SIAM Review最新文献

{"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}

{"title":"Survey and Review","authors":"Marlis Hochbruck","doi":"10.1137/25m1766504","DOIUrl":"https://doi.org/10.1137/25m1766504","url":null,"abstract":"SIAM Review, Volume 67, Issue 4, Page 659-659, December 2025. <br/>","PeriodicalId":49525,"journal":{"name":"SIAM Review","volume":"75 1","pages":""},"PeriodicalIF":10.2,"publicationDate":"2025-11-06","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"145448290","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":"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}

{"title":"Education","authors":"Hélène Frankowska","doi":"10.1137/25m1766528","DOIUrl":"https://doi.org/10.1137/25m1766528","url":null,"abstract":"SIAM Review, Volume 67, Issue 4, Page 863-863, December 2025. <br/>","PeriodicalId":49525,"journal":{"name":"SIAM Review","volume":"9 1","pages":""},"PeriodicalIF":10.2,"publicationDate":"2025-11-06","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"145448283","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":"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}

{"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}

{"title":"SIGEST","authors":"The Editors","doi":"10.1137/25m1766541","DOIUrl":"https://doi.org/10.1137/25m1766541","url":null,"abstract":"SIAM Review, Volume 67, Issue 4, Page 799-799, December 2025. <br/>","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":"145448285","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":"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}

{"title":"Research Spotlights","authors":"Stefan M. Wild","doi":"10.1137/25m1766516","DOIUrl":"https://doi.org/10.1137/25m1766516","url":null,"abstract":"SIAM Review, Volume 67, Issue 4, Page 735-735, December 2025. <br/>","PeriodicalId":49525,"journal":{"name":"SIAM Review","volume":"1 1","pages":""},"PeriodicalIF":10.2,"publicationDate":"2025-11-06","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"145448291","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":"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}