Gabriele Ciaramella, Martin J. Gander, Tommaso Vanzan
SIAM Review, Volume 68, Issue 1, Page 172-203, February 2026. Abstract. This paper offers a self-contained exposition of the fundamental mathematical and computational tools for interpolation on the Grassmann manifold, including detailed derivations of geodesics and explicit formulations of the exponential and logarithmic maps. The presentation emphasizes intuition and draws continuous parallels with the Euclidean setting. This pedagogical approach facilitates the understanding of linear, piecewise linear, and high-order interpolation algorithms, as well as their extension to more general manifolds. Two numerical examples are finally used to illustrate the potential of these algorithms: one in the context of parametric model order reduction, and another drawn from stationary iterative methods for linear systems.
{"title":"A Gentle Introduction to Interpolation on the Grassmann Manifold","authors":"Gabriele Ciaramella, Martin J. Gander, Tommaso Vanzan","doi":"10.1137/24m1628591","DOIUrl":"https://doi.org/10.1137/24m1628591","url":null,"abstract":"SIAM Review, Volume 68, Issue 1, Page 172-203, February 2026. <br/> Abstract. This paper offers a self-contained exposition of the fundamental mathematical and computational tools for interpolation on the Grassmann manifold, including detailed derivations of geodesics and explicit formulations of the exponential and logarithmic maps. The presentation emphasizes intuition and draws continuous parallels with the Euclidean setting. This pedagogical approach facilitates the understanding of linear, piecewise linear, and high-order interpolation algorithms, as well as their extension to more general manifolds. Two numerical examples are finally used to illustrate the potential of these algorithms: one in the context of parametric model order reduction, and another drawn from stationary iterative methods for linear systems.","PeriodicalId":49525,"journal":{"name":"SIAM Review","volume":"385 1","pages":""},"PeriodicalIF":10.2,"publicationDate":"2026-02-09","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"146138552","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 68, Issue 1, Page 213-214, February 2026. Stellarators are devices used in plasma physics to confine very hot plasmas (i.e., ionized gases) with magnetic fields to sustain nuclear fusion reactions. Fusion is the Sun’s energy source, and the achievement of sustained fusion on Earth has been studied for several decades as a promising source of clean and safe energy. Unlike tokamaks, which use a combination of simple magnetic fields and plasma current to cage the plasma, stellarators rely solely on external magnetic fields. This has potential advantages for sustained fusion energy production, but requires the design of complicated magnetic fields and expensive-to-build, complex electromagnetic coils.
{"title":"Book Review:; An Introduction to Stellarators","authors":"Georg Stadler","doi":"10.1137/25m1728582","DOIUrl":"https://doi.org/10.1137/25m1728582","url":null,"abstract":"SIAM Review, Volume 68, Issue 1, Page 213-214, February 2026. <br/> Stellarators are devices used in plasma physics to confine very hot plasmas (i.e., ionized gases) with magnetic fields to sustain nuclear fusion reactions. Fusion is the Sun’s energy source, and the achievement of sustained fusion on Earth has been studied for several decades as a promising source of clean and safe energy. Unlike tokamaks, which use a combination of simple magnetic fields and plasma current to cage the plasma, stellarators rely solely on external magnetic fields. This has potential advantages for sustained fusion energy production, but requires the design of complicated magnetic fields and expensive-to-build, complex electromagnetic coils.","PeriodicalId":49525,"journal":{"name":"SIAM Review","volume":"16 1","pages":""},"PeriodicalIF":10.2,"publicationDate":"2026-02-09","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"146138545","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 68, Issue 1, Page 93-123, February 2026. Abstract. In order to compute the Fourier transform of a function [math] on the real line numerically, one samples [math] on a grid and then takes the discrete Fourier transform. We derive exact error estimates for this procedure in terms of the decay and smoothness of [math]. The analysis provides an asymptotically optimal recipe for how to relate the number of samples, the sampling interval, and the grid size.
{"title":"Quantitative Estimates: How Well Does the Discrete Fourier Transform Approximate the Fourier Transform on [math]","authors":"Martin Ehler, Karlheinz Gröchenig, Andreas Klotz","doi":"10.1137/24m1650399","DOIUrl":"https://doi.org/10.1137/24m1650399","url":null,"abstract":"SIAM Review, Volume 68, Issue 1, Page 93-123, February 2026. <br/> Abstract. In order to compute the Fourier transform of a function [math] on the real line numerically, one samples [math] on a grid and then takes the discrete Fourier transform. We derive exact error estimates for this procedure in terms of the decay and smoothness of [math]. The analysis provides an asymptotically optimal recipe for how to relate the number of samples, the sampling interval, and the grid size.","PeriodicalId":49525,"journal":{"name":"SIAM Review","volume":"6 1","pages":""},"PeriodicalIF":10.2,"publicationDate":"2026-02-09","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"146138554","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 68, Issue 1, Page 207-208, February 2026. The two volumes of Feller’s Introduction to Probability are classic and comprehensive. They span a huge range of topics, starting from measure theory and basic probability distributions and moving on to topics such as Markov chains and the central limit theorem. The books are very well written and accessible. It seems that most of the theory that is being employed in 2025 by applied modelers and analysts is touched on in these books in some way. Of course, the emphases are different. Much of the foundational material on probability theory has not changed a lot since Feller’s two volumes, including basic measure theory, the law of large numbers and the central limit theorem, conditional probabilities, and theorems such as the Borel Cantelli Lemma. In contrast, it seems that Feller spends a lot of time surveying a range of special probability models and distributions whereas more modern treatments would perhaps spend more time in surveying general analytic techniques. This is probably partly due to the fact that numerical simulation techniques were not nearly as powerful when the book was written, so scholars tended to focus more on specific tractable models.
{"title":"Featured Review:; Introduction to Probability and Its Applications","authors":"James N. MacLaurin","doi":"10.1137/24m1717579","DOIUrl":"https://doi.org/10.1137/24m1717579","url":null,"abstract":"SIAM Review, Volume 68, Issue 1, Page 207-208, February 2026. <br/> The two volumes of Feller’s Introduction to Probability are classic and comprehensive. They span a huge range of topics, starting from measure theory and basic probability distributions and moving on to topics such as Markov chains and the central limit theorem. The books are very well written and accessible. It seems that most of the theory that is being employed in 2025 by applied modelers and analysts is touched on in these books in some way. Of course, the emphases are different. Much of the foundational material on probability theory has not changed a lot since Feller’s two volumes, including basic measure theory, the law of large numbers and the central limit theorem, conditional probabilities, and theorems such as the Borel Cantelli Lemma. In contrast, it seems that Feller spends a lot of time surveying a range of special probability models and distributions whereas more modern treatments would perhaps spend more time in surveying general analytic techniques. This is probably partly due to the fact that numerical simulation techniques were not nearly as powerful when the book was written, so scholars tended to focus more on specific tractable models.","PeriodicalId":49525,"journal":{"name":"SIAM Review","volume":"72 1","pages":""},"PeriodicalIF":10.2,"publicationDate":"2026-02-09","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"146138546","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 68, Issue 1, Page 153-171, February 2026. Abstract. Interpreting data with mathematical models is an important aspect of real-world industrial and applied mathematical modeling. Often we are interested to understand the extent to which a particular set of data informs and constrains model parameters. This question is closely related to the concept of parameter identifiability, and in this article we present a series of computational exercises to introduce tools that can be used to assess parameter identifiability, estimate parameters, and generate model predictions. Taking a likelihood-based approach, we show that very similar ideas and algorithms can be used to deal with a range of different mathematical modeling frameworks. The exercises and results presented in this article are supported by a suite of open access codes that can be accessed on GitHub.
{"title":"Parameter Identifiability, Parameter Estimation, and Model Prediction for Differential Equation Models","authors":"Matthew J. Simpson, Ruth E. Baker","doi":"10.1137/24m1667968","DOIUrl":"https://doi.org/10.1137/24m1667968","url":null,"abstract":"SIAM Review, Volume 68, Issue 1, Page 153-171, February 2026. <br/> Abstract. Interpreting data with mathematical models is an important aspect of real-world industrial and applied mathematical modeling. Often we are interested to understand the extent to which a particular set of data informs and constrains model parameters. This question is closely related to the concept of parameter identifiability, and in this article we present a series of computational exercises to introduce tools that can be used to assess parameter identifiability, estimate parameters, and generate model predictions. Taking a likelihood-based approach, we show that very similar ideas and algorithms can be used to deal with a range of different mathematical modeling frameworks. The exercises and results presented in this article are supported by a suite of open access codes that can be accessed on GitHub.","PeriodicalId":49525,"journal":{"name":"SIAM Review","volume":"3 1","pages":""},"PeriodicalIF":10.2,"publicationDate":"2026-02-09","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"146146213","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 68, Issue 1, Page 208-210, February 2026. Linear algebra is often viewed as one of the most foundational courses in a mathematics or computer science curriculum, yet it is also one that can intimidate students with its abstract formalism and steep learning curve. In Linear Algebra: A Problem-Centered Approach, Róbert Freud reimagines the subject by presenting it not as a procession of theorems and proofs, but as an unfolding narrative of problems, motivations, and applications. Published as part of the AMS’s Pure and Applied Undergraduate Texts series, this book brings together the rigor of traditional mathematics with the accessibility and playfulness of the Hungarian problem-solving tradition.
{"title":"Book Review:; Linear Algebra: A Problem-Centered Approach","authors":"Anita T. Layton","doi":"10.1137/25m1799192","DOIUrl":"https://doi.org/10.1137/25m1799192","url":null,"abstract":"SIAM Review, Volume 68, Issue 1, Page 208-210, February 2026. <br/> Linear algebra is often viewed as one of the most foundational courses in a mathematics or computer science curriculum, yet it is also one that can intimidate students with its abstract formalism and steep learning curve. In Linear Algebra: A Problem-Centered Approach, Róbert Freud reimagines the subject by presenting it not as a procession of theorems and proofs, but as an unfolding narrative of problems, motivations, and applications. Published as part of the AMS’s Pure and Applied Undergraduate Texts series, this book brings together the rigor of traditional mathematics with the accessibility and playfulness of the Hungarian problem-solving tradition.","PeriodicalId":49525,"journal":{"name":"SIAM Review","volume":"9 1","pages":""},"PeriodicalIF":10.2,"publicationDate":"2026-02-09","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"146138547","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 68, Issue 1, Page 210-211, February 2026. Control in Finite and Infinite Dimension is an excellent textbook based on many years of in-depth teaching experience, as well as on the author’s expertise in the field. It provides a concise and (mostly) self-contained introduction to mathematical control theory, for both finite- and infinite-dimensional systems. It is written at the level of a Master’s/Ph.D. program, but for more experienced researchers it can also serve as a good overview of the basic results and the main tools used in the field, as well as material for lectures in specialised courses on the topic. It covers the most important parts of the classical control theory: controllability, observability and their duality, optimal controls, and stabilization (the latter two only in the finite-dimensional case).
{"title":"Book Review:; Control in Finite and Infinite Dimension","authors":"Martin Lazar","doi":"10.1137/25m1732787","DOIUrl":"https://doi.org/10.1137/25m1732787","url":null,"abstract":"SIAM Review, Volume 68, Issue 1, Page 210-211, February 2026. <br/> Control in Finite and Infinite Dimension is an excellent textbook based on many years of in-depth teaching experience, as well as on the author’s expertise in the field. It provides a concise and (mostly) self-contained introduction to mathematical control theory, for both finite- and infinite-dimensional systems. It is written at the level of a Master’s/Ph.D. program, but for more experienced researchers it can also serve as a good overview of the basic results and the main tools used in the field, as well as material for lectures in specialised courses on the topic. It covers the most important parts of the classical control theory: controllability, observability and their duality, optimal controls, and stabilization (the latter two only in the finite-dimensional case).","PeriodicalId":49525,"journal":{"name":"SIAM Review","volume":"1 1","pages":""},"PeriodicalIF":10.2,"publicationDate":"2026-02-09","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"146138544","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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