SIAM Review, Volume 67, Issue 3, Page 543-575, August 2025. Abstract.This paper analyzes hierarchical Bayesian inverse problems using techniques from high-dimensional statistics. Our analysis leverages a property of hierarchical Bayesian regularizers that we call approximate decomposability to obtain nonasymptotic bounds on the reconstruction error attained by maximum a posteriori estimators. The new theory explains how hierarchical Bayesian models that exploit sparsity, group sparsity, and sparse representations of the unknown parameter can achieve accurate reconstructions in high-dimensional settings.
{"title":"Hierarchical Bayesian Inverse Problems: A High-Dimensional Statistics Viewpoint","authors":"Daniel Sanz-Alonso, Nathan Waniorek","doi":"10.1137/24m1629328","DOIUrl":"https://doi.org/10.1137/24m1629328","url":null,"abstract":"SIAM Review, Volume 67, Issue 3, Page 543-575, August 2025. <br/> Abstract.This paper analyzes hierarchical Bayesian inverse problems using techniques from high-dimensional statistics. Our analysis leverages a property of hierarchical Bayesian regularizers that we call approximate decomposability to obtain nonasymptotic bounds on the reconstruction error attained by maximum a posteriori estimators. The new theory explains how hierarchical Bayesian models that exploit sparsity, group sparsity, and sparse representations of the unknown parameter can achieve accurate reconstructions in high-dimensional settings.","PeriodicalId":49525,"journal":{"name":"SIAM Review","volume":"27 1","pages":""},"PeriodicalIF":10.2,"publicationDate":"2025-08-07","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"144792658","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 3, Page 579-604, August 2025. Abstract.Diffusive flows, and their discrete counterparts, are ubiquitous in the physical and engineering sciences. In many important examples, the total mass is not preserved and therefore standard probabilistic models are not suitable. Examples include electrons which may be absorbed by the medium in which they travel. In population genetics, some individuals may “disappear” due to their genotype. In traffic flows over a network, some vehicles might simply exit the circulation and park. In this more general situation, where some of the mass may be lost, it is of particular interest to reconcile the observed initial and final marginal distributions with a given prior. In the case when the two marginals are probability distributions, and thus of equal mass, this problem was posed and, to a considerable extent, solved by E. Schrödinger in 1931/32. It is now known as the Schrödinger Bridge Problem (SBP). It turns out that Schrödinger’s problem can be viewed as both a modeling and a control problem. Due to the fundamental significance of this problem, interest in the SBP and in its deterministic (zero-noise limit) counterpart of optimal mass transport (OMT) has in recent years enticed scientists from a broad spectrum of disciplines, including physics, stochastic control, computer science, probability theory, and geometry. Yet, while the mathematics and applications of SBP/OMT have been developing at a considerable pace, accounting for marginals of unequal mass has received scant attention. The problem of interpolating between “unbalanced” marginals has been approached by introducing source/sink terms into the transport equations in an ad hoc manner, chiefly driven by applications in image registration. Nevertheless, as hinted at above, losses are inherent in many physical processes and, thereby, models that account for lossy transport may also need to be reconciled with observed marginals following Schrödinger’s quest, that is, to adjust the probability of trajectories of particles, including those that do not make it to the terminal observation point, so that the updated evolution represents the most likely way that particles may have been transported, or vanished, at some intermediate point. Thus, the purpose of this work is to develop such a natural generalization of the SBP for diffusive evolution with losses, whereupon particles are “killed” (jump into a coffin/extinction state) according to a probabilistic law, and thereby mass is gradually lost along their stochastically driven flow. Through a suitable embedding, which appears to be novel, we turn the problem into an SBP for stochastic processes that combine diffusive and jump characteristics. Then, following a large-deviations formalism in the style of E. Schrödinger, given a prior law that allows for losses, we ask for the most probable evolution of particles along with the most likely killing rate as the particles transition
{"title":"Optimal Survival Strategies for Diffusive Flows: A Schrödinger Bridge Approach to Unbalanced Transport","authors":"Yongxin Chen, Tryphon T. Georgiou, Michele Pavon","doi":"10.1137/25m176581x","DOIUrl":"https://doi.org/10.1137/25m176581x","url":null,"abstract":"SIAM Review, Volume 67, Issue 3, Page 579-604, August 2025. <br/> Abstract.Diffusive flows, and their discrete counterparts, are ubiquitous in the physical and engineering sciences. In many important examples, the total mass is not preserved and therefore standard probabilistic models are not suitable. Examples include electrons which may be absorbed by the medium in which they travel. In population genetics, some individuals may “disappear” due to their genotype. In traffic flows over a network, some vehicles might simply exit the circulation and park. In this more general situation, where some of the mass may be lost, it is of particular interest to reconcile the observed initial and final marginal distributions with a given prior. In the case when the two marginals are probability distributions, and thus of equal mass, this problem was posed and, to a considerable extent, solved by E. Schrödinger in 1931/32. It is now known as the Schrödinger Bridge Problem (SBP). It turns out that Schrödinger’s problem can be viewed as both a modeling and a control problem. Due to the fundamental significance of this problem, interest in the SBP and in its deterministic (zero-noise limit) counterpart of optimal mass transport (OMT) has in recent years enticed scientists from a broad spectrum of disciplines, including physics, stochastic control, computer science, probability theory, and geometry. Yet, while the mathematics and applications of SBP/OMT have been developing at a considerable pace, accounting for marginals of unequal mass has received scant attention. The problem of interpolating between “unbalanced” marginals has been approached by introducing source/sink terms into the transport equations in an ad hoc manner, chiefly driven by applications in image registration. Nevertheless, as hinted at above, losses are inherent in many physical processes and, thereby, models that account for lossy transport may also need to be reconciled with observed marginals following Schrödinger’s quest, that is, to adjust the probability of trajectories of particles, including those that do not make it to the terminal observation point, so that the updated evolution represents the most likely way that particles may have been transported, or vanished, at some intermediate point. Thus, the purpose of this work is to develop such a natural generalization of the SBP for diffusive evolution with losses, whereupon particles are “killed” (jump into a coffin/extinction state) according to a probabilistic law, and thereby mass is gradually lost along their stochastically driven flow. Through a suitable embedding, which appears to be novel, we turn the problem into an SBP for stochastic processes that combine diffusive and jump characteristics. Then, following a large-deviations formalism in the style of E. Schrödinger, given a prior law that allows for losses, we ask for the most probable evolution of particles along with the most likely killing rate as the particles transition","PeriodicalId":49525,"journal":{"name":"SIAM Review","volume":"95 1","pages":""},"PeriodicalIF":10.2,"publicationDate":"2025-08-07","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"144792663","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 3, Page 650-651, August 2025. Inverse Optimal Control and Inverse Noncooperative Dynamic Game Theory is a detailed exploration of inverse problems in control theory, particularly suited to researchers and practitioners working on multiagent systems, robotics, and economics. The book systematically builds up the theory and techniques necessary for recovering cost functions, which can explain observed behavior in both individual agents and systems of agents operating with competing objectives. The authors provide a deep dive into the mathematical foundations, addressing both discrete and continuous systems in optimal control and dynamic game theory.
{"title":"Book Review:; Inverse Optimal Control and Inverse Noncooperative Dynamic Game Theory: A Minimum-Principle Approach","authors":"Sebastián Zamorano Aliaga","doi":"10.1137/24m1635120","DOIUrl":"https://doi.org/10.1137/24m1635120","url":null,"abstract":"SIAM Review, Volume 67, Issue 3, Page 650-651, August 2025. <br/> Inverse Optimal Control and Inverse Noncooperative Dynamic Game Theory is a detailed exploration of inverse problems in control theory, particularly suited to researchers and practitioners working on multiagent systems, robotics, and economics. The book systematically builds up the theory and techniques necessary for recovering cost functions, which can explain observed behavior in both individual agents and systems of agents operating with competing objectives. The authors provide a deep dive into the mathematical foundations, addressing both discrete and continuous systems in optimal control and dynamic game theory.","PeriodicalId":49525,"journal":{"name":"SIAM Review","volume":"21 1","pages":""},"PeriodicalIF":10.2,"publicationDate":"2025-08-07","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"144792662","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 3, Page 651-654, August 2025. In this book the authors present the variational analysis via [math]-convergence of functionals defined on functions [math], where [math] is a small parameter finally tending to zero, [math] is a so-called lattice (typically [math]), and [math] is a finite state-space. The functionals (often called energies due to applications in physics) are of many different types, but share the common feature that when the lattice spacing [math] tends to zero, functions [math] with bounded energy (or suitable transformations) give rise to a finite partition.
{"title":"Book Review:; Discrete Variational Problems with Interfaces","authors":"Matthias Ruf","doi":"10.1137/24m1677964","DOIUrl":"https://doi.org/10.1137/24m1677964","url":null,"abstract":"SIAM Review, Volume 67, Issue 3, Page 651-654, August 2025. <br/> In this book the authors present the variational analysis via [math]-convergence of functionals defined on functions [math], where [math] is a small parameter finally tending to zero, [math] is a so-called lattice (typically [math]), and [math] is a finite state-space. The functionals (often called energies due to applications in physics) are of many different types, but share the common feature that when the lattice spacing [math] tends to zero, functions [math] with bounded energy (or suitable transformations) give rise to a finite partition.","PeriodicalId":49525,"journal":{"name":"SIAM Review","volume":"1 1","pages":""},"PeriodicalIF":10.2,"publicationDate":"2025-08-07","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"144792666","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 2, Page 408-411, May 2025. Optimal transport was originally invented by Gaspard Monge [“Mémoire sur la théorie des déblais et des remblais,” Mem. Math. Phys. Acad. Royale Sci., (1781), pp. 666–704] to model the problem of optimally mapping one distribution of mass onto another. This was later reformulated by Leonid Kantorovich as a well-posed linear program using the notion of transport plans instead of maps in [“On the translocation of masses,” Dokl. Akad. Nauk. USSR (N.S.), 37 (1942), pp. 199–201], which earned him the Nobel Memorial Prize in Economic Sciences. In the past four decades the field of optimal transport has grown far beyond its original purpose and has evolved into a driving force for applications both within mathematics and in other sciences. This book review deals with the new monograph Optimal Mass Transport on Euclidean Spaces by Francesco Maggi.
SIAM评论，第67卷，第2期，第408-411页，2025年5月。最优运输最初是由加斯帕德·蒙格（Gaspard Monge）发明的，他曾说过：“msammoire sur la thsamorie des dsamblais et des remblais”。数学。理论物理。皇家科学院院士（1781），第666-704页)，以模拟最佳映射一个质量分布到另一个的问题。这后来被列昂尼德·坎托罗维奇（Leonid Kantorovich）重新表述为一个完备的线性规划，使用交通计划的概念而不是地图[On the translocation of mass， Dokl]。Akad。研究。苏联（N.S.）， 37(1942)，第199-201页)，他因此获得了诺贝尔经济学奖。在过去的四十年中，最优运输领域的发展远远超出了其最初的目的，并已发展成为数学和其他科学应用的推动力。这篇书评是关于弗朗西斯科·马吉在欧几里得空间上的最优质量输运的新专著。
{"title":"Book Review:; Optimal Mass Transport on Euclidean Spaces","authors":"Leon Bungert","doi":"10.1137/24m1637854","DOIUrl":"https://doi.org/10.1137/24m1637854","url":null,"abstract":"SIAM Review, Volume 67, Issue 2, Page 408-411, May 2025. <br/> Optimal transport was originally invented by Gaspard Monge [“Mémoire sur la théorie des déblais et des remblais,” Mem. Math. Phys. Acad. Royale Sci., (1781), pp. 666–704] to model the problem of optimally mapping one distribution of mass onto another. This was later reformulated by Leonid Kantorovich as a well-posed linear program using the notion of transport plans instead of maps in [“On the translocation of masses,” Dokl. Akad. Nauk. USSR (N.S.), 37 (1942), pp. 199–201], which earned him the Nobel Memorial Prize in Economic Sciences. In the past four decades the field of optimal transport has grown far beyond its original purpose and has evolved into a driving force for applications both within mathematics and in other sciences. This book review deals with the new monograph Optimal Mass Transport on Euclidean Spaces by Francesco Maggi.","PeriodicalId":49525,"journal":{"name":"SIAM Review","volume":"228 1","pages":""},"PeriodicalIF":10.2,"publicationDate":"2025-05-08","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"143920675","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 2, Page 411-411, May 2025. A short discussion on stochastic calculus is given under the assumption that the fundamentals of probability theory are known to readers. Some related basic details on probability theory should have been included to make the book more self-contained. Further, analytic solutions of some stochastic differential equations (SDEs), which are used in modeling real-life events, are given. However, author should have included well-posedness under the general assumptions and then should have either discussed these SDEs as a special case or provided an explanation for the necessity of dealing with such equations separately.
{"title":"Book Review:; Stochastic Integral and Differential Equations in Mathematical Modelling","authors":"Chaman Kumar","doi":"10.1137/24m163548x","DOIUrl":"https://doi.org/10.1137/24m163548x","url":null,"abstract":"SIAM Review, Volume 67, Issue 2, Page 411-411, May 2025. <br/> A short discussion on stochastic calculus is given under the assumption that the fundamentals of probability theory are known to readers. Some related basic details on probability theory should have been included to make the book more self-contained. Further, analytic solutions of some stochastic differential equations (SDEs), which are used in modeling real-life events, are given. However, author should have included well-posedness under the general assumptions and then should have either discussed these SDEs as a special case or provided an explanation for the necessity of dealing with such equations separately.","PeriodicalId":49525,"journal":{"name":"SIAM Review","volume":"16 1","pages":""},"PeriodicalIF":10.2,"publicationDate":"2025-05-08","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"143920676","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 2, Page 375-398, May 2025. Abstract.We present and discuss the Streeter–Phelps equations, which were the first river quality model. If the parameters are constants, then the model in its linear formulation can be solved explicitly. This reveals, however, that depending on parameters and initial data, the model might predict negative oxygen concentrations, which marks a breakdown of the model. To address this shortcoming, we introduce a nonlinear modification which, in the case of constant parameters, we can study in the phase plane. In real-world applications, parameters are never constant and are usually known not exactly, but instead with some uncertainty. We show how we can use the solutions for the constant parameter case to obtain estimates for the unknown solutions from estimates of the model parameters, using differential inequalities.
{"title":"Uncertainty Analysis of a Simple River Quality Model Using Differential Inequalities","authors":"Grace D’Agostino, Hermann J. Eberl","doi":"10.1137/23m1616406","DOIUrl":"https://doi.org/10.1137/23m1616406","url":null,"abstract":"SIAM Review, Volume 67, Issue 2, Page 375-398, May 2025. <br/> Abstract.We present and discuss the Streeter–Phelps equations, which were the first river quality model. If the parameters are constants, then the model in its linear formulation can be solved explicitly. This reveals, however, that depending on parameters and initial data, the model might predict negative oxygen concentrations, which marks a breakdown of the model. To address this shortcoming, we introduce a nonlinear modification which, in the case of constant parameters, we can study in the phase plane. In real-world applications, parameters are never constant and are usually known not exactly, but instead with some uncertainty. We show how we can use the solutions for the constant parameter case to obtain estimates for the unknown solutions from estimates of the model parameters, using differential inequalities.","PeriodicalId":49525,"journal":{"name":"SIAM Review","volume":"23 1","pages":""},"PeriodicalIF":10.2,"publicationDate":"2025-05-08","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"143920680","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 2, Page 401-403, May 2025. It’s 7.30 am when my alarm wakes me up and I am greeted by my notifications. While eating breakfast, I watch videos YouTube recommends to me: sometimes news stories, sometimes my guilty pleasure of a new “Say Yes to the Dress” clip. On my way to campus, I play my daylist, a curated playlist from Spotify based on what I normally listen to on a given weekday and time. Apparently, as I write this, “Nostalgia 2010s Tuesday Afternoon” is waiting for me. In the classroom, I teach students how to load in data, visualize it, and run a regression.
{"title":"Featured Review:; How Data Happened: A History from the Age of Reason to the Age of Algorithms","authors":"Rachel Roca","doi":"10.1137/24m1635521","DOIUrl":"https://doi.org/10.1137/24m1635521","url":null,"abstract":"SIAM Review, Volume 67, Issue 2, Page 401-403, May 2025. <br/> It’s 7.30 am when my alarm wakes me up and I am greeted by my notifications. While eating breakfast, I watch videos YouTube recommends to me: sometimes news stories, sometimes my guilty pleasure of a new “Say Yes to the Dress” clip. On my way to campus, I play my daylist, a curated playlist from Spotify based on what I normally listen to on a given weekday and time. Apparently, as I write this, “Nostalgia 2010s Tuesday Afternoon” is waiting for me. In the classroom, I teach students how to load in data, visualize it, and run a regression.","PeriodicalId":49525,"journal":{"name":"SIAM Review","volume":"287 1","pages":""},"PeriodicalIF":10.2,"publicationDate":"2025-05-08","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"143920666","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 2, Page 404-405, May 2025. “Math is like a drag queen: marvelous, whimsical, at times even controversial, but never boring!” That it how the preface of Math in Drag begins. It is also an excellent description of the book. Math in Drag was authored by Kyne Santos, who often goes by Kyne. Kyne studied mathematics at the University of Waterloo and went viral teaching math on TikTok. Indeed, over a million people have flocked to Kyne’s @onlinekyne account for camp explanations of quadratic equations and square roots. Kyne is also a drag queen and competed in the first season of Canada’s Drag Race.
{"title":"Book Review:; Math in Drag","authors":"Laura W. Layton","doi":"10.1137/24m1668767","DOIUrl":"https://doi.org/10.1137/24m1668767","url":null,"abstract":"SIAM Review, Volume 67, Issue 2, Page 404-405, May 2025. <br/> “Math is like a drag queen: marvelous, whimsical, at times even controversial, but never boring!” That it how the preface of Math in Drag begins. It is also an excellent description of the book. Math in Drag was authored by Kyne Santos, who often goes by Kyne. Kyne studied mathematics at the University of Waterloo and went viral teaching math on TikTok. Indeed, over a million people have flocked to Kyne’s @onlinekyne account for camp explanations of quadratic equations and square roots. Kyne is also a drag queen and competed in the first season of Canada’s Drag Race.","PeriodicalId":49525,"journal":{"name":"SIAM Review","volume":"64 1","pages":""},"PeriodicalIF":10.2,"publicationDate":"2025-05-08","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"143920679","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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