Carlos Améndola, Kathlén Kohn, Philipp Reichenbach, Anna Seigal
SIAM Review, Volume 66, Issue 4, Page 721-747, November 2024. We uncover connections between maximum likelihood estimation in statistics and norm minimization over a group orbit in invariant theory. We present a dictionary that relates notions of stability from geometric invariant theory to the existence and uniqueness of a maximum likelihood estimate. Our dictionary holds for both discrete and continuous statistical models: we discuss log-linear models and Gaussian models, including matrix normal models and directed Gaussian graphical models. Our approach reveals promising consequences of the interplay between invariant theory and statistics. For instance, algorithms from statistics can be used in invariant theory, and vice versa.
{"title":"A Bridge between Invariant Theory and Maximum Likelihood Estimation","authors":"Carlos Améndola, Kathlén Kohn, Philipp Reichenbach, Anna Seigal","doi":"10.1137/24m1661753","DOIUrl":"https://doi.org/10.1137/24m1661753","url":null,"abstract":"SIAM Review, Volume 66, Issue 4, Page 721-747, November 2024. <br/> We uncover connections between maximum likelihood estimation in statistics and norm minimization over a group orbit in invariant theory. We present a dictionary that relates notions of stability from geometric invariant theory to the existence and uniqueness of a maximum likelihood estimate. Our dictionary holds for both discrete and continuous statistical models: we discuss log-linear models and Gaussian models, including matrix normal models and directed Gaussian graphical models. Our approach reveals promising consequences of the interplay between invariant theory and statistics. For instance, algorithms from statistics can be used in invariant theory, and vice versa.","PeriodicalId":49525,"journal":{"name":"SIAM Review","volume":"3 1","pages":""},"PeriodicalIF":10.2,"publicationDate":"2024-11-07","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142594680","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}
Ariel Cintrón-Arias, Ryan Andrew Nivens, Anant Godbole, Calvin B. Purvis
SIAM Review, Volume 66, Issue 4, Page 778-792, November 2024. Mathematicians have traditionally been a select group of academics who produce high-impact ideas enabling substantial results in several fields of science. Throughout the past 35 years, undergraduates enrolling in mathematics or statistics have represented a nearly constant proportion of approximately 1% of bachelor degrees awarded in the United States. Even within STEM majors, mathematics or statistics only constitute about 6% of undergraduate degrees awarded nationally. However, the need for STEM professionals continues to grow, and the list of required occupational skills rests heavily in foundational concepts of mathematical modeling curricula, where the interplay of data, computer simulation, and underlying theoretical frameworks takes center stage. It is not viable to expect a majority of these STEM undergraduates to pursue a double major that includes mathematics. Here we present our solution, some early results of its implementation, and a vision for possible nationwide adoption.
{"title":"Developing Workforce with Mathematical Modeling Skills","authors":"Ariel Cintrón-Arias, Ryan Andrew Nivens, Anant Godbole, Calvin B. Purvis","doi":"10.1137/19m1277643","DOIUrl":"https://doi.org/10.1137/19m1277643","url":null,"abstract":"SIAM Review, Volume 66, Issue 4, Page 778-792, November 2024. <br/> Mathematicians have traditionally been a select group of academics who produce high-impact ideas enabling substantial results in several fields of science. Throughout the past 35 years, undergraduates enrolling in mathematics or statistics have represented a nearly constant proportion of approximately 1% of bachelor degrees awarded in the United States. Even within STEM majors, mathematics or statistics only constitute about 6% of undergraduate degrees awarded nationally. However, the need for STEM professionals continues to grow, and the list of required occupational skills rests heavily in foundational concepts of mathematical modeling curricula, where the interplay of data, computer simulation, and underlying theoretical frameworks takes center stage. It is not viable to expect a majority of these STEM undergraduates to pursue a double major that includes mathematics. Here we present our solution, some early results of its implementation, and a vision for possible nationwide adoption.","PeriodicalId":49525,"journal":{"name":"SIAM Review","volume":"45 1","pages":""},"PeriodicalIF":10.2,"publicationDate":"2024-11-07","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142594677","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 749-749, November 2024. <br/> In this issue the Education section presents two contributions. The first paper, “Sandpiles and Dunes: Mathematical Models for Granular Matter,” by Piermarco Cannarsa and Stefano Finzi Vita, presents a review of mathematical models for formation of sand piles and dunes. In nature and everyday life various materials appear as conglomerates of particles, like, for instance, sand, gravel, fresh snow, rice, sugar, etc. On larger scales, granular material exhibits new and more complex phenomena which are still not fully understood. It is very different from that of a solid, liquid, or gas in the sense that it can show characteristics similar to one or the other depending on the energy of the system. Its modeling can help in understanding complex natural phenomena such as dune migration, erosion, landslides, and avalanches, and can contribute to the development of environmental protection programs. Such models are also important in various applications in agriculture, construction, energy production, as well as in the chemical, pharmaceutical, food, and metallurgical industries. Even if a sufficiently consolidated general model for the dynamics of granular materials is not available yet, significant progress has been made recently with the introduction of new theoretical models adapted to more specific situations. In this article, after a general description and some historical comments, the authors limit themselves to considering the problem of the growth of a pile of sand on a table under the action of a vertical source of small intensity, neglecting the effects of wind, which has an important role in dune formation. Still, even for such an apparently simpler case, many interesting phenomena do arise and are described in an easily accessible way. Accompanying pictures of real-life experiences make the reading truly enjoyable, and numerical illustrations bring even better intuition on the complexity of phenomena. The authors also indicate literature for further learning. This article is well organized, neatly written, and presents the subject highlighting some of the major aspects. This review of existing models can become a starting point for research projects in a Master's program of applied mathematics and partial differential equations. It could also be used by advanced mathematics students to learn differential models of granular material in an affordable way. The second paper, “Developing Workforce with Mathematical Modeling Skills,” is presented by Ariel Cintrón-Arias, Ryan Andrew Nivens, Anant Godbole and Calvin B. Purvis. Undergraduate mathematics degrees constitute a very small portion of all awarded degrees in the U.S., and this portion is stagnating, while the job growth between 2016 and 2026 for Statisticians and Mathematicians is expected to be substantial. So the need for growth in mathematical training becomes imperative. The authors discuss the nationwide production o
{"title":"Education","authors":"Hélène Frankowska","doi":"10.1137/24n976018","DOIUrl":"https://doi.org/10.1137/24n976018","url":null,"abstract":"SIAM Review, Volume 66, Issue 4, Page 749-749, November 2024. <br/> In this issue the Education section presents two contributions. The first paper, “Sandpiles and Dunes: Mathematical Models for Granular Matter,” by Piermarco Cannarsa and Stefano Finzi Vita, presents a review of mathematical models for formation of sand piles and dunes. In nature and everyday life various materials appear as conglomerates of particles, like, for instance, sand, gravel, fresh snow, rice, sugar, etc. On larger scales, granular material exhibits new and more complex phenomena which are still not fully understood. It is very different from that of a solid, liquid, or gas in the sense that it can show characteristics similar to one or the other depending on the energy of the system. Its modeling can help in understanding complex natural phenomena such as dune migration, erosion, landslides, and avalanches, and can contribute to the development of environmental protection programs. Such models are also important in various applications in agriculture, construction, energy production, as well as in the chemical, pharmaceutical, food, and metallurgical industries. Even if a sufficiently consolidated general model for the dynamics of granular materials is not available yet, significant progress has been made recently with the introduction of new theoretical models adapted to more specific situations. In this article, after a general description and some historical comments, the authors limit themselves to considering the problem of the growth of a pile of sand on a table under the action of a vertical source of small intensity, neglecting the effects of wind, which has an important role in dune formation. Still, even for such an apparently simpler case, many interesting phenomena do arise and are described in an easily accessible way. Accompanying pictures of real-life experiences make the reading truly enjoyable, and numerical illustrations bring even better intuition on the complexity of phenomena. The authors also indicate literature for further learning. This article is well organized, neatly written, and presents the subject highlighting some of the major aspects. This review of existing models can become a starting point for research projects in a Master's program of applied mathematics and partial differential equations. It could also be used by advanced mathematics students to learn differential models of granular material in an affordable way. The second paper, “Developing Workforce with Mathematical Modeling Skills,” is presented by Ariel Cintrón-Arias, Ryan Andrew Nivens, Anant Godbole and Calvin B. Purvis. Undergraduate mathematics degrees constitute a very small portion of all awarded degrees in the U.S., and this portion is stagnating, while the job growth between 2016 and 2026 for Statisticians and Mathematicians is expected to be substantial. So the need for growth in mathematical training becomes imperative. The authors discuss the nationwide production o","PeriodicalId":49525,"journal":{"name":"SIAM Review","volume":"14 1","pages":""},"PeriodicalIF":10.2,"publicationDate":"2024-11-07","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142594679","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 694-718, November 2024. We analyze an inverse problem for water waves posed by Richard Feynman in the BBC documentary Fun to Imagine. We show that the problem can be modeled as an inverse Cauchy problem for gravity-capillary waves, conduct a detailed analysis of the Cauchy problem, and give a uniqueness proof for the inverse problem. Somewhat surprisingly, this results in a positive answer to Feynman's question. In addition, we derive stability estimates for the inverse problem for both continuous and discrete measurements, propose a simple inversion method, and conduct numerical experiments to verify our results.
{"title":"Feynman's Inverse Problem","authors":"Adrian Kirkeby","doi":"10.1137/23m1611488","DOIUrl":"https://doi.org/10.1137/23m1611488","url":null,"abstract":"SIAM Review, Volume 66, Issue 4, Page 694-718, November 2024. <br/> We analyze an inverse problem for water waves posed by Richard Feynman in the BBC documentary Fun to Imagine. We show that the problem can be modeled as an inverse Cauchy problem for gravity-capillary waves, conduct a detailed analysis of the Cauchy problem, and give a uniqueness proof for the inverse problem. Somewhat surprisingly, this results in a positive answer to Feynman's question. In addition, we derive stability estimates for the inverse problem for both continuous and discrete measurements, propose a simple inversion method, and conduct numerical experiments to verify our results.","PeriodicalId":49525,"journal":{"name":"SIAM Review","volume":"109 1","pages":""},"PeriodicalIF":10.2,"publicationDate":"2024-11-07","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142594682","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}
Stephen Coombes, Mustafa Şayli, Rüdiger Thul, Rachel Nicks, Mason A. Porter, Yi Ming Lai
SIAM Review, Volume 66, Issue 4, Page 619-679, November 2024. There is enormous interest---both mathematically and in diverse applications---in understanding the dynamics of coupled-oscillator networks. The real-world motivation of such networks arises from studies of the brain, the heart, ecology, and more. It is common to describe the rich emergent behavior in these systems in terms of complex patterns of network activity that reflect both the connectivity and the nonlinear dynamics of the network components. Such behavior is often organized around phase-locked periodic states and their instabilities. However, the explicit calculation of periodic orbits in nonlinear systems (even in low dimensions) is notoriously hard, so network-level insights often require the numerical construction of some underlying periodic component. In this paper, we review powerful techniques for studying coupled-oscillator networks. We discuss phase reductions, phase--amplitude reductions, and the master stability function for smooth dynamical systems. We then focus, in particular, on the augmentation of these methods to analyze piecewise-linear systems, for which one can readily construct periodic orbits. This yields useful insights into network behavior, but the cost is that one needs to study nonsmooth dynamical systems. The study of nonsmooth systems is well developed when focusing on the interacting units (i.e., at the node level) of a system, and we give a detailed presentation of how to use saltation operators, which can treat the propagation of perturbations through switching manifolds, to understand dynamics and bifurcations at the network level. We illustrate this merger of tools and techniques from network science and nonsmooth dynamical systems with applications to neural systems, cardiac systems, networks of electromechanical oscillators, and cooperation in cattle herds.
{"title":"Oscillatory Networks: Insights from Piecewise-Linear Modeling","authors":"Stephen Coombes, Mustafa Şayli, Rüdiger Thul, Rachel Nicks, Mason A. Porter, Yi Ming Lai","doi":"10.1137/22m1534365","DOIUrl":"https://doi.org/10.1137/22m1534365","url":null,"abstract":"SIAM Review, Volume 66, Issue 4, Page 619-679, November 2024. <br/> There is enormous interest---both mathematically and in diverse applications---in understanding the dynamics of coupled-oscillator networks. The real-world motivation of such networks arises from studies of the brain, the heart, ecology, and more. It is common to describe the rich emergent behavior in these systems in terms of complex patterns of network activity that reflect both the connectivity and the nonlinear dynamics of the network components. Such behavior is often organized around phase-locked periodic states and their instabilities. However, the explicit calculation of periodic orbits in nonlinear systems (even in low dimensions) is notoriously hard, so network-level insights often require the numerical construction of some underlying periodic component. In this paper, we review powerful techniques for studying coupled-oscillator networks. We discuss phase reductions, phase--amplitude reductions, and the master stability function for smooth dynamical systems. We then focus, in particular, on the augmentation of these methods to analyze piecewise-linear systems, for which one can readily construct periodic orbits. This yields useful insights into network behavior, but the cost is that one needs to study nonsmooth dynamical systems. The study of nonsmooth systems is well developed when focusing on the interacting units (i.e., at the node level) of a system, and we give a detailed presentation of how to use saltation operators, which can treat the propagation of perturbations through switching manifolds, to understand dynamics and bifurcations at the network level. We illustrate this merger of tools and techniques from network science and nonsmooth dynamical systems with applications to neural systems, cardiac systems, networks of electromechanical oscillators, and cooperation in cattle herds.","PeriodicalId":49525,"journal":{"name":"SIAM Review","volume":"145 1","pages":""},"PeriodicalIF":10.2,"publicationDate":"2024-11-07","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142594686","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}
Liliana Borcea, Josselin Garnier, Alexander V. Mamonov, Jörn Zimmerling
SIAM Review, Volume 66, Issue 3, Page 501-532, May 2024. Waveform inversion is concerned with estimating a heterogeneous medium, modeled by variable coefficients of wave equations, using sources that emit probing signals and receivers that record the generated waves. It is an old and intensively studied inverse problem with a wide range of applications, but the existing inversion methodologies are still far from satisfactory. The typical mathematical formulation is a nonlinear least squares data fit optimization and the difficulty stems from the nonconvexity of the objective function that displays numerous local minima at which local optimization approaches stagnate. This pathological behavior has at least three unavoidable causes: (1) The mapping from the unknown coefficients to the wave field is nonlinear and complicated. (2) The sources and receivers typically lie on a single side of the medium, so only backscattered waves are measured. (3) The probing signals are band limited and with high frequency content. There is a lot of activity in the computational science and engineering communities that seeks to mitigate the difficulty of estimating the medium by data fitting. In this paper we present a different point of view, based on reduced order models (ROMs) of two operators that control the wave propagation. The ROMs are called data driven because they are computed directly from the measurements, without any knowledge of the wave field inside the inaccessible medium. This computation is noniterative and uses standard numerical linear algebra methods. The resulting ROMs capture features of the physics of wave propagation in a complementary way and have surprisingly good approximation properties that facilitate waveform inversion.
{"title":"When Data Driven Reduced Order Modeling Meets Full Waveform Inversion","authors":"Liliana Borcea, Josselin Garnier, Alexander V. Mamonov, Jörn Zimmerling","doi":"10.1137/23m1552826","DOIUrl":"https://doi.org/10.1137/23m1552826","url":null,"abstract":"SIAM Review, Volume 66, Issue 3, Page 501-532, May 2024. <br/> Waveform inversion is concerned with estimating a heterogeneous medium, modeled by variable coefficients of wave equations, using sources that emit probing signals and receivers that record the generated waves. It is an old and intensively studied inverse problem with a wide range of applications, but the existing inversion methodologies are still far from satisfactory. The typical mathematical formulation is a nonlinear least squares data fit optimization and the difficulty stems from the nonconvexity of the objective function that displays numerous local minima at which local optimization approaches stagnate. This pathological behavior has at least three unavoidable causes: (1) The mapping from the unknown coefficients to the wave field is nonlinear and complicated. (2) The sources and receivers typically lie on a single side of the medium, so only backscattered waves are measured. (3) The probing signals are band limited and with high frequency content. There is a lot of activity in the computational science and engineering communities that seeks to mitigate the difficulty of estimating the medium by data fitting. In this paper we present a different point of view, based on reduced order models (ROMs) of two operators that control the wave propagation. The ROMs are called data driven because they are computed directly from the measurements, without any knowledge of the wave field inside the inaccessible medium. This computation is noniterative and uses standard numerical linear algebra methods. The resulting ROMs capture features of the physics of wave propagation in a complementary way and have surprisingly good approximation properties that facilitate waveform inversion.","PeriodicalId":49525,"journal":{"name":"SIAM Review","volume":"24 1","pages":""},"PeriodicalIF":10.2,"publicationDate":"2024-08-08","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"141904641","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 3, Page 479-479, May 2024. <br/> Equitable distribution of geographically dispersed resources presents a significant challenge, particularly in defining quantifiable measures of equity. How can we optimally allocate polling sites or hospitals to serve their constituencies? This issue's first Research Spotlight, “Persistent Homology for Resource Coverage: A Case Study of Access to Polling Sites," addresses these questions by demonstrating the application of topological data analysis to identify holes in resource accessibility and coverage. Authors Abigail Hickok, Benjamin Jarman, Michael Johnson, Jiajie Luo, and Mason A. Porter employ persistent homology, a technique that tracks the formation and disappearance of these holes as spatial scales vary. To make matters concrete, the authors consider a case study on access to polling sites and use a non-Euclidean distance that accounts for both travel and waiting times. In their case study, the authors use a weighted Vietoris--Rips filtration based on a symmetrized form of this distance and limit their examination to instances where the approximations underlying the filtration are less likely to lead to approximation-based artifacts. Details, as well as source code, are provided on the estimation of the various quantities, such as travel time, waiting time, and demographics (e.g., age, vehicle access). The result is a homology class that “dies" at time $t$ if it takes $t$ total minutes to cast a vote. The paper concludes with an exposition of potential limitations and future directions that serve to encourage additional investigation into this class of problems (which includes settings where one wants to deploy different sensors to cover a spatial domain) and related techniques. What secrets lurk within? From flaws in human-made infrastructure to materials deep beneath the Earth's land and ocean surfaces to anomalies in patients, our next Research Spotlight, “When Data Driven Reduced Order Modeling Meets Full Waveform Inversion," addresses math and methods to recover the unknown. Authors Liliana Borcea, Josselin Garnier, Alexander V. Mamonov, and Jörn Zimmerling show how tools from numerical linear algebra and reduced-order modeling can be brought to bear on inverse wave scattering problems. Their setup encapsulates a wide variety of sensing modalities, wherein receivers emit a signal (such as an acoustic wave) and a time series of wavefield measurements is subsequently captured at one or more sources. Full waveform inversion refers to the recovery of the unknown “within" and is typically addressed via iterative, nonlinear equations/least-squares solvers. However, it is often plagued by a notoriously nonconvex, ill-conditioned optimization landscape. The authors show how some of the challenges typically encountered in this inversion can be mitigated with the use of reduced-order models. These models employ observed data snapshots to form lower-dimensional, computationally a
{"title":"Research Spotlights","authors":"Stefan M. Wild","doi":"10.1137/24n975931","DOIUrl":"https://doi.org/10.1137/24n975931","url":null,"abstract":"SIAM Review, Volume 66, Issue 3, Page 479-479, May 2024. <br/> Equitable distribution of geographically dispersed resources presents a significant challenge, particularly in defining quantifiable measures of equity. How can we optimally allocate polling sites or hospitals to serve their constituencies? This issue's first Research Spotlight, “Persistent Homology for Resource Coverage: A Case Study of Access to Polling Sites,\" addresses these questions by demonstrating the application of topological data analysis to identify holes in resource accessibility and coverage. Authors Abigail Hickok, Benjamin Jarman, Michael Johnson, Jiajie Luo, and Mason A. Porter employ persistent homology, a technique that tracks the formation and disappearance of these holes as spatial scales vary. To make matters concrete, the authors consider a case study on access to polling sites and use a non-Euclidean distance that accounts for both travel and waiting times. In their case study, the authors use a weighted Vietoris--Rips filtration based on a symmetrized form of this distance and limit their examination to instances where the approximations underlying the filtration are less likely to lead to approximation-based artifacts. Details, as well as source code, are provided on the estimation of the various quantities, such as travel time, waiting time, and demographics (e.g., age, vehicle access). The result is a homology class that “dies\" at time $t$ if it takes $t$ total minutes to cast a vote. The paper concludes with an exposition of potential limitations and future directions that serve to encourage additional investigation into this class of problems (which includes settings where one wants to deploy different sensors to cover a spatial domain) and related techniques. What secrets lurk within? From flaws in human-made infrastructure to materials deep beneath the Earth's land and ocean surfaces to anomalies in patients, our next Research Spotlight, “When Data Driven Reduced Order Modeling Meets Full Waveform Inversion,\" addresses math and methods to recover the unknown. Authors Liliana Borcea, Josselin Garnier, Alexander V. Mamonov, and Jörn Zimmerling show how tools from numerical linear algebra and reduced-order modeling can be brought to bear on inverse wave scattering problems. Their setup encapsulates a wide variety of sensing modalities, wherein receivers emit a signal (such as an acoustic wave) and a time series of wavefield measurements is subsequently captured at one or more sources. Full waveform inversion refers to the recovery of the unknown “within\" and is typically addressed via iterative, nonlinear equations/least-squares solvers. However, it is often plagued by a notoriously nonconvex, ill-conditioned optimization landscape. The authors show how some of the challenges typically encountered in this inversion can be mitigated with the use of reduced-order models. These models employ observed data snapshots to form lower-dimensional, computationally a","PeriodicalId":49525,"journal":{"name":"SIAM Review","volume":"76 1","pages":""},"PeriodicalIF":10.2,"publicationDate":"2024-08-08","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"141908890","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 3, Page 573-573, May 2024. In this issue the Education section presents “Combinatorial and Hodge Laplacians: Similarities and Differences,” by Emily Ribando-Gros, Rui Wang, Jiahui Chen, Yiying Tong, and Guo-Wei Wei. Combinatorial Laplacians and their spectra are important tools in the study of molecular stability, electrical networks, neuroscience, deep learning, signal processing, etc. The continuous Hodge Laplacian allows one, in some cases, to generate an unknown shape from only its Laplacian spectrum. In particular, both combinatorial Laplacians and continuous Hodge Laplacian are useful in describing the topology of data; see, for instance, [L.-H. Lim, “Hodge Laplacians on graphs,” SIAM Rev., 62 (2020), pp. 685--715]. Since nowadays computations frequently involve these Laplacians, it is important to have a good understanding of the differences and relations between them. Indeed, though the Hodge Laplacian and the combinatorial Laplacian share similarities in revealing the topological dimension and geometric shape of data, at the same time they are intrinsically different in their domains of definitions and applicability to specific data formats. To facilitate comparisons, the authors introduce boundary-induced graph (BIG) Laplacians, the purpose of which is “to put the combinatorial Laplacians and Hodge Laplacian on equal footing.” BIG Laplacian brings, in fact, the combinatorial Laplacian closer to the continuous Hodge Laplacian. In this paper similarities and differences between combinatorial Laplacian, BIG Laplacian, and Hodge Laplacian are examined. Some elements of spectral analysis related to topological data analysis (TDA) are also provided. TDA and connected ideas have recently gained a lot of interest, and so this paper is timely. It is written in a way that should make it accessible for early career researchers; the reader should already have a good understanding of some notions of graph theory, spectral geometry, differential geometry, and algebraic topology. The paper is not self-contained and eventually could be used by group-based research projects in a Master's program for advanced mathematics students.
{"title":"Education","authors":"Hélène Frankowska","doi":"10.1137/24n975955","DOIUrl":"https://doi.org/10.1137/24n975955","url":null,"abstract":"SIAM Review, Volume 66, Issue 3, Page 573-573, May 2024. <br/> In this issue the Education section presents “Combinatorial and Hodge Laplacians: Similarities and Differences,” by Emily Ribando-Gros, Rui Wang, Jiahui Chen, Yiying Tong, and Guo-Wei Wei. Combinatorial Laplacians and their spectra are important tools in the study of molecular stability, electrical networks, neuroscience, deep learning, signal processing, etc. The continuous Hodge Laplacian allows one, in some cases, to generate an unknown shape from only its Laplacian spectrum. In particular, both combinatorial Laplacians and continuous Hodge Laplacian are useful in describing the topology of data; see, for instance, [L.-H. Lim, “Hodge Laplacians on graphs,” SIAM Rev., 62 (2020), pp. 685--715]. Since nowadays computations frequently involve these Laplacians, it is important to have a good understanding of the differences and relations between them. Indeed, though the Hodge Laplacian and the combinatorial Laplacian share similarities in revealing the topological dimension and geometric shape of data, at the same time they are intrinsically different in their domains of definitions and applicability to specific data formats. To facilitate comparisons, the authors introduce boundary-induced graph (BIG) Laplacians, the purpose of which is “to put the combinatorial Laplacians and Hodge Laplacian on equal footing.” BIG Laplacian brings, in fact, the combinatorial Laplacian closer to the continuous Hodge Laplacian. In this paper similarities and differences between combinatorial Laplacian, BIG Laplacian, and Hodge Laplacian are examined. Some elements of spectral analysis related to topological data analysis (TDA) are also provided. TDA and connected ideas have recently gained a lot of interest, and so this paper is timely. It is written in a way that should make it accessible for early career researchers; the reader should already have a good understanding of some notions of graph theory, spectral geometry, differential geometry, and algebraic topology. The paper is not self-contained and eventually could be used by group-based research projects in a Master's program for advanced mathematics students.","PeriodicalId":49525,"journal":{"name":"SIAM Review","volume":"38 1","pages":""},"PeriodicalIF":10.2,"publicationDate":"2024-08-08","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"141909212","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}
Abigail Hickok, Benjamin Jarman, Michael Johnson, Jiajie Luo, Mason A. Porter
SIAM Review, Volume 66, Issue 3, Page 481-500, May 2024. It is important to choose the geographical distributions of public resources in a fair and equitable manner. However, it is complicated to quantify the equity of such a distribution; important factors include distances to resource sites, availability of transportation, and ease of travel. We use persistent homology, which is a tool from topological data analysis, to study the availability and coverage of polling sites. The information from persistent homology allows us to infer holes in a distribution of polling sites. We analyze and compare the coverage of polling sites in Los Angeles County and five cities (Atlanta, Chicago, Jacksonville, New York City, and Salt Lake City), and we conclude that computation of persistent homology appears to be a reasonable approach to analyzing resource coverage.
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SIAM Review, Volume 66, Issue 3, Page 535-571, May 2024. Supervised operator learning centers on the use of training data, in the form of input-output pairs, to estimate maps between infinite-dimensional spaces. It is emerging as a powerful tool to complement traditional scientific computing, which may often be framed in terms of operators mapping between spaces of functions. Building on the classical random features methodology for scalar regression, this paper introduces the function-valued random features method. This leads to a supervised operator learning architecture that is practical for nonlinear problems yet is structured enough to facilitate efficient training through the optimization of a convex, quadratic cost. Due to the quadratic structure, the trained model is equipped with convergence guarantees and error and complexity bounds, properties that are not readily available for most other operator learning architectures. At its core, the proposed approach builds a linear combination of random operators. This turns out to be a low-rank approximation of an operator-valued kernel ridge regression algorithm, and hence the method also has strong connections to Gaussian process regression. The paper designs function-valued random features that are tailored to the structure of two nonlinear operator learning benchmark problems arising from parametric partial differential equations. Numerical results demonstrate the scalability, discretization invariance, and transferability of the function-valued random features method.
{"title":"Operator Learning Using Random Features: A Tool for Scientific Computing","authors":"Nicholas H. Nelsen, Andrew M. Stuart","doi":"10.1137/24m1648703","DOIUrl":"https://doi.org/10.1137/24m1648703","url":null,"abstract":"SIAM Review, Volume 66, Issue 3, Page 535-571, May 2024. <br/> Supervised operator learning centers on the use of training data, in the form of input-output pairs, to estimate maps between infinite-dimensional spaces. It is emerging as a powerful tool to complement traditional scientific computing, which may often be framed in terms of operators mapping between spaces of functions. Building on the classical random features methodology for scalar regression, this paper introduces the function-valued random features method. This leads to a supervised operator learning architecture that is practical for nonlinear problems yet is structured enough to facilitate efficient training through the optimization of a convex, quadratic cost. Due to the quadratic structure, the trained model is equipped with convergence guarantees and error and complexity bounds, properties that are not readily available for most other operator learning architectures. At its core, the proposed approach builds a linear combination of random operators. This turns out to be a low-rank approximation of an operator-valued kernel ridge regression algorithm, and hence the method also has strong connections to Gaussian process regression. The paper designs function-valued random features that are tailored to the structure of two nonlinear operator learning benchmark problems arising from parametric partial differential equations. Numerical results demonstrate the scalability, discretization invariance, and transferability of the function-valued random features method.","PeriodicalId":49525,"journal":{"name":"SIAM Review","volume":"68 1","pages":""},"PeriodicalIF":10.2,"publicationDate":"2024-08-08","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"141909214","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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