Acta NumericaPub Date : 2026-06-19DOI: 10.1017/s0962492926100294
Jan S. Hesthaven, Benjamin Peherstorfer, Benjamin Unger
{"title":"Nonlinear model reduction for transport-dominated problems","authors":"Jan S. Hesthaven, Benjamin Peherstorfer, Benjamin Unger","doi":"10.1017/s0962492926100294","DOIUrl":"https://doi.org/10.1017/s0962492926100294","url":null,"abstract":"This article surveys nonlinear model reduction methods that remain effective in regimes where linear reduced-space approximations are intrinsically inefficient, such as transport-dominated problems with wave-like phenomena and moving coherent structures, which are commonly associated with the Kolmogorov barrier. The article organizes nonlinear model reduction techniques around three key elements – nonlinear parametrizations, reduced dynamics and online solvers – and categorizes existing approaches into transformation-based methods, online adaptive techniques, and formulations that combine generic nonlinear parametrizations with instantaneous residual minimization.","PeriodicalId":48863,"journal":{"name":"Acta Numerica","volume":"30 1","pages":"173-272"},"PeriodicalIF":14.2,"publicationDate":"2026-06-19","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"148288214","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}
Acta NumericaPub Date : 2026-06-19DOI: 10.1017/s0962492925100263
Yuji Nakatsukasa, Lloyd N. Trefethen
{"title":"Applications of AAA rational approximation","authors":"Yuji Nakatsukasa, Lloyd N. Trefethen","doi":"10.1017/s0962492925100263","DOIUrl":"https://doi.org/10.1017/s0962492925100263","url":null,"abstract":"The AAA algorithm for rational approximation is employed to illustrate applications of rational functions all across numerical analysis. For example, rational functions enable strikingly effective methods for numerical differentiation and integration; interpolation of equispaced data; locating zeros, poles, and branch points; analytic continuation; computation of inverse functions; imputation of missing data; computation of nonlinear eigenvalues and resonances; model order reduction/reduced order modelling; solution of Wiener–Hopf and Hilbert transform problems; conformal mapping; and solution of the two-dimensional Laplace, biharmonic and Helmholtz equations. The paper also surveys various improvements and generalizations of the AAA algorithm that have developed since its original appearance in 2018.","PeriodicalId":48863,"journal":{"name":"Acta Numerica","volume":"14 1","pages":"459-601"},"PeriodicalIF":14.2,"publicationDate":"2026-06-19","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"148288212","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}
Acta NumericaPub Date : 2026-06-19DOI: 10.1017/s0962492925100275
Jeffrey Galkowski, Euan A. Spence
{"title":"Numerical analysis of the high-frequency Helmholtz equation using semiclassical analysis","authors":"Jeffrey Galkowski, Euan A. Spence","doi":"10.1017/s0962492925100275","DOIUrl":"https://doi.org/10.1017/s0962492925100275","url":null,"abstract":"We consider the numerical solution of high-frequency scattering problems modelled by the Helmholtz equation with a bounded obstacle. Although the analysis of this problem dates back at least 50 years, over the past decade or so, tools and techniques from <jats:italic>semiclassical analysis</jats:italic> have provided a new perspective and been used to settle several long-standing open problems in this area. Semiclassical analysis works in phase space (i.e. position and frequency) and describes rigorously the extent to which solutions of high-frequency PDEs are dictated by the properties of the corresponding geometric optics rays. The goals of the article are (i) to give a introduction to semiclassical analysis aimed at non-experts and (ii) to showcase some of the numerical analysis results about finite element methods, boundary element methods and domain decomposition methods obtained using semiclassical techniques.","PeriodicalId":48863,"journal":{"name":"Acta Numerica","volume":"3 1","pages":"1-171"},"PeriodicalIF":14.2,"publicationDate":"2026-06-19","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"148288215","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}
Acta NumericaPub Date : 2026-06-19DOI: 10.1017/s0962492925100287
David Krieg, Mario Ullrich
{"title":"Approximation of functions: Optimal sampling and complexity","authors":"David Krieg, Mario Ullrich","doi":"10.1017/s0962492925100287","DOIUrl":"https://doi.org/10.1017/s0962492925100287","url":null,"abstract":"We consider the approximation or recovery of functions based on a finite number of function evaluations. This is a well-studied problem in optimal recovery, machine learning and numerical analysis in general, but many fundamental insights were obtained only recently. We discuss different aspects of the information-theoretic limit that appears because of the limited data available, as well as algorithms and sampling strategies that come as close to it as possible. We also discuss (optimal) sampling in a broader sense, allowing other types of measurements that may be nonlinear, adaptive and random, and present several relations between the different settings in the spirit of information-based complexity. We hope that this article will provide both a basic introduction to the subject and a contemporary summary of the current state of research.","PeriodicalId":48863,"journal":{"name":"Acta Numerica","volume":"20 1","pages":"273-457"},"PeriodicalIF":14.2,"publicationDate":"2026-06-19","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"148288213","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}
Acta NumericaPub Date : 2025-07-01DOI: 10.1017/s0962492924000072
Martin J. Gander, Shu-Lin Wu, Tao Zhou
{"title":"Time parallelization for hyperbolic and parabolic problems","authors":"Martin J. Gander, Shu-Lin Wu, Tao Zhou","doi":"10.1017/s0962492924000072","DOIUrl":"https://doi.org/10.1017/s0962492924000072","url":null,"abstract":"<p>Time parallelization, also known as PinT (parallel-in-time), is a new research direction for the development of algorithms used for solving very large-scale evolution problems on highly parallel computing architectures. Despite the fact that interesting theoretical work on PinT appeared as early as 1964, it was not until 2004, when processor clock speeds reached their physical limit, that research in PinT took off. A distinctive characteristic of parallelization in time is that information flow only goes forward in time, meaning that time evolution processes seem necessarily to be sequential. Nevertheless, many algorithms have been developed for PinT computations over the past two decades, and they are often grouped into four basic classes according to how the techniques work and are used: shooting-type methods; waveform relaxation methods based on domain decomposition; multigrid methods in space–time; and direct time parallel methods. However, over the past few years, it has been recognized that highly successful PinT algorithms for parabolic problems struggle when applied to hyperbolic problems. We will therefore focus on this important aspect, first by providing a summary of the fundamental differences between parabolic and hyperbolic problems for time parallelization. We then group PinT algorithms into two basic groups. The first group contains four effective PinT techniques for hyperbolic problems: Schwarz waveform relaxation (SWR) with its relation to tent pitching; parallel integral deferred correction; ParaExp; and ParaDiag. While the methods in the first group also work well for parabolic problems, we then present PinT methods specifically designed for parabolic problems in the second group: Parareal; the parallel full approximation scheme in space–time (PFASST); multigrid reduction in time (MGRiT); and space–time multigrid (STMG). We complement our analysis with numerical illustrations using four time-dependent PDEs: the heat equation; the advection–diffusion equation; Burgers’ equation; and the second-order wave equation.</p>","PeriodicalId":48863,"journal":{"name":"Acta Numerica","volume":"26 1","pages":""},"PeriodicalIF":14.2,"publicationDate":"2025-07-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"144520756","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}
Acta NumericaPub Date : 2025-07-01DOI: 10.1017/s0962492924000060
Edoardo Calvello, Sebastian Reich, Andrew M. Stuart
{"title":"Ensemble Kalman methods: A mean-field perspective","authors":"Edoardo Calvello, Sebastian Reich, Andrew M. Stuart","doi":"10.1017/s0962492924000060","DOIUrl":"https://doi.org/10.1017/s0962492924000060","url":null,"abstract":"<p>Ensemble Kalman methods, introduced in 1994 in the context of ocean state estimation, are now widely used for state estimation and parameter estimation (inverse problems) in many arenae. Their success stems from the fact that they take an underlying computational model as a black box to provide a systematic, derivative-free methodology for incorporating observations; furthermore the ensemble approach allows for sensitivities and uncertainties to be calculated. Analysis of the accuracy of ensemble Kalman methods, especially in terms of uncertainty quantification, is lagging behind empirical success; this paper provides a unifying mean-field-based framework for their analysis. Both state estimation and parameter estimation problems are considered, and formulations in both discrete and continuous time are employed. For state estimation problems, both the control and filtering approaches are considered; analogously for parameter estimation problems, the optimization and Bayesian perspectives are both studied. As well as providing an elegant framework, the mean-field perspective also allows for the derivation of a variety of methods used in practice. In addition it unifies a wide-ranging literature in the field and suggests open problems.</p>","PeriodicalId":48863,"journal":{"name":"Acta Numerica","volume":"19 1","pages":""},"PeriodicalIF":14.2,"publicationDate":"2025-07-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"144521054","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}
Acta NumericaPub Date : 2025-07-01DOI: 10.1017/s0962492924000059
Jennifer Scott, Miroslav Tůma
{"title":"Sparse linear least-squares problems","authors":"Jennifer Scott, Miroslav Tůma","doi":"10.1017/s0962492924000059","DOIUrl":"https://doi.org/10.1017/s0962492924000059","url":null,"abstract":"<p>Least-squares problems are a cornerstone of computational science and engineering. Over the years, the size of the problems that researchers and practitioners face has constantly increased, making it essential that sparsity is exploited in the solution process. The goal of this article is to present a broad review of key algorithms for solving large-scale linear least-squares problems. This includes sparse direct methods and algebraic preconditioners that are used in combination with iterative solvers. Where software is available, this is highlighted.</p>","PeriodicalId":48863,"journal":{"name":"Acta Numerica","volume":"11 1","pages":""},"PeriodicalIF":14.2,"publicationDate":"2025-07-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"144520751","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}
Acta NumericaPub Date : 2025-07-01DOI: 10.1017/s0962492925000017
Erik Burman, Peter Hansbo, Mats G. Larson, Sara Zahedi
{"title":"Cut finite element methods","authors":"Erik Burman, Peter Hansbo, Mats G. Larson, Sara Zahedi","doi":"10.1017/s0962492925000017","DOIUrl":"https://doi.org/10.1017/s0962492925000017","url":null,"abstract":"<p>Cut finite element methods (CutFEM) extend the standard finite element method to unfitted meshes, enabling the accurate resolution of domain boundaries and interfaces without requiring the mesh to conform to them. This approach preserves the key properties and accuracy of the standard method while addressing challenges posed by complex geometries and moving interfaces.</p><p>In recent years, CutFEM has gained significant attention for its ability to discretize partial differential equations in domains with intricate geometries. This paper provides a comprehensive review of the core concepts and key developments in CutFEM, beginning with its formulation for common model problems and the presentation of fundamental analytical results, including error estimates and condition number estimates for the resulting algebraic systems. Stabilization techniques for cut elements, which ensure numerical robustness, are also explored. Finally, extensions to methods involving Lagrange multipliers and applications to time-dependent problems are discussed.</p>","PeriodicalId":48863,"journal":{"name":"Acta Numerica","volume":"1 1","pages":""},"PeriodicalIF":14.2,"publicationDate":"2025-07-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"144520754","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}
Acta NumericaPub Date : 2025-07-01DOI: 10.1017/s0962492924000084
Daniel Kuhn, Soroosh Shafiee, Wolfram Wiesemann
{"title":"Distributionally robust optimization","authors":"Daniel Kuhn, Soroosh Shafiee, Wolfram Wiesemann","doi":"10.1017/s0962492924000084","DOIUrl":"https://doi.org/10.1017/s0962492924000084","url":null,"abstract":"<p>Distributionally robust optimization (DRO) studies decision problems under uncertainty where the probability distribution governing the uncertain problem parameters is itself uncertain. A key component of any DRO model is its ambiguity set, that is, a family of probability distributions consistent with any available structural or statistical information. DRO seeks decisions that perform best under the worst distribution in the ambiguity set. This worst case criterion is supported by findings in psychology and neuroscience, which indicate that many decision-makers have a low tolerance for distributional ambiguity. DRO is rooted in statistics, operations research and control theory, and recent research has uncovered its deep connections to regularization techniques and adversarial training in machine learning. This survey presents the key findings of the field in a unified and self-contained manner.</p>","PeriodicalId":48863,"journal":{"name":"Acta Numerica","volume":"19 1","pages":""},"PeriodicalIF":14.2,"publicationDate":"2025-07-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"144520825","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}
Acta NumericaPub Date : 2025-07-01DOI: 10.1017/s0962492924000102
Leszek Demkowicz, Jay Gopalakrishnan
{"title":"The discontinuous Petrov–Galerkin method","authors":"Leszek Demkowicz, Jay Gopalakrishnan","doi":"10.1017/s0962492924000102","DOIUrl":"https://doi.org/10.1017/s0962492924000102","url":null,"abstract":"<p>The discontinuous Petrov–Galerkin (DPG) method is a Petrov–Galerkin finite element method with test functions designed for obtaining stability. These test functions are computable locally, element by element, and are motivated by optimal test functions which attain the supremum in an inf-sup condition. A profound consequence of the use of nearly optimal test functions is that the DPG method can inherit the stability of the (undiscretized) variational formulation, be it coercive or not. This paper combines a presentation of the fundamentals of the DPG ideas with a review of the ongoing research on theory and applications of the DPG methodology. The scope of the presented theory is restricted to linear problems on Hilbert spaces, but pointers to extensions are provided. Multiple viewpoints to the basic theory are provided. They show that the DPG method is equivalent to a method which minimizes a residual in a dual norm, as well as to a mixed method where one solution component is an approximate error representation function. Being a residual minimization method, the DPG method yields Hermitian positive definite stiffness matrix systems even for non-self-adjoint boundary value problems. Having a built-in error representation, the method has the out-of-the-box feature that it can immediately be used in automatic adaptive algorithms. Contrary to standard Galerkin methods, which are uninformed about test and trial norms, the DPG method must be equipped with a concrete test norm which enters the computations. Of particular interest are variational formulations in which one can tailor the norm to obtain robust stability. Key techniques to rigorously prove convergence of DPG schemes, including construction of Fortin operators, which in the DPG case can be done element by element, are discussed in detail. Pointers to open frontiers are presented.</p>","PeriodicalId":48863,"journal":{"name":"Acta Numerica","volume":"633 1","pages":""},"PeriodicalIF":14.2,"publicationDate":"2025-07-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"144520755","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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