SIAM Review, Volume 67, Issue 4, Page 801-861, December 2025. Abstract.We develop and analyze a method to reduce the size of a very large set of data points in a high-dimensional Euclidean space [math] to a small set of weighted points such that the result of a predetermined data analysis task on the reduced set is approximately the same as that for the original point set. For example, computing the first [math] principal components of the reduced set will return approximately the first [math] principal components of the original set, or computing the centers of a [math]-means clustering on the reduced set will return an approximation for the original set. Such a reduced set is also known as a coreset. The main new features of our construction are that the cardinality of the reduced set is independent of the dimension [math] of the input space and that the sets are mergeable [P. K. Agarwal et al., Proceedings of the 31st ACM SIGMOD-SIGACT-SIGAI Symposium on Principals of Database Systems, 2012, pp. 23–34]. The latter property means that the union of two reduced sets is a reduced set for the union of the two original sets. It allows us to turn our methods into streaming or distributed algorithms using standard approaches. For problems such as [math]-means and subspace approximation the coreset sizes are also independent of the number of input points. Our method is based on data-dependently projecting the points on a low-dimensional subspace and reducing the cardinality of the points inside this subspace using known methods. The proposed approach works for a wide range of data analysis techniques including [math]-means clustering, principal component analysis, and subspace clustering. The main conceptual contribution is a new coreset definition that allows charging for the costs that appear for every solution to an additive constant.
SIAM评论，第67卷，第4期，801-861页，2025年12月。摘要。我们开发并分析了一种方法，将高维欧几里德空间（数学）中非常大的数据点集的大小减少到一个小的加权点集，从而使预定的数据分析任务的结果与原始点集的结果大致相同。例如，计算约简集的第一个[math]主成分将近似返回原始集的第一个[math]主成分，或者计算约简集上的[math]均值聚类的中心将返回原始集的近似值。这样的约简集也被称为核集。我们的构造的主要新特征是，约简集的基数与输入空间的维数[math]无关，并且集合是可合并的[P]。K. Agarwal et al.，第31届ACM SIGMOD-SIGACT-SIGAI数据库系统研讨会论文集，2012,pp. 23-34]。后一个性质意味着两个简化集的并集是两个原始集的并集的简化集。它允许我们使用标准方法将我们的方法转换为流或分布式算法。对于像[math]-means和子空间近似这样的问题，核心集的大小也与输入点的数量无关。我们的方法是基于基于数据的低维子空间上的点投影，并使用已知方法减少该子空间内点的基数。所提出的方法适用于广泛的数据分析技术，包括[数学]均值聚类、主成分分析和子空间聚类。主要的概念贡献是一个新的核心定义，允许对每个解决方案出现的成本收取一个附加常数。
{"title":"Turning Big Data Into Tiny Data: Coresets for Unsupervised Learning Problems","authors":"Dan Feldman, Melanie Schmidt, Christian Sohler","doi":"10.1137/25m1799684","DOIUrl":"https://doi.org/10.1137/25m1799684","url":null,"abstract":"SIAM Review, Volume 67, Issue 4, Page 801-861, December 2025. <br/> Abstract.We develop and analyze a method to reduce the size of a very large set of data points in a high-dimensional Euclidean space [math] to a small set of weighted points such that the result of a predetermined data analysis task on the reduced set is approximately the same as that for the original point set. For example, computing the first [math] principal components of the reduced set will return approximately the first [math] principal components of the original set, or computing the centers of a [math]-means clustering on the reduced set will return an approximation for the original set. Such a reduced set is also known as a coreset. The main new features of our construction are that the cardinality of the reduced set is independent of the dimension [math] of the input space and that the sets are mergeable [P. K. Agarwal et al., Proceedings of the 31st ACM SIGMOD-SIGACT-SIGAI Symposium on Principals of Database Systems, 2012, pp. 23–34]. The latter property means that the union of two reduced sets is a reduced set for the union of the two original sets. It allows us to turn our methods into streaming or distributed algorithms using standard approaches. For problems such as [math]-means and subspace approximation the coreset sizes are also independent of the number of input points. Our method is based on data-dependently projecting the points on a low-dimensional subspace and reducing the cardinality of the points inside this subspace using known methods. The proposed approach works for a wide range of data analysis techniques including [math]-means clustering, principal component analysis, and subspace clustering. The main conceptual contribution is a new coreset definition that allows charging for the costs that appear for every solution to an additive constant.","PeriodicalId":49525,"journal":{"name":"SIAM Review","volume":"53 1","pages":""},"PeriodicalIF":10.2,"publicationDate":"2025-11-06","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"145448284","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 4, Page 661-733, December 2025. Abstract.The Mathieu function is a special function satisfying the Mathieu differential equation. Since its inception in 1868, numerous algorithms and programs have been published to calculate it, and so it is about time to review the performance of available software. First, the fundamentals of Mathieu functions are summarized such as definition, normalization, nomenclature, and methods of solution. Then, we review several programs for Mathieu functions of integer orders with real parameters and compare the results numerically by running individual software; in addition, Bessel function routines are also compared. Finally, a straightforward algorithm is recommended with codes written in MATLAB and GNU Octave.
{"title":"Numerical Review of Mathieu Function Programs for Integer Orders and Real Parameters","authors":"Ho-Chul Shin","doi":"10.1137/23m1572726","DOIUrl":"https://doi.org/10.1137/23m1572726","url":null,"abstract":"SIAM Review, Volume 67, Issue 4, Page 661-733, December 2025. <br/> Abstract.The Mathieu function is a special function satisfying the Mathieu differential equation. Since its inception in 1868, numerous algorithms and programs have been published to calculate it, and so it is about time to review the performance of available software. First, the fundamentals of Mathieu functions are summarized such as definition, normalization, nomenclature, and methods of solution. Then, we review several programs for Mathieu functions of integer orders with real parameters and compare the results numerically by running individual software; in addition, Bessel function routines are also compared. Finally, a straightforward algorithm is recommended with codes written in MATLAB and GNU Octave.","PeriodicalId":49525,"journal":{"name":"SIAM Review","volume":"27 1","pages":""},"PeriodicalIF":10.2,"publicationDate":"2025-11-06","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"145448289","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 4, Page 873-902, December 2025. Abstract.Equations, particularly differential equations, are fundamental for understanding natural phenomena and predicting complex dynamics across various scientific and engineering disciplines. However, the governing equations for many complex systems remain unknown due to intricate underlying mechanisms. Recent advancements in machine learning and data science offer a new paradigm for modeling unknown equations from measurement or simulation data. This paradigm shift, known as data-driven discovery or modeling, stands at the forefront of artificial intelligence for science (AI4Science), with significant progress made in recent years. In this paper, we introduce a systematic educational framework for data-driven modeling of unknown equations using deep learning. This versatile framework is capable of learning unknown ordinary differential equations (ODEs), partial differential equations (PDEs), differential-algebraic equations (DAEs), integro-differential equations (IDEs), stochastic differential equations (SDEs), reduced or partially observed systems, and nonautonomous differential equations. Based on this framework, we have developed Deep Unknown Equations (DUE), an open-source software package designed to facilitate the data-driven modeling of unknown equations using modern deep learning techniques. DUE serves as an educational tool for classroom instruction, enabling students and newcomers to gain hands-on experience with differential equations, data-driven modeling, and contemporary deep learning approaches such as fully connected neural networks (FNNs), residual neural networks (ResNet), generalized ResNet (gResNet), operator semigroup networks (OSG-Net), and transformers from large language models (LLMs). Additionally, DUE is a versatile and accessible toolkit for researchers across various scientific and engineering fields. It is applicable not only for learning unknown equations from data, but also for surrogate modeling of known, yet complex equations that are costly to solve using traditional numerical methods. We provide detailed descriptions of DUE and demonstrate its capabilities through diverse examples which serve as templates that can be easily adapted for other applications. The source code for DUE is available at https://github.com/AI4Equations/due.
{"title":"DUE: A Deep Learning Framework and Library for Modeling Unknown Equations","authors":"Junfeng Chen, Kailiang Wu, Dongbin Xiu","doi":"10.1137/24m1671827","DOIUrl":"https://doi.org/10.1137/24m1671827","url":null,"abstract":"SIAM Review, Volume 67, Issue 4, Page 873-902, December 2025. <br/> Abstract.Equations, particularly differential equations, are fundamental for understanding natural phenomena and predicting complex dynamics across various scientific and engineering disciplines. However, the governing equations for many complex systems remain unknown due to intricate underlying mechanisms. Recent advancements in machine learning and data science offer a new paradigm for modeling unknown equations from measurement or simulation data. This paradigm shift, known as data-driven discovery or modeling, stands at the forefront of artificial intelligence for science (AI4Science), with significant progress made in recent years. In this paper, we introduce a systematic educational framework for data-driven modeling of unknown equations using deep learning. This versatile framework is capable of learning unknown ordinary differential equations (ODEs), partial differential equations (PDEs), differential-algebraic equations (DAEs), integro-differential equations (IDEs), stochastic differential equations (SDEs), reduced or partially observed systems, and nonautonomous differential equations. Based on this framework, we have developed Deep Unknown Equations (DUE), an open-source software package designed to facilitate the data-driven modeling of unknown equations using modern deep learning techniques. DUE serves as an educational tool for classroom instruction, enabling students and newcomers to gain hands-on experience with differential equations, data-driven modeling, and contemporary deep learning approaches such as fully connected neural networks (FNNs), residual neural networks (ResNet), generalized ResNet (gResNet), operator semigroup networks (OSG-Net), and transformers from large language models (LLMs). Additionally, DUE is a versatile and accessible toolkit for researchers across various scientific and engineering fields. It is applicable not only for learning unknown equations from data, but also for surrogate modeling of known, yet complex equations that are costly to solve using traditional numerical methods. We provide detailed descriptions of DUE and demonstrate its capabilities through diverse examples which serve as templates that can be easily adapted for other applications. The source code for DUE is available at https://github.com/AI4Equations/due.","PeriodicalId":49525,"journal":{"name":"SIAM Review","volume":"109 1","pages":""},"PeriodicalIF":10.2,"publicationDate":"2025-11-06","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"145448281","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 4, Page 905-909, December 2025. Optimal control theory has long been a cornerstone of mathematical modeling and decision-making across disciplines such as engineering, economics, and the physical sciences. Yet, as the complexity of control systems continues to grow, so does the demand for more robust and efficient computational techniques to solve these problems. Alfio Borzì’s The Sequential Quadratic Hamiltonian Method: Solving Optimal Control Problems addresses this challenge head-on, introducing a groundbreaking numerical optimization procedure, the sequential quadratic Hamiltonian (SQH) method. This book not only builds upon the theoretical framework established by the Pontryagin maximum principle (PMP), but also offers a practical computational tool that is both versatile and robust. With applications ranging from differential Nash games to deep learning via residual neural networks, the book is as much a testament to the SQH method’s adaptability as it is to its computational power. In this review, we describe the book’s structure, its significant contributions to the field of applied and computational mathematics, and its interdisciplinary relevance. We explore how the SQH method redefines the landscape of optimal control, offering new pathways for both theoretical investigation and practical implementation.
{"title":"Featured Review:; The Sequential Quadratic Hamiltonian Method: Solving Optimal Control Problems","authors":"Souvik Roy","doi":"10.1137/24m1700958","DOIUrl":"https://doi.org/10.1137/24m1700958","url":null,"abstract":"SIAM Review, Volume 67, Issue 4, Page 905-909, December 2025. <br/> Optimal control theory has long been a cornerstone of mathematical modeling and decision-making across disciplines such as engineering, economics, and the physical sciences. Yet, as the complexity of control systems continues to grow, so does the demand for more robust and efficient computational techniques to solve these problems. Alfio Borzì’s The Sequential Quadratic Hamiltonian Method: Solving Optimal Control Problems addresses this challenge head-on, introducing a groundbreaking numerical optimization procedure, the sequential quadratic Hamiltonian (SQH) method. This book not only builds upon the theoretical framework established by the Pontryagin maximum principle (PMP), but also offers a practical computational tool that is both versatile and robust. With applications ranging from differential Nash games to deep learning via residual neural networks, the book is as much a testament to the SQH method’s adaptability as it is to its computational power. In this review, we describe the book’s structure, its significant contributions to the field of applied and computational mathematics, and its interdisciplinary relevance. We explore how the SQH method redefines the landscape of optimal control, offering new pathways for both theoretical investigation and practical implementation.","PeriodicalId":49525,"journal":{"name":"SIAM Review","volume":"54 1","pages":""},"PeriodicalIF":10.2,"publicationDate":"2025-11-06","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"145448286","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 4, Page 909-912, December 2025. Paul Smaldino’s take on math modeling is unique. Many math modeling textbooks lean into engineering and physics applications, including topics like mechanics, optimization, and operations research. Modeling textbooks in the life or social sciences typically focus on biology or economics. The text Modeling Social Behavior uses simple agent-based, discrete, network, and probabilistic models of social animals (especially humans) to explore phenomena as varied as flocking, segregation, contagion, opinion dynamics, and cultural evolution. The book is an eclectic survey of applications and basic methods in math modeling of social dynamics.
{"title":"Book Review:; Modeling Social Behavior: Mathematical and Agent-Based Models of Social Dynamics and Cultural Evolution","authors":"Sara Clifton","doi":"10.1137/24m1700922","DOIUrl":"https://doi.org/10.1137/24m1700922","url":null,"abstract":"SIAM Review, Volume 67, Issue 4, Page 909-912, December 2025. <br/> Paul Smaldino’s take on math modeling is unique. Many math modeling textbooks lean into engineering and physics applications, including topics like mechanics, optimization, and operations research. Modeling textbooks in the life or social sciences typically focus on biology or economics. The text Modeling Social Behavior uses simple agent-based, discrete, network, and probabilistic models of social animals (especially humans) to explore phenomena as varied as flocking, segregation, contagion, opinion dynamics, and cultural evolution. The book is an eclectic survey of applications and basic methods in math modeling of social dynamics.","PeriodicalId":49525,"journal":{"name":"SIAM Review","volume":"93 1","pages":""},"PeriodicalIF":10.2,"publicationDate":"2025-11-06","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"145448280","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 4, Page 771-798, December 2025. Abstract.Ensemble Kalman inversion (EKI) methods are a family of iterative methods for solving weighted least squares problems, especially those arising in scientific and engineering inverse problems in which unknown parameters or states are estimated from observed data by minimizing the weighted square norm of the data misfit. Implementation of EKI only requires the evaluation of the forward model mapping the unknown to the data, and does not require derivatives or adjoints of the forward model. The methods therefore offer an attractive alternative to gradient-based optimization approaches in inverse problem settings where evaluating derivatives or adjoints of the forward model is computationally intractable. This work presents a new analysis of the behavior of both deterministic and stochastic versions of basic EKI for linear observation operators, resulting in a natural interpretation of EKI’s convergence properties in terms of “fundamental subspaces” analogous to Strang’s fundamental subspaces of linear algebra. Our analysis directly examines the discrete EKI iterations instead of their continuous-time limits considered in previous analyses, and it provides spectral decompositions that define six fundamental subspaces of EKI spanning both observation and state spaces. This approach verifies convergence rates previously derived for continuous-time limits, and yields new results describing both deterministic and stochastic EKI convergence behavior with respect to the standard minimum-norm weighted least squares solution in terms of the fundamental subspaces. Numerical experiments illustrate our theoretical results.
{"title":"The Fundamental Subspaces of Ensemble Kalman Inversion","authors":"Elizabeth Qian, Christopher Beattie","doi":"10.1137/24m1693143","DOIUrl":"https://doi.org/10.1137/24m1693143","url":null,"abstract":"SIAM Review, Volume 67, Issue 4, Page 771-798, December 2025. <br/> Abstract.Ensemble Kalman inversion (EKI) methods are a family of iterative methods for solving weighted least squares problems, especially those arising in scientific and engineering inverse problems in which unknown parameters or states are estimated from observed data by minimizing the weighted square norm of the data misfit. Implementation of EKI only requires the evaluation of the forward model mapping the unknown to the data, and does not require derivatives or adjoints of the forward model. The methods therefore offer an attractive alternative to gradient-based optimization approaches in inverse problem settings where evaluating derivatives or adjoints of the forward model is computationally intractable. This work presents a new analysis of the behavior of both deterministic and stochastic versions of basic EKI for linear observation operators, resulting in a natural interpretation of EKI’s convergence properties in terms of “fundamental subspaces” analogous to Strang’s fundamental subspaces of linear algebra. Our analysis directly examines the discrete EKI iterations instead of their continuous-time limits considered in previous analyses, and it provides spectral decompositions that define six fundamental subspaces of EKI spanning both observation and state spaces. This approach verifies convergence rates previously derived for continuous-time limits, and yields new results describing both deterministic and stochastic EKI convergence behavior with respect to the standard minimum-norm weighted least squares solution in terms of the fundamental subspaces. Numerical experiments illustrate our theoretical results.","PeriodicalId":49525,"journal":{"name":"SIAM Review","volume":"35 1","pages":""},"PeriodicalIF":10.2,"publicationDate":"2025-11-06","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"145448287","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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