SIAM Review, Volume 65, Issue 3, Page 869-886, August 2023. Peixoto's structural stability and density theorems represent milestones in the modern theory of dynamical systems and their applications. Despite the importance of these theorems, they are often treated rather superficially, if at all, in upper level undergraduate courses on dynamical systems or differential equations. This is mainly because of the depth and length of the proofs. In this module, we formulate and prove the one-dimensional analogues of Peixoto's theorems in an intuitive and fairly simple way using only concepts and results that for the most part should be familiar to upper level undergraduate students in the mathematical sciences or related fields. The intention is to provide students who may be interested in further study in dynamical systems with an accessible one-dimensional treatment of structural stability theory that should help make Peixoto's theorems and their more recent generalizations easier to appreciate and understand. Further, we believe it is important and interesting for students to know the historical context of these discoveries since the mathematics was not done in isolation. The historical context is perhaps even more appropriate as it is the 100th anniversary of Marília Chaves Peixoto's and Maurício Matos Peixoto's births, February 24th and April 15th, 1921, respectively.
{"title":"The One-Dimensional Version of Peixoto's Structural Stability Theorem: A Calculus-Based Proof","authors":"Aminur Rahman, D. Blackmore","doi":"10.1137/21m1426572","DOIUrl":"https://doi.org/10.1137/21m1426572","url":null,"abstract":"SIAM Review, Volume 65, Issue 3, Page 869-886, August 2023. <br/> Peixoto's structural stability and density theorems represent milestones in the modern theory of dynamical systems and their applications. Despite the importance of these theorems, they are often treated rather superficially, if at all, in upper level undergraduate courses on dynamical systems or differential equations. This is mainly because of the depth and length of the proofs. In this module, we formulate and prove the one-dimensional analogues of Peixoto's theorems in an intuitive and fairly simple way using only concepts and results that for the most part should be familiar to upper level undergraduate students in the mathematical sciences or related fields. The intention is to provide students who may be interested in further study in dynamical systems with an accessible one-dimensional treatment of structural stability theory that should help make Peixoto's theorems and their more recent generalizations easier to appreciate and understand. Further, we believe it is important and interesting for students to know the historical context of these discoveries since the mathematics was not done in isolation. The historical context is perhaps even more appropriate as it is the 100th anniversary of Marília Chaves Peixoto's and Maurício Matos Peixoto's births, February 24th and April 15th, 1921, respectively.","PeriodicalId":49525,"journal":{"name":"SIAM Review","volume":"31 8","pages":""},"PeriodicalIF":10.2,"publicationDate":"2023-08-08","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"71509659","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":1,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
SIAM Review, Volume 65, Issue 3, Page 829-829, August 2023. The SIGEST article in this issue, which comes from the SIAM/ASA Journal on Uncertainty Quantification, is “Bayesian Inverse Problems Are Usually Well-Posed,” by Jonas Latz. The author investigates the well-posedness of Bayesian approaches to inverse problems, generalizing the framework of well-posedness introduced by Andrew Stuart to a set of weaker assumptions. Well-posedness here is understood in the sense of Hadamard, that is, a solution exists, is unique, and continuously depends on the input data. Inverse problems are typically ill-posed due to properties of the model, a lack of data, and measurement noise. The Bayesian approach to inverse problems reformulates the quest for a solution to the inverse problem in terms of a quest for its posterior distribution, which is determined by the data likelihood and prior distribution of the solution, and which in contrast to the inverse problem itself should be well-posed. In the Bayesian context, well-posedness typically relates to existence, uniqueness, and Lipschitz continuity of the posterior distribution with respect to the data in the so-called Hellinger distance. In many practical applications such well-posedness is difficult, if not impossible, to verify. Moreover, the choice of the Hellinger distance as the right metric might not always be the best fitted depending on the problem at hand. This sets the starting point for the paper where the author introduces a new framework for well-posedness of Bayesian inverse problems in which he shows existence, uniqueness, and continuity with respect to various metrics for a large class of Bayesian inverse problems, with conditions that are either nonrestrictive or verifiable in practical settings. This paper gives a strong new mathematical foundation for Bayesian inverse problems. The underlying statistical and probabilistic concepts are explained comprehensively and comprehensibly and, thus, in a way that opens up the Bayesian approach for a large readership. For the SIGEST version of the paper the author introduced more background material to make it more accessible to a general audience and extended the conclusion and outlook section, summarizing developments in the field that happened since the publication of the original work and discussing future research directions.
{"title":"SIGEST","authors":"The Editors","doi":"10.1137/23n975740","DOIUrl":"https://doi.org/10.1137/23n975740","url":null,"abstract":"SIAM Review, Volume 65, Issue 3, Page 829-829, August 2023. <br/> The SIGEST article in this issue, which comes from the SIAM/ASA Journal on Uncertainty Quantification, is “Bayesian Inverse Problems Are Usually Well-Posed,” by Jonas Latz. The author investigates the well-posedness of Bayesian approaches to inverse problems, generalizing the framework of well-posedness introduced by Andrew Stuart to a set of weaker assumptions. Well-posedness here is understood in the sense of Hadamard, that is, a solution exists, is unique, and continuously depends on the input data. Inverse problems are typically ill-posed due to properties of the model, a lack of data, and measurement noise. The Bayesian approach to inverse problems reformulates the quest for a solution to the inverse problem in terms of a quest for its posterior distribution, which is determined by the data likelihood and prior distribution of the solution, and which in contrast to the inverse problem itself should be well-posed. In the Bayesian context, well-posedness typically relates to existence, uniqueness, and Lipschitz continuity of the posterior distribution with respect to the data in the so-called Hellinger distance. In many practical applications such well-posedness is difficult, if not impossible, to verify. Moreover, the choice of the Hellinger distance as the right metric might not always be the best fitted depending on the problem at hand. This sets the starting point for the paper where the author introduces a new framework for well-posedness of Bayesian inverse problems in which he shows existence, uniqueness, and continuity with respect to various metrics for a large class of Bayesian inverse problems, with conditions that are either nonrestrictive or verifiable in practical settings. This paper gives a strong new mathematical foundation for Bayesian inverse problems. The underlying statistical and probabilistic concepts are explained comprehensively and comprehensibly and, thus, in a way that opens up the Bayesian approach for a large readership. For the SIGEST version of the paper the author introduced more background material to make it more accessible to a general audience and extended the conclusion and outlook section, summarizing developments in the field that happened since the publication of the original work and discussing future research directions.","PeriodicalId":49525,"journal":{"name":"SIAM Review","volume":"31 11","pages":""},"PeriodicalIF":10.2,"publicationDate":"2023-08-08","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"71509719","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":1,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
SIAM Review, Volume 65, Issue 3, Page 733-733, August 2023. <br/> The three articles in this issue's Research Spotlight section highlight the breadth of problems and approaches that have differential equations as a central component. In the first article, “Neural ODE Control for Classification, Approximation, and Transport,” authors Domènec Ruiz-Balet and Enrique Zuazua seek to expand understanding of some of the main properties of deep neural networks. To this end, the authors develop a dynamical control theoretical analysis of neural ordinary differential equations, a discretization of which is commonly known as a ResNet in machine learning. In this approach, time-dependent parameters are defined by piecewise-constant controls used to achieve targets associated with classification and regression tasks. A key aspect of the article's treatment is the reliance on an activation function characterization that only deforms one half space, leaving the other half space invariant; the rectified linear unit (ReLU) is a popular example of such an activation function. The authors derive constructive universal approximation results that can be used to understand how the complexity of the control depends on the target function's properties. Among other applications, these results are used to control a neural transport equation with the Wasserstein distance, common in optimal transport problems, measuring the approximation quality. Ruiz-Balet and Zuazua conclude with a number of open problems. Differential equation--based compartment models date back at least a century, when they were used to model the dynamics of malaria in a mixed population of humans and mosquitoes. Since then, compartment models have been used in areas far beyond epidemiology, typically with the simplifying assumption that each compartment is internally well mixed. As a consequence, all members in a compartment are treated the same, independent of how long they have resided in the compartment. In “Compartment Models with Memory,” authors Timothy Ginn and Lynn Schreyer expand the fields for which compartment models can provide insight by incorporating age in compartment in the underlying rate coefficients. This has the benefit of being able to account for a wide array of residence time distributions and comes at a cost of having to numerically solve a system of Volterra integral equations instead of a system of ordinary differential equations. The authors demonstrate and validate this approach on a number of examples and conclude by incorporating a delay in contagiousness of infected persons in a nonlinear SARS-CoV-2 transmission model. The authors also summarize several open questions based on this approach of allowing model parameters to be written as functions of age in compartment. “Does the Helmholtz Boundary Element Method Suffer from the Pollution Effect?” This is the question posed by (and the title of) the final Research Spotlights article in this issue. Authors Jeffrey Galkowski
{"title":"Research Spotlights","authors":"Stefan M. Wild","doi":"10.1137/23n975739","DOIUrl":"https://doi.org/10.1137/23n975739","url":null,"abstract":"SIAM Review, Volume 65, Issue 3, Page 733-733, August 2023. <br/> The three articles in this issue's Research Spotlight section highlight the breadth of problems and approaches that have differential equations as a central component. In the first article, “Neural ODE Control for Classification, Approximation, and Transport,” authors Domènec Ruiz-Balet and Enrique Zuazua seek to expand understanding of some of the main properties of deep neural networks. To this end, the authors develop a dynamical control theoretical analysis of neural ordinary differential equations, a discretization of which is commonly known as a ResNet in machine learning. In this approach, time-dependent parameters are defined by piecewise-constant controls used to achieve targets associated with classification and regression tasks. A key aspect of the article's treatment is the reliance on an activation function characterization that only deforms one half space, leaving the other half space invariant; the rectified linear unit (ReLU) is a popular example of such an activation function. The authors derive constructive universal approximation results that can be used to understand how the complexity of the control depends on the target function's properties. Among other applications, these results are used to control a neural transport equation with the Wasserstein distance, common in optimal transport problems, measuring the approximation quality. Ruiz-Balet and Zuazua conclude with a number of open problems. Differential equation--based compartment models date back at least a century, when they were used to model the dynamics of malaria in a mixed population of humans and mosquitoes. Since then, compartment models have been used in areas far beyond epidemiology, typically with the simplifying assumption that each compartment is internally well mixed. As a consequence, all members in a compartment are treated the same, independent of how long they have resided in the compartment. In “Compartment Models with Memory,” authors Timothy Ginn and Lynn Schreyer expand the fields for which compartment models can provide insight by incorporating age in compartment in the underlying rate coefficients. This has the benefit of being able to account for a wide array of residence time distributions and comes at a cost of having to numerically solve a system of Volterra integral equations instead of a system of ordinary differential equations. The authors demonstrate and validate this approach on a number of examples and conclude by incorporating a delay in contagiousness of infected persons in a nonlinear SARS-CoV-2 transmission model. The authors also summarize several open questions based on this approach of allowing model parameters to be written as functions of age in compartment. “Does the Helmholtz Boundary Element Method Suffer from the Pollution Effect?” This is the question posed by (and the title of) the final Research Spotlights article in this issue. Authors Jeffrey Galkowski ","PeriodicalId":49525,"journal":{"name":"SIAM Review","volume":"32 2","pages":""},"PeriodicalIF":10.2,"publicationDate":"2023-08-08","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"71509716","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":1,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
SIAM Review, Volume 65, Issue 3, Page 601-685, August 2023. Over the last 30 years a plethora of variational regularization models for image reconstruction have been proposed and thoroughly inspected by the applied mathematics community. Among them, the pioneering prototype often taught and learned in basic courses in mathematical image processing is the celebrated Rudin--Osher--Fatemi (ROF) model [L. I. Rudin, S. Osher, and E. Fatemi, Phys. D, 60 (1992), pp. 259--268], which relies on the minimization of the edge-preserving total variation (TV) seminorm as a regularization term. Despite its (often limiting) simplicity, this model is still very much employed in many applications and used as a benchmark for assessing the performance of modern learning-based image reconstruction approaches, thanks to its thorough analytical and numerical understanding. Among the many extensions to TV proposed over the years, a large class is based on the concept of space variance. Space-variant models can indeed overcome the intrinsic inability of TV to describe local features (strength, sharpness, directionality) by means of an adaptive mathematical modeling which accommodates local regularization weighting, variable smoothness, and anisotropy. Those ideas can further be cast in the flexible Bayesian framework of generalized Gaussian distributions and combined with maximum likelihood and hierarchical optimization approaches for efficient hyperparameter estimation. In this work, we review and connect the major contributions in the field of space-variant TV-type image reconstruction models, focusing, in particular, on their Bayesian interpretation which paves the way to new exciting and unexplored research directions.
{"title":"On and Beyond Total Variation Regularization in Imaging: The Role of Space Variance","authors":"Monica Pragliola, Luca Calatroni, Alessandro Lanza, Fiorella Sgallari","doi":"10.1137/21m1410683","DOIUrl":"https://doi.org/10.1137/21m1410683","url":null,"abstract":"SIAM Review, Volume 65, Issue 3, Page 601-685, August 2023. <br/> Over the last 30 years a plethora of variational regularization models for image reconstruction have been proposed and thoroughly inspected by the applied mathematics community. Among them, the pioneering prototype often taught and learned in basic courses in mathematical image processing is the celebrated Rudin--Osher--Fatemi (ROF) model [L. I. Rudin, S. Osher, and E. Fatemi, Phys. D, 60 (1992), pp. 259--268], which relies on the minimization of the edge-preserving total variation (TV) seminorm as a regularization term. Despite its (often limiting) simplicity, this model is still very much employed in many applications and used as a benchmark for assessing the performance of modern learning-based image reconstruction approaches, thanks to its thorough analytical and numerical understanding. Among the many extensions to TV proposed over the years, a large class is based on the concept of space variance. Space-variant models can indeed overcome the intrinsic inability of TV to describe local features (strength, sharpness, directionality) by means of an adaptive mathematical modeling which accommodates local regularization weighting, variable smoothness, and anisotropy. Those ideas can further be cast in the flexible Bayesian framework of generalized Gaussian distributions and combined with maximum likelihood and hierarchical optimization approaches for efficient hyperparameter estimation. In this work, we review and connect the major contributions in the field of space-variant TV-type image reconstruction models, focusing, in particular, on their Bayesian interpretation which paves the way to new exciting and unexplored research directions.","PeriodicalId":49525,"journal":{"name":"SIAM Review","volume":"32 5","pages":""},"PeriodicalIF":10.2,"publicationDate":"2023-08-08","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"71509713","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}
Christian Bick, Elizabeth Gross, Heather A. Harrington, Michael T. Schaub
SIAM Review, Volume 65, Issue 3, Page 686-731, August 2023. Network-based modeling of complex systems and data using the language of graphs has become an essential topic across a range of different disciplines. Arguably, this graph-based perspective derives its success from the relative simplicity of graphs: A graph consists of nothing more than a set of vertices and a set of edges, describing relationships between pairs of such vertices. This simple combinatorial structure makes graphs interpretable and flexible modeling tools. The simplicity of graphs as system models, however, has been scrutinized in the literature recently. Specifically, it has been argued from a variety of different angles that there is a need for higher-order networks, which go beyond the paradigm of modeling pairwise relationships, as encapsulated by graphs. In this survey article we take stock of these recent developments. Our goals are to clarify (i) what higher-order networks are, (ii) why these are interesting objects of study, and (iii) how they can be used in applications.
{"title":"What Are Higher-Order Networks?","authors":"Christian Bick, Elizabeth Gross, Heather A. Harrington, Michael T. Schaub","doi":"10.1137/21m1414024","DOIUrl":"https://doi.org/10.1137/21m1414024","url":null,"abstract":"SIAM Review, Volume 65, Issue 3, Page 686-731, August 2023. <br/> Network-based modeling of complex systems and data using the language of graphs has become an essential topic across a range of different disciplines. Arguably, this graph-based perspective derives its success from the relative simplicity of graphs: A graph consists of nothing more than a set of vertices and a set of edges, describing relationships between pairs of such vertices. This simple combinatorial structure makes graphs interpretable and flexible modeling tools. The simplicity of graphs as system models, however, has been scrutinized in the literature recently. Specifically, it has been argued from a variety of different angles that there is a need for higher-order networks, which go beyond the paradigm of modeling pairwise relationships, as encapsulated by graphs. In this survey article we take stock of these recent developments. Our goals are to clarify (i) what higher-order networks are, (ii) why these are interesting objects of study, and (iii) how they can be used in applications.","PeriodicalId":49525,"journal":{"name":"SIAM Review","volume":"32 4","pages":""},"PeriodicalIF":10.2,"publicationDate":"2023-08-08","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"71509714","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":1,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
SIAM Review, Volume 65, Issue 3, Page 867-867, August 2023. <br/> In this issue, the Education section presents two contributions. “The One-Dimensional Version of Peixoto's Structural Stability Theorem: A Calculus-Based Proof,” by Aminur Rahman and D. Blackmore, proposes, in the one-dimensional setting, a novel proof of Peixoto's structural stability and density theorem, which is fundamental in dynamical systems theory. In this framework the structural stability theorem says that a $C^1$ dynamical system $dot x =f(x)$ on $mathbb{S} ^1$ is structurally stable if and only if it has finitely many equilibrium points, all of which are hyperbolic. In the above $mathbb{S} ^1$ denotes the unit circle in $mathbb{R}^2$ and a point $x_star$ is called hyperbolic if $f'(x_star) neq 0$. The Peixto density result says that the set of all $C^1$ structurally stable systems on $mathbb{S}^1$ is open and dense in the space of all $C^1$ dynamical systems on $mathbb{S} ^1$ endowed with the $C^1$ norm. The original Peixoto's theorem is more complex and is valid for any smooth closed surface. Its proof, however, is long and not accessible using the tools available to advanced undergraduates, in contrast with the proposed one-dimensional proof, which an undergraduate could follow. This does not mean that the proof itself is elementary. Preliminaries recall all the basic definitions that are needed to successfully conduct the task. The style is rigorous and self-contained. The article also provides some historical comments, making the reading lively and encouraging further learning. The second paper, “Piecewise Smooth Models of Pumping a Child's Swing,” is presented by Brigid Murphy and Paul Glendinning. It concerns models of a child, in either a seated or standing position, swinging on a playground swing. In the article, which arose from the MSc dissertation by one of the authors, these models are analyzed using Lagrangian mechanics and may serve as an introduction to the different ways in which piecewise smooth systems do arise in modeling. The authors describe control strategies of swingers, and, in particular, whether it is possible for the swing to go through a full turn over its pivot. Piecewise smooth terms do naturally appear while discussing the strategies, and this future is analyzed in detail. Indeed the instantaneous reposition of the body of the swinger leads to a jump in the configuration of the swing. Numerical simulations are performed with a standard software package. These investigations would be suitable for undergraduate projects related to classical mechanics courses. At a higher degree level, projects could include further refinement of the existing methods and/or getting more accurate numerical solutions using available specialized software packages. The final section also discusses various related mathematical questions that would be interesting to investigate in this context and mentions other models involving jumps described using piecewise smoot
{"title":"Education","authors":"Hélène Frankowska","doi":"10.1137/23n975752","DOIUrl":"https://doi.org/10.1137/23n975752","url":null,"abstract":"SIAM Review, Volume 65, Issue 3, Page 867-867, August 2023. <br/> In this issue, the Education section presents two contributions. “The One-Dimensional Version of Peixoto's Structural Stability Theorem: A Calculus-Based Proof,” by Aminur Rahman and D. Blackmore, proposes, in the one-dimensional setting, a novel proof of Peixoto's structural stability and density theorem, which is fundamental in dynamical systems theory. In this framework the structural stability theorem says that a $C^1$ dynamical system $dot x =f(x)$ on $mathbb{S} ^1$ is structurally stable if and only if it has finitely many equilibrium points, all of which are hyperbolic. In the above $mathbb{S} ^1$ denotes the unit circle in $mathbb{R}^2$ and a point $x_star$ is called hyperbolic if $f'(x_star) neq 0$. The Peixto density result says that the set of all $C^1$ structurally stable systems on $mathbb{S}^1$ is open and dense in the space of all $C^1$ dynamical systems on $mathbb{S} ^1$ endowed with the $C^1$ norm. The original Peixoto's theorem is more complex and is valid for any smooth closed surface. Its proof, however, is long and not accessible using the tools available to advanced undergraduates, in contrast with the proposed one-dimensional proof, which an undergraduate could follow. This does not mean that the proof itself is elementary. Preliminaries recall all the basic definitions that are needed to successfully conduct the task. The style is rigorous and self-contained. The article also provides some historical comments, making the reading lively and encouraging further learning. The second paper, “Piecewise Smooth Models of Pumping a Child's Swing,” is presented by Brigid Murphy and Paul Glendinning. It concerns models of a child, in either a seated or standing position, swinging on a playground swing. In the article, which arose from the MSc dissertation by one of the authors, these models are analyzed using Lagrangian mechanics and may serve as an introduction to the different ways in which piecewise smooth systems do arise in modeling. The authors describe control strategies of swingers, and, in particular, whether it is possible for the swing to go through a full turn over its pivot. Piecewise smooth terms do naturally appear while discussing the strategies, and this future is analyzed in detail. Indeed the instantaneous reposition of the body of the swinger leads to a jump in the configuration of the swing. Numerical simulations are performed with a standard software package. These investigations would be suitable for undergraduate projects related to classical mechanics courses. At a higher degree level, projects could include further refinement of the existing methods and/or getting more accurate numerical solutions using available specialized software packages. The final section also discusses various related mathematical questions that would be interesting to investigate in this context and mentions other models involving jumps described using piecewise smoot","PeriodicalId":49525,"journal":{"name":"SIAM Review","volume":"31 9","pages":""},"PeriodicalIF":10.2,"publicationDate":"2023-08-08","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"71509721","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":1,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
SIAM Review, Volume 65, Issue 3, Page 831-865, August 2023. Inverse problems describe the task of blending a mathematical model with observational data---a fundamental task in many scientific and engineering disciplines. The solvability of such a task is usually classified through its well-posedness. A problem is well-posed if it has a unique solution that depends continuously on input or data. Inverse problems are usually ill-posed, but can sometimes be approached through a methodology that formulates a possibly well-posed problem. Usual methodologies are the variational and the Bayesian approach to inverse problems. For the Bayesian approach, Stuart [Acta Numer., 19 (2010), pp. 451--559] has given assumptions under which the posterior measure---the Bayesian inverse problem's solution---exists, is unique, and is Lipschitz continuous with respect to the Hellinger distance and, thus, well-posed. In this work, we simplify and generalize this concept: Indeed, we show well-posedness by proving existence, uniqueness, and continuity in Hellinger distance, Wasserstein distance, and total variation distance, and with respect to weak convergence, respectively, under significantly weaker assumptions. An immense class of practically relevant Bayesian inverse problems satisfies those conditions. The conditions can often be verified without analyzing the underlying mathematical model---the model can be treated as a black box.
{"title":"Bayesian Inverse Problems Are Usually Well-Posed","authors":"Jonas Latz","doi":"10.1137/23m1556435","DOIUrl":"https://doi.org/10.1137/23m1556435","url":null,"abstract":"SIAM Review, Volume 65, Issue 3, Page 831-865, August 2023. <br/> Inverse problems describe the task of blending a mathematical model with observational data---a fundamental task in many scientific and engineering disciplines. The solvability of such a task is usually classified through its well-posedness. A problem is well-posed if it has a unique solution that depends continuously on input or data. Inverse problems are usually ill-posed, but can sometimes be approached through a methodology that formulates a possibly well-posed problem. Usual methodologies are the variational and the Bayesian approach to inverse problems. For the Bayesian approach, Stuart [Acta Numer., 19 (2010), pp. 451--559] has given assumptions under which the posterior measure---the Bayesian inverse problem's solution---exists, is unique, and is Lipschitz continuous with respect to the Hellinger distance and, thus, well-posed. In this work, we simplify and generalize this concept: Indeed, we show well-posedness by proving existence, uniqueness, and continuity in Hellinger distance, Wasserstein distance, and total variation distance, and with respect to weak convergence, respectively, under significantly weaker assumptions. An immense class of practically relevant Bayesian inverse problems satisfies those conditions. The conditions can often be verified without analyzing the underlying mathematical model---the model can be treated as a black box.","PeriodicalId":49525,"journal":{"name":"SIAM Review","volume":"31 10","pages":""},"PeriodicalIF":10.2,"publicationDate":"2023-08-08","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"71509720","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 Comprehensive Proof of Bertrand's Theorem","authors":"P. Leenheer, Jack W. Musgrove, Tyler Schimleck","doi":"10.1137/21m1436658","DOIUrl":"https://doi.org/10.1137/21m1436658","url":null,"abstract":"","PeriodicalId":49525,"journal":{"name":"SIAM Review","volume":"21 1","pages":"563-588"},"PeriodicalIF":10.2,"publicationDate":"2023-05-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"73066658","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}
The Education section in this issue presents two contributions. In `"Nesterov's Method for Convex Optimization," Noel J. Walkington proposes a teaching guide for a first course in optimization of this well-known algorithm for computing the minimum of a convex function. This algorithm, first proposed in 1983 by Yuri Nesterov, though well recognized in computational optimization in the presence of large data as a more efficient tool than the steepest descent method, is still absent in most modern textbooks on optimization. The author of the present article develops an elementary analysis of Nesterov's first order algorithm that parallels that of steepest descent but with an additional requirement proposed by Nesterov. Two cases are discussed. The first concerns an unconstrained minimization problem, while the second includes closed convex constraints represented using infinite penalization of the cost. More generally, the cost function becomes the sum of a smooth convex function and a lower semicontinuous convex function. Several student-level exercises are included in this paper. Results are nicely illustrated by an example of a signal recovery problem and a discussion of the Uzawa algorithm for optimization problems with constraints defined by inequalities involving convex functions. The second paper, "A Comprehensive Proof of Bertrand's Theorem," is presented by Patrick De Leenheer, John Musgrove, and Tyler Schimleck. It concerns the behavior of the solutions of the classical two-body problem and states that, among all possible gravitational laws, there are only two exhibiting the property that all bounded orbits are closed: Newtonian gravitation and Hookean gravitation. Historically, even if Newton was aware that there are to specific gravitational laws having the above property, it was only two centuries later, in 1873, that Bertrand realized that these are the only ones. Bertrand's theorem, due to its important consequences, has been integrated into the undergraduate curriculum in theoretical mechanics, but its proof, accessible to undergraduate mathematics or physics students, seems to be absent from modern textbooks. Although Bertrand's original paper did not contain a precise proof, V. Arnold proposed a sketch of it based on six subproblems. Among other contributions, this article provides a complete proof of the sixth subproblem under a specific assumption imposed on the magnitude of the force in the motion model. Under this assumption, a complete proof of Bertrand's theorem is then given, incorporating also earlier contributions by other authors. Still, comprehensive does not mean simple here, and this paper may be used to conceive several research projects for advanced-level undergraduate students in mathematics or physics.
本期的“教育”部分有两篇文章。在“Nesterov的凸优化方法”中，Noel J. Walkington提出了一个关于这个著名算法优化的第一门课程的教学指南，用于计算凸函数的最小值。该算法由Yuri Nesterov于1983年首次提出，虽然在大数据存在的计算优化中被公认为比最陡下降法更有效的工具，但在大多数现代优化教科书中仍然没有。本文的作者对Nesterov的一阶算法进行了初步分析，该算法与最陡下降算法相似，但带有Nesterov提出的附加要求。讨论了两个案例。第一个问题涉及无约束最小化问题，而第二个问题包括使用无限代价惩罚表示的闭合凸约束。更一般地说，代价函数变成光滑凸函数和下半连续凸函数的和。本文包含了几个学生水平的练习。通过一个信号恢复问题的例子和讨论Uzawa算法对包含凸函数的不等式定义约束的优化问题的结果很好地说明了这一点。第二篇论文，“伯特兰定理的全面证明”，由Patrick De Leenheer, John Musgrove和Tyler Schimleck提出。它关注经典二体问题解的行为，并指出，在所有可能的引力定律中，只有两个定律表现出所有有界轨道都闭合的性质:牛顿引力和胡克引力。从历史上看，即使牛顿意识到有特定的万有引力定律具有上述性质，直到两个世纪后的1873年，伯特兰才意识到这些定律是唯一的。由于其重要的结果，伯特兰定理已经被纳入了理论力学的本科课程，但是它的证明，对于数学或物理专业的本科生来说，似乎没有在现代教科书中出现。尽管Bertrand的原始论文没有包含精确的证明，V. Arnold还是提出了一个基于六个子问题的草图。在其他贡献中，本文提供了在运动模型中对力的大小施加的特定假设下的第六子问题的完整证明。在此假设下，结合其他作者的早期贡献，给出了伯特兰定理的完整证明。然而，全面并不意味着简单，本文可以用来设想几个研究项目的高等水平的本科生数学或物理。
{"title":"Education","authors":"Hélène Frankowska","doi":"10.1137/23n975703","DOIUrl":"https://doi.org/10.1137/23n975703","url":null,"abstract":"The Education section in this issue presents two contributions. In `\"Nesterov's Method for Convex Optimization,\" Noel J. Walkington proposes a teaching guide for a first course in optimization of this well-known algorithm for computing the minimum of a convex function. This algorithm, first proposed in 1983 by Yuri Nesterov, though well recognized in computational optimization in the presence of large data as a more efficient tool than the steepest descent method, is still absent in most modern textbooks on optimization. The author of the present article develops an elementary analysis of Nesterov's first order algorithm that parallels that of steepest descent but with an additional requirement proposed by Nesterov. Two cases are discussed. The first concerns an unconstrained minimization problem, while the second includes closed convex constraints represented using infinite penalization of the cost. More generally, the cost function becomes the sum of a smooth convex function and a lower semicontinuous convex function. Several student-level exercises are included in this paper. Results are nicely illustrated by an example of a signal recovery problem and a discussion of the Uzawa algorithm for optimization problems with constraints defined by inequalities involving convex functions. The second paper, \"A Comprehensive Proof of Bertrand's Theorem,\" is presented by Patrick De Leenheer, John Musgrove, and Tyler Schimleck. It concerns the behavior of the solutions of the classical two-body problem and states that, among all possible gravitational laws, there are only two exhibiting the property that all bounded orbits are closed: Newtonian gravitation and Hookean gravitation. Historically, even if Newton was aware that there are to specific gravitational laws having the above property, it was only two centuries later, in 1873, that Bertrand realized that these are the only ones. Bertrand's theorem, due to its important consequences, has been integrated into the undergraduate curriculum in theoretical mechanics, but its proof, accessible to undergraduate mathematics or physics students, seems to be absent from modern textbooks. Although Bertrand's original paper did not contain a precise proof, V. Arnold proposed a sketch of it based on six subproblems. Among other contributions, this article provides a complete proof of the sixth subproblem under a specific assumption imposed on the magnitude of the force in the motion model. Under this assumption, a complete proof of Bertrand's theorem is then given, incorporating also earlier contributions by other authors. Still, comprehensive does not mean simple here, and this paper may be used to conceive several research projects for advanced-level undergraduate students in mathematics or physics.","PeriodicalId":49525,"journal":{"name":"SIAM Review","volume":"16 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2023-05-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"136338577","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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