laws. This SIGEST article is a revised and extended version of the article [R. M. Pereira, N. Nguyen van yen, K. Schneider, and M. Farge, Multiscale Model. Simul., 20 (2022), pp. 1147--1166].","PeriodicalId":49525,"journal":{"name":"SIAM Review","volume":"11 8","pages":""},"PeriodicalIF":10.2,"publicationDate":"2023-11-07","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"71473805","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":1,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}

{"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}

{"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}

{"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}

{"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}

{"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}

{"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}