SIAM Review, Volume 67, Issue 3, Page 415-539, August 2025. Abstract.We provide a tutorial-style review of stochastic dual dynamic programming (SDDP), one of the state-of-the-art solution methods for large-scale multistage stochastic programs. Since it was introduced about 30 years ago for solving large-scale multistage stochastic linear programming problems in energy planning, SDDP has been applied to practical problems from several fields and has been enriched by various improvements and enhancements to address broader problem classes. We begin with a detailed introduction to SDDP, with special focus on its motivation, complexity, and required assumptions. Then, we present and discuss in depth the existing enhancements as well as current research trends that allow for the alleviation of those assumptions.
{"title":"Stochastic Dual Dynamic Programming and Its Variants: A Review","authors":"Christian Füllner, Steffen Rebennack","doi":"10.1137/23m1575093","DOIUrl":"https://doi.org/10.1137/23m1575093","url":null,"abstract":"SIAM Review, Volume 67, Issue 3, Page 415-539, August 2025. <br/> Abstract.We provide a tutorial-style review of stochastic dual dynamic programming (SDDP), one of the state-of-the-art solution methods for large-scale multistage stochastic programs. Since it was introduced about 30 years ago for solving large-scale multistage stochastic linear programming problems in energy planning, SDDP has been applied to practical problems from several fields and has been enriched by various improvements and enhancements to address broader problem classes. We begin with a detailed introduction to SDDP, with special focus on its motivation, complexity, and required assumptions. Then, we present and discuss in depth the existing enhancements as well as current research trends that allow for the alleviation of those assumptions.","PeriodicalId":49525,"journal":{"name":"SIAM Review","volume":"16 1","pages":""},"PeriodicalIF":10.2,"publicationDate":"2025-08-07","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"144797304","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 656-658, August 2025. In insurance mathematics and actuarial sciences, modeling the dynamics of insured events is a pivotal challenge that demands advanced and sophisticated techniques due to the growing complexity of insurance markets. This complexity, coupled with the exponential growth in data availability in recent years, has acted as a catalyst for the adoption of datacentric approaches in forecasting random phenomena.
{"title":"Book Review:; Statistical Foundations of Actuarial Learning and Its Applications","authors":"Olivier Menoukeu-Pamen","doi":"10.1137/24m1651575","DOIUrl":"https://doi.org/10.1137/24m1651575","url":null,"abstract":"SIAM Review, Volume 67, Issue 3, Page 656-658, August 2025. <br/> In insurance mathematics and actuarial sciences, modeling the dynamics of insured events is a pivotal challenge that demands advanced and sophisticated techniques due to the growing complexity of insurance markets. This complexity, coupled with the exponential growth in data availability in recent years, has acted as a catalyst for the adoption of datacentric approaches in forecasting random phenomena.","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":"144792665","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 624-641, August 2025. Abstract.We guide the reader on a journey through mathematical modeling and numerical analysis, emphasizing the crucial interplay of both disciplines. Targeting undergraduate students with basic knowledge of dynamical systems and numerical methods for ordinary differential equations, we explore a model from mathematical biology where numerical methods fail badly due to catastrophic floating point errors. We analyze the reasons for this behavior by studying the steady states of the model and use the theory of invariants to develop an alternative model suited for numerical simulations. Our story is intended to motivate the combining of analytical knowledge and numerical knowledge, even in those cases where the world looks fine at first sight. We have set up an online repository containing an interactive notebook with all the numerical experiments in this article to make this study fully reproducible and useful for classroom teaching.
{"title":"Modeling Still Matters: A Surprising Instance of Catastrophic Floating Point Errors in Mathematical Biology and Numerical Methods for ODEs","authors":"Cordula Reisch, Hendrik Ranocha","doi":"10.1137/23m1563967","DOIUrl":"https://doi.org/10.1137/23m1563967","url":null,"abstract":"SIAM Review, Volume 67, Issue 3, Page 624-641, August 2025. <br/> Abstract.We guide the reader on a journey through mathematical modeling and numerical analysis, emphasizing the crucial interplay of both disciplines. Targeting undergraduate students with basic knowledge of dynamical systems and numerical methods for ordinary differential equations, we explore a model from mathematical biology where numerical methods fail badly due to catastrophic floating point errors. We analyze the reasons for this behavior by studying the steady states of the model and use the theory of invariants to develop an alternative model suited for numerical simulations. Our story is intended to motivate the combining of analytical knowledge and numerical knowledge, even in those cases where the world looks fine at first sight. We have set up an online repository containing an interactive notebook with all the numerical experiments in this article to make this study fully reproducible and useful for classroom teaching.","PeriodicalId":49525,"journal":{"name":"SIAM Review","volume":"11 1","pages":""},"PeriodicalIF":10.2,"publicationDate":"2025-08-07","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"144792657","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 655-656, August 2025. Lattice dynamical systems (LDS) are infinite-dimensional systems of ordinary differential equations (ODEs). They can be formulated as ODEs on a Banach space of bi-infinite sequences. They may arise in various ways: some are obtained as discretizations of partial differential equations or integral equations, and others are infinite-dimensional counterparts of finite-dimensional ODE models such as the Hopfield neural network model. This book studies various kinds of LDS that might be of autonomous, nonautonomous, or random nature. It focuses on first showing that the underlying LDS induces an autonomous, nonautonomous, or random semidynamical system, then providing sufficient criteria for the existence of a global, pullback, or random attractor.
{"title":"Book Review:; Dissipative Lattice Dynamical Systems","authors":"Ábel Garab","doi":"10.1137/24m1675606","DOIUrl":"https://doi.org/10.1137/24m1675606","url":null,"abstract":"SIAM Review, Volume 67, Issue 3, Page 655-656, August 2025. <br/> Lattice dynamical systems (LDS) are infinite-dimensional systems of ordinary differential equations (ODEs). They can be formulated as ODEs on a Banach space of bi-infinite sequences. They may arise in various ways: some are obtained as discretizations of partial differential equations or integral equations, and others are infinite-dimensional counterparts of finite-dimensional ODE models such as the Hopfield neural network model. This book studies various kinds of LDS that might be of autonomous, nonautonomous, or random nature. It focuses on first showing that the underlying LDS induces an autonomous, nonautonomous, or random semidynamical system, then providing sufficient criteria for the existence of a global, pullback, or random attractor.","PeriodicalId":49525,"journal":{"name":"SIAM Review","volume":"69 1","pages":""},"PeriodicalIF":10.2,"publicationDate":"2025-08-07","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"144792661","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 645-650, August 2025. Democracy: What’s math got to do with it? If you were to ask someone who has not previously studied this topic, you’d likely receive the obvious and simple answer: We count votes, and whoever gets the most votes wins. Okay. If you find someone who follows politics, you might get another answer: We need statistics to take good polls and make predictions. A little more satisfying, and true, but you could say that this type of analysis is more commentary on who is winning and losing in the political process rather than an analysis of the process itself. In the recent book Making Democracy Count: How Mathematics Improves Voting, Electoral Maps, and Representation, Ismar Volić gives another answer: Math has everything to do with democracy.
{"title":"Featured Review:; Making Democracy Count: How Mathematics Improves Voting, Electoral Maps, and Representation","authors":"Beth Malmskog","doi":"10.1137/24m1675655","DOIUrl":"https://doi.org/10.1137/24m1675655","url":null,"abstract":"SIAM Review, Volume 67, Issue 3, Page 645-650, August 2025. <br/> Democracy: What’s math got to do with it? If you were to ask someone who has not previously studied this topic, you’d likely receive the obvious and simple answer: We count votes, and whoever gets the most votes wins. Okay. If you find someone who follows politics, you might get another answer: We need statistics to take good polls and make predictions. A little more satisfying, and true, but you could say that this type of analysis is more commentary on who is winning and losing in the political process rather than an analysis of the process itself. In the recent book Making Democracy Count: How Mathematics Improves Voting, Electoral Maps, and Representation, Ismar Volić gives another answer: Math has everything to do with democracy.","PeriodicalId":49525,"journal":{"name":"SIAM Review","volume":"32 1","pages":""},"PeriodicalIF":10.2,"publicationDate":"2025-08-07","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"144792664","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}
Catherine Higham, Desmond J. Higham, Peter Grindrod
SIAM Review, Volume 67, Issue 3, Page 607-623, August 2025. Abstract.Generative artificial intelligence (GAI) refers to algorithms that create synthetic but realistic output. Diffusion models currently offer state-of-the-art performance in GAI for images. They also form a key component in more general tools, including text-to-image generators and large language models. Diffusion models work by adding noise to the available training data and then learning how to reverse the process. The reverse operation may then be applied to new random data in order to produce new outputs. We provide a brief introduction to diffusion models for applied mathematicians and statisticians. Our key aims are to (a) present illustrative computational examples, (b) give a careful derivation of the underlying mathematical formulas involved, and (c) draw a connection with partial differential equation (PDE) diffusion models. We provide code for the computational experiments. We hope that this topic will be of interest to advanced undergraduate and postgraduate students. Portions of the material may also provide useful motivational examples for those who teach courses in stochastic processes, inference, machine learning, PDEs, or scientific computing.
{"title":"Diffusion Models for Generative Artificial Intelligence: An Introduction for Applied Mathematicians","authors":"Catherine Higham, Desmond J. Higham, Peter Grindrod","doi":"10.1137/23m1626232","DOIUrl":"https://doi.org/10.1137/23m1626232","url":null,"abstract":"SIAM Review, Volume 67, Issue 3, Page 607-623, August 2025. <br/> Abstract.Generative artificial intelligence (GAI) refers to algorithms that create synthetic but realistic output. Diffusion models currently offer state-of-the-art performance in GAI for images. They also form a key component in more general tools, including text-to-image generators and large language models. Diffusion models work by adding noise to the available training data and then learning how to reverse the process. The reverse operation may then be applied to new random data in order to produce new outputs. We provide a brief introduction to diffusion models for applied mathematicians and statisticians. Our key aims are to (a) present illustrative computational examples, (b) give a careful derivation of the underlying mathematical formulas involved, and (c) draw a connection with partial differential equation (PDE) diffusion models. We provide code for the computational experiments. We hope that this topic will be of interest to advanced undergraduate and postgraduate students. Portions of the material may also provide useful motivational examples for those who teach courses in stochastic processes, inference, machine learning, PDEs, or scientific computing.","PeriodicalId":49525,"journal":{"name":"SIAM Review","volume":"115 1","pages":""},"PeriodicalIF":10.2,"publicationDate":"2025-08-07","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"144792656","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 654-655, August 2025. Some books on differential equations or computational methods for differential equations present the mathematical theories or numerical algorithms in detail, but include only a few illustrative codes. In contrast, the book under review places the emphasis on Mathematica codes. In other words, this book is a collection of Mathematica codes for the solutions of various types of differential equations.
{"title":"Book Review:; Differential Equations: Solving Ordinary and Partial Differential Equations with Mathematica","authors":"Hao Chen","doi":"10.1137/24m170096x","DOIUrl":"https://doi.org/10.1137/24m170096x","url":null,"abstract":"SIAM Review, Volume 67, Issue 3, Page 654-655, August 2025. <br/> Some books on differential equations or computational methods for differential equations present the mathematical theories or numerical algorithms in detail, but include only a few illustrative codes. In contrast, the book under review places the emphasis on Mathematica codes. In other words, this book is a collection of Mathematica codes for the solutions of various types of differential equations.","PeriodicalId":49525,"journal":{"name":"SIAM Review","volume":"16 1","pages":""},"PeriodicalIF":10.2,"publicationDate":"2025-08-07","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"144797132","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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