SIAM Review, Volume 67, Issue 1, Page 3-70, March 2025. Abstract.Uncertainty is prevalent in engineering design and data-driven problems and, more broadly, in decision making. Due to inherent risk-averseness and ambiguity about assumptions, it is common to address uncertainty by formulating and solving conservative optimization models expressed using measures of risk and related concepts. We survey the rapid development of risk measures over the last quarter century. From their beginning in financial engineering, we recount their spread to nearly all areas of engineering and applied mathematics. Solidly rooted in convex analysis, risk measures furnish a general framework for handling uncertainty with significant computational and theoretical advantages. We describe the key facts, list several concrete algorithms, and provide an extensive list of references for further reading. The survey recalls connections with utility theory and distributionally robust optimization, points to emerging applications areas such as fair machine learning, and defines measures of reliability.
{"title":"Risk-Adaptive Approaches to Stochastic Optimization: A Survey","authors":"Johannes O. Royset","doi":"10.1137/22m1538946","DOIUrl":"https://doi.org/10.1137/22m1538946","url":null,"abstract":"SIAM Review, Volume 67, Issue 1, Page 3-70, March 2025. <br/> Abstract.Uncertainty is prevalent in engineering design and data-driven problems and, more broadly, in decision making. Due to inherent risk-averseness and ambiguity about assumptions, it is common to address uncertainty by formulating and solving conservative optimization models expressed using measures of risk and related concepts. We survey the rapid development of risk measures over the last quarter century. From their beginning in financial engineering, we recount their spread to nearly all areas of engineering and applied mathematics. Solidly rooted in convex analysis, risk measures furnish a general framework for handling uncertainty with significant computational and theoretical advantages. We describe the key facts, list several concrete algorithms, and provide an extensive list of references for further reading. The survey recalls connections with utility theory and distributionally robust optimization, points to emerging applications areas such as fair machine learning, and defines measures of reliability.","PeriodicalId":49525,"journal":{"name":"SIAM Review","volume":"128 1","pages":""},"PeriodicalIF":10.2,"publicationDate":"2025-02-06","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"143258254","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}
Thomas Nagel, Tymofiy Gerasimov, Jere Remes, Dominik Kern
SIAM Review, Volume 67, Issue 1, Page 176-193, March 2025. Abstract.This paper is intended to serve as a low-hurdle introduction to nonlocality for graduate students and researchers with an engineering mechanics or physics background who did not have a formal introduction to the underlying mathematical basis. We depart from simple examples motivated by structural mechanics to form a physical intuition and demonstrate nonlocality using concepts familiar to most engineers. We then show how concepts of nonlocality are at the core of one of the most active current research fields in applied mechanics, namely, in phase-field modeling of fracture. From a mathematical perspective, these developments rest on the concept of convolution in both its discrete and its continuous forms. The previous mechanical examples may thus serve as an intuitive explanation of what convolution implies from a physical perspective. In the supplementary material we highlight a broader range of applications of the concepts of nonlocality and convolution in other branches of science and engineering by generalizing from the examples explained in detail in the main body of the article.
{"title":"Neighborhood Watch in Mechanics: Nonlocal Models and Convolution","authors":"Thomas Nagel, Tymofiy Gerasimov, Jere Remes, Dominik Kern","doi":"10.1137/22m1541721","DOIUrl":"https://doi.org/10.1137/22m1541721","url":null,"abstract":"SIAM Review, Volume 67, Issue 1, Page 176-193, March 2025. <br/> Abstract.This paper is intended to serve as a low-hurdle introduction to nonlocality for graduate students and researchers with an engineering mechanics or physics background who did not have a formal introduction to the underlying mathematical basis. We depart from simple examples motivated by structural mechanics to form a physical intuition and demonstrate nonlocality using concepts familiar to most engineers. We then show how concepts of nonlocality are at the core of one of the most active current research fields in applied mechanics, namely, in phase-field modeling of fracture. From a mathematical perspective, these developments rest on the concept of convolution in both its discrete and its continuous forms. The previous mechanical examples may thus serve as an intuitive explanation of what convolution implies from a physical perspective. In the supplementary material we highlight a broader range of applications of the concepts of nonlocality and convolution in other branches of science and engineering by generalizing from the examples explained in detail in the main body of the article.","PeriodicalId":49525,"journal":{"name":"SIAM Review","volume":"47 1","pages":""},"PeriodicalIF":10.2,"publicationDate":"2025-02-06","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"143258570","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 67, Issue 1, Page 208-209, March 2025. The book Mathematical Pictures at a Data Science Exhibition aims to introduce the reader to the many mathematical ideas that congregate under the ever-expanding umbrella of data science. Given the meteoric rise of this field and the immense speed at which it often moves, this book acts as a welcome road map for graduate students and researchers in the field. Given its focus on theory, the book should be most appreciated by mathematicians as well as theoretical statisticians and computer scientists. While algorithms are the main focus of the book, the exposition is by no means a hands-on tutorial in data science, but rather an introductory text on the theoretical ideas behind data science algorithms and problems.
{"title":"Book Review:; Mathematical Pictures at a Data Science Exhibition","authors":"Bamdad Hosseini","doi":"10.1137/24m1635077","DOIUrl":"https://doi.org/10.1137/24m1635077","url":null,"abstract":"SIAM Review, Volume 67, Issue 1, Page 208-209, March 2025. <br/> The book Mathematical Pictures at a Data Science Exhibition aims to introduce the reader to the many mathematical ideas that congregate under the ever-expanding umbrella of data science. Given the meteoric rise of this field and the immense speed at which it often moves, this book acts as a welcome road map for graduate students and researchers in the field. Given its focus on theory, the book should be most appreciated by mathematicians as well as theoretical statisticians and computer scientists. While algorithms are the main focus of the book, the exposition is by no means a hands-on tutorial in data science, but rather an introductory text on the theoretical ideas behind data science algorithms and problems.","PeriodicalId":49525,"journal":{"name":"SIAM Review","volume":"11 1","pages":""},"PeriodicalIF":10.2,"publicationDate":"2025-02-06","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"143258245","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 4, Page 751-777, November 2024. Granular materials are everywhere, in the environment but also in our pantry. Their properties are different from those of any solid material, due to the possibility of sudden phenomena such as avalanches or landslides. Here we present a brief survey on their characteristics and on what can be found (from the past thirty years) in the recent mathematics literature in order to reproduce their behavior. We discuss, in particular, differential models proposed for the growth of a sandpile on a table and, when wind comes into play, for the formation and dynamics of sand dunes. This field of research is still of great interest since there is no consolidated general model for the dynamics of granular matter, but rather only standalone models adapted to specific situations.
{"title":"Sandpiles and Dunes: Mathematical Models for Granular Matter","authors":"Piermarco Cannarsa, Stefano Finzi Vita","doi":"10.1137/23m1583673","DOIUrl":"https://doi.org/10.1137/23m1583673","url":null,"abstract":"SIAM Review, Volume 66, Issue 4, Page 751-777, November 2024. <br/> Granular materials are everywhere, in the environment but also in our pantry. Their properties are different from those of any solid material, due to the possibility of sudden phenomena such as avalanches or landslides. Here we present a brief survey on their characteristics and on what can be found (from the past thirty years) in the recent mathematics literature in order to reproduce their behavior. We discuss, in particular, differential models proposed for the growth of a sandpile on a table and, when wind comes into play, for the formation and dynamics of sand dunes. This field of research is still of great interest since there is no consolidated general model for the dynamics of granular matter, but rather only standalone models adapted to specific situations.","PeriodicalId":49525,"journal":{"name":"SIAM Review","volume":"18 1","pages":""},"PeriodicalIF":10.2,"publicationDate":"2024-11-07","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142594678","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 4, Page 719-719, November 2024. The SIGEST article in this issue, “A Bridge between Invariant Theory and Maximum Likelihood Estimation,” by Carlos Améndola, Kathlén Kohn, Philipp Reichenbach, and Anna Seigal, uncovers the deep connections between geometric invariant theory and statistical methods, specifically maximum likelihood estimation (MLE) by connecting it to norm minimization over group orbits. The authors develop a dictionary relating stability notions in geometric invariant theory to the existence and uniqueness of MLEs, which applies to both Gaussian and log-linear models. In comparison to the original 2021 version of the paper that appeared in the SIAM Journal on Applied Algebra and Geometry, for the SIGEST version, the authors added new content on log-linear models, simplified technical proofs, removed detailed appendices, and incorporated new examples and figures for accessibility. In particular, the focus was primarily on Gaussian models, whereas this updated SIGEST version expands the coverage by incorporating results from the authors' companion paper on log-linear models. Furthermore, a new figure (Fig. 1) visually illustrates the two core concepts of invariant theory and MLE. Significant changes include the removal of technical details and appendices to streamline the content and make it more accessible to a broader audience. The introduction of examples, particularly for the Kempf--Ness Theorem, further aids understanding. This paper makes several key contributions of broad mathematical interest. MLE is a key statistical technique that is widely used. Having a new handle on its well-posedness analysis deepens the understanding of the mechanisms behind this technique as well as potentially paves the way to extending existing theory for MLE models. Also, on the computational side, algorithms from the optimization over orbits can be used for MLE, and vice versa, which could possibly lead to new and more efficient algorithms in both fields. Overall, the work beautifully highlights how techniques from one field can be applied to the other, with applications to generalization bounds, group actions, and optimization landscapes. In the last section of their SIGEST paper the authors discuss possible future research directions that capitalize on the dictionary they have uncovered.
{"title":"SIGEST","authors":"The Editors","doi":"10.1137/24n976006","DOIUrl":"https://doi.org/10.1137/24n976006","url":null,"abstract":"SIAM Review, Volume 66, Issue 4, Page 719-719, November 2024. <br/> The SIGEST article in this issue, “A Bridge between Invariant Theory and Maximum Likelihood Estimation,” by Carlos Améndola, Kathlén Kohn, Philipp Reichenbach, and Anna Seigal, uncovers the deep connections between geometric invariant theory and statistical methods, specifically maximum likelihood estimation (MLE) by connecting it to norm minimization over group orbits. The authors develop a dictionary relating stability notions in geometric invariant theory to the existence and uniqueness of MLEs, which applies to both Gaussian and log-linear models. In comparison to the original 2021 version of the paper that appeared in the SIAM Journal on Applied Algebra and Geometry, for the SIGEST version, the authors added new content on log-linear models, simplified technical proofs, removed detailed appendices, and incorporated new examples and figures for accessibility. In particular, the focus was primarily on Gaussian models, whereas this updated SIGEST version expands the coverage by incorporating results from the authors' companion paper on log-linear models. Furthermore, a new figure (Fig. 1) visually illustrates the two core concepts of invariant theory and MLE. Significant changes include the removal of technical details and appendices to streamline the content and make it more accessible to a broader audience. The introduction of examples, particularly for the Kempf--Ness Theorem, further aids understanding. This paper makes several key contributions of broad mathematical interest. MLE is a key statistical technique that is widely used. Having a new handle on its well-posedness analysis deepens the understanding of the mechanisms behind this technique as well as potentially paves the way to extending existing theory for MLE models. Also, on the computational side, algorithms from the optimization over orbits can be used for MLE, and vice versa, which could possibly lead to new and more efficient algorithms in both fields. Overall, the work beautifully highlights how techniques from one field can be applied to the other, with applications to generalization bounds, group actions, and optimization landscapes. In the last section of their SIGEST paper the authors discuss possible future research directions that capitalize on the dictionary they have uncovered.","PeriodicalId":49525,"journal":{"name":"SIAM Review","volume":"69 1","pages":""},"PeriodicalIF":10.2,"publicationDate":"2024-11-07","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142594681","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 4, Page 617-617, November 2024. Neural oscillations are periodic activities of neurons in the central nervous system of eumetazoa. In an oscillatory neural network, neurons are modeled by coupled oscillators. Oscillatory networks are employed for describing the behavior of complex systems in biology or ecology with respect to the connectivity of the network components or the nonlinear dynamics of the individual units. Phase-locked periodic states and their instabilities are core features in the analysis of oscillatory networks. In “Oscillatory Networks: Insights from Piecewise-Linear Modeling,” Stephen Coombes, Mustafa Şayli, Rüdiger Thul, Rachel Nicks, Mason A. Porter, and Yi Ming Lai review techniques for studying coupled oscillatory networks. They first discuss phase reductions, phase-amplitude reductions, and the master stability function for smooth dynamical systems. Then they consider nonsmooth piecewise-linear (PWL) systems, for which periodic orbits are easily obtained. Saltation operators are used for modeling the propagation of perturbations through switching manifolds in the analysis of the dynamics and bifurcations at the network level. Applications to neural systems, cardiac systems, networks of electromechanical oscillators, and cooperation in cattle herds illustrate the power of these methods. PWL modeling has been applied for a long time in engineering. Recently, it has been introduced in other fields, such as social sciences, finance, and biology. For many modern applications in science, piecewise models are much more versatile than the classical smooth dynamical systems. In neuroscience, PWL functions enable explicit calculations which are infeasible in the original smooth system. This includes discontinuous dynamical systems, which are used to model impacting mechanical oscillators, integrate-and-fire models of spiking neurons, and cardiac oscillators. On the other hand, the price to pay is the retrieval of new conditions for the existence, uniqueness, and stability of solutions. The paper discusses the application of PWL models to a large variety of applications from engineering and biology. It will be of interest to many readers.
SIAM Review》，第 66 卷第 4 期，第 617-617 页，2024 年 11 月。 神经振荡是真尾目动物中枢神经系统中神经元的周期性活动。在振荡神经网络中，神经元由耦合振荡器建模。振荡网络用于描述生物学或生态学中复杂系统的行为，涉及网络组件的连接性或单个单元的非线性动态。锁相周期状态及其不稳定性是振荡网络分析的核心特征。在《振荡网络：中，Stephen Coombes、Mustafa Şayli、Rüdiger Thul、Rachel Nicks、Mason A. Porter 和 Yi Ming Lai 回顾了研究耦合振荡网络的技术。他们首先讨论了平滑动力系统的相位还原、相幅还原和主稳定函数。然后，他们考虑了非光滑的片线性 (PWL) 系统，对于这些系统，周期轨道很容易获得。在分析网络层面的动力学和分岔时，盐化算子用于模拟扰动通过开关流形的传播。在神经系统、心脏系统、机电振荡器网络和牛群合作中的应用说明了这些方法的威力。PWL 建模在工程领域应用已久。最近，它又被引入其他领域，如社会科学、金融和生物学。对于许多现代科学应用来说，片断模型比经典的平滑动态系统用途更广。在神经科学中，PWL 函数可以进行在原始平稳系统中不可行的显式计算。这包括用于模拟冲击机械振荡器、尖峰神经元的积分-发射模型和心脏振荡器的非连续动力系统。另一方面，所付出的代价是要检索解的存在性、唯一性和稳定性的新条件。本文讨论了 PWL 模型在工程学和生物学中的大量应用。它将引起许多读者的兴趣。
{"title":"Survey and Review","authors":"Marlis Hochbruck","doi":"10.1137/24n975980","DOIUrl":"https://doi.org/10.1137/24n975980","url":null,"abstract":"SIAM Review, Volume 66, Issue 4, Page 617-617, November 2024. <br/> Neural oscillations are periodic activities of neurons in the central nervous system of eumetazoa. In an oscillatory neural network, neurons are modeled by coupled oscillators. Oscillatory networks are employed for describing the behavior of complex systems in biology or ecology with respect to the connectivity of the network components or the nonlinear dynamics of the individual units. Phase-locked periodic states and their instabilities are core features in the analysis of oscillatory networks. In “Oscillatory Networks: Insights from Piecewise-Linear Modeling,” Stephen Coombes, Mustafa Şayli, Rüdiger Thul, Rachel Nicks, Mason A. Porter, and Yi Ming Lai review techniques for studying coupled oscillatory networks. They first discuss phase reductions, phase-amplitude reductions, and the master stability function for smooth dynamical systems. Then they consider nonsmooth piecewise-linear (PWL) systems, for which periodic orbits are easily obtained. Saltation operators are used for modeling the propagation of perturbations through switching manifolds in the analysis of the dynamics and bifurcations at the network level. Applications to neural systems, cardiac systems, networks of electromechanical oscillators, and cooperation in cattle herds illustrate the power of these methods. PWL modeling has been applied for a long time in engineering. Recently, it has been introduced in other fields, such as social sciences, finance, and biology. For many modern applications in science, piecewise models are much more versatile than the classical smooth dynamical systems. In neuroscience, PWL functions enable explicit calculations which are infeasible in the original smooth system. This includes discontinuous dynamical systems, which are used to model impacting mechanical oscillators, integrate-and-fire models of spiking neurons, and cardiac oscillators. On the other hand, the price to pay is the retrieval of new conditions for the existence, uniqueness, and stability of solutions. The paper discusses the application of PWL models to a large variety of applications from engineering and biology. It will be of interest to many readers.","PeriodicalId":49525,"journal":{"name":"SIAM Review","volume":"9 1","pages":""},"PeriodicalIF":10.2,"publicationDate":"2024-11-07","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142594685","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 4, Page 683-693, November 2024. In this short, conceptual paper we observe that closely related mathematics applies in four contexts with disparate literatures: (1) sigmoidal and RBF approximation of smooth functions, (2) rational approximation of analytic functions with singularities, (3) $hpkern .7pt$-mesh refinement for solution of pdes, and (4) double exponential (DE) and generalized Gauss quadrature. The relationships start from the change of variables $s = log(x)$, and they suggest possibilities for new analyses and new methods in several areas. Concerning (2) and (3), we show that both problems feature the same effect of “linear tapering” near the singularity---of clustered poles in rational approximation and of polynomial orders in $hpkern .7pt$-mesh refinement. Concerning (4), we note that the tapering effect appears here too, and that the change of variables interpretation sheds new light on why the DE and generalized Gauss methods are effective at integrating arbitrary singularities.
{"title":"Sigmoid Functions, Multiscale Resolution of Singularities, and $hp$-Mesh Refinement","authors":"Daan Huybrechs, Lloyd N. Trefethen","doi":"10.1137/23m1556629","DOIUrl":"https://doi.org/10.1137/23m1556629","url":null,"abstract":"SIAM Review, Volume 66, Issue 4, Page 683-693, November 2024. <br/> In this short, conceptual paper we observe that closely related mathematics applies in four contexts with disparate literatures: (1) sigmoidal and RBF approximation of smooth functions, (2) rational approximation of analytic functions with singularities, (3) $hpkern .7pt$-mesh refinement for solution of pdes, and (4) double exponential (DE) and generalized Gauss quadrature. The relationships start from the change of variables $s = log(x)$, and they suggest possibilities for new analyses and new methods in several areas. Concerning (2) and (3), we show that both problems feature the same effect of “linear tapering” near the singularity---of clustered poles in rational approximation and of polynomial orders in $hpkern .7pt$-mesh refinement. Concerning (4), we note that the tapering effect appears here too, and that the change of variables interpretation sheds new light on why the DE and generalized Gauss methods are effective at integrating arbitrary singularities.","PeriodicalId":49525,"journal":{"name":"SIAM Review","volume":"95 1","pages":""},"PeriodicalIF":10.2,"publicationDate":"2024-11-07","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142594683","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 4, Page 681-681, November 2024. Logarithmic transformations are used broadly in data science, mathematics, and engineering, and yet they can still reveal surprising connections between seemingly unrelated disciplines. This issue's first research spotlight, “Sigmoid Functions, Multiscale Resolution of Singularities, and $hp$-Mesh Refinement,” illuminates how the change of variables $s = log(x)$ connects different areas of computational mathematics. Authors Daan Huybrechs and Lloyd “Nick” Trefethen show new relationships between smooth approximation, rational approximation theory, adaptive mesh refinement, and numerical quadrature. For example, the authors show that this change of variables can be naturally tied to a “linear tapering” effect near singularities, which is a common feature in both rational approximation and $hp$-mesh refinement. Through a number of effective examples, the authors illustrate the power of these relationships across areas that have seen relatively independent lines of development. In doing so, the authors suggest opportunities for developing and analyzing new methods by leveraging the new connections, including mesh refinement strategies, techniques for multivariate approximation, and hybrid approaches that combine the strengths of disparate methods. How well can information be recovered from water waves? This question is at the heart of this issue's second research spotlight, “Feynman's Inverse Problem.” Author Adrian Kirkeby is motivated by a thought experiment posed by the physicist and iconoclast Richard Feynman wherein an insect floating in a swimming pool wants to determine where and when others have jumped into the pool, causing the waves the insect observes. Kirkeby constructs and analyzes a linear 2D-3D system of partial differential equations (PDEs) for the forward model. Leveraging the nonlocality of this system of PDEs, Kirkeby shows conditions under which the insect can determine the source of the waves---in fact, uniquely---simply by observing the wave amplitude and water velocity in any small area of the surface. This model is then extended to capture settings where noisy observations and observations at a finite number of time and space points are collected, and establishes stability properties and error bounds for the reconstruction. The paper concludes with illustrative numerical experiments based on a nonharmonic Fourier inversion method. Kirkeby also highlights several avenues for future research, noting that inverse problems for water or other surface waves have received less attention than those involving acoustic or electromagnetic waves. As an added bonus, the referenced video of Feynman is not to be missed.
{"title":"Research Spotlights","authors":"Stefan M. Wild","doi":"10.1137/24n975992","DOIUrl":"https://doi.org/10.1137/24n975992","url":null,"abstract":"SIAM Review, Volume 66, Issue 4, Page 681-681, November 2024. <br/> Logarithmic transformations are used broadly in data science, mathematics, and engineering, and yet they can still reveal surprising connections between seemingly unrelated disciplines. This issue's first research spotlight, “Sigmoid Functions, Multiscale Resolution of Singularities, and $hp$-Mesh Refinement,” illuminates how the change of variables $s = log(x)$ connects different areas of computational mathematics. Authors Daan Huybrechs and Lloyd “Nick” Trefethen show new relationships between smooth approximation, rational approximation theory, adaptive mesh refinement, and numerical quadrature. For example, the authors show that this change of variables can be naturally tied to a “linear tapering” effect near singularities, which is a common feature in both rational approximation and $hp$-mesh refinement. Through a number of effective examples, the authors illustrate the power of these relationships across areas that have seen relatively independent lines of development. In doing so, the authors suggest opportunities for developing and analyzing new methods by leveraging the new connections, including mesh refinement strategies, techniques for multivariate approximation, and hybrid approaches that combine the strengths of disparate methods. How well can information be recovered from water waves? This question is at the heart of this issue's second research spotlight, “Feynman's Inverse Problem.” Author Adrian Kirkeby is motivated by a thought experiment posed by the physicist and iconoclast Richard Feynman wherein an insect floating in a swimming pool wants to determine where and when others have jumped into the pool, causing the waves the insect observes. Kirkeby constructs and analyzes a linear 2D-3D system of partial differential equations (PDEs) for the forward model. Leveraging the nonlocality of this system of PDEs, Kirkeby shows conditions under which the insect can determine the source of the waves---in fact, uniquely---simply by observing the wave amplitude and water velocity in any small area of the surface. This model is then extended to capture settings where noisy observations and observations at a finite number of time and space points are collected, and establishes stability properties and error bounds for the reconstruction. The paper concludes with illustrative numerical experiments based on a nonharmonic Fourier inversion method. Kirkeby also highlights several avenues for future research, noting that inverse problems for water or other surface waves have received less attention than those involving acoustic or electromagnetic waves. As an added bonus, the referenced video of Feynman is not to be missed.","PeriodicalId":49525,"journal":{"name":"SIAM Review","volume":"243 1","pages":""},"PeriodicalIF":10.2,"publicationDate":"2024-11-07","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142594684","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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