Acta Numerica最新文献

Splitting methods for differential equations 微分方程的分割方法

IF 14.2 1区数学

Acta Numerica Pub Date : 2024-09-04 DOI: 10.1017/s0962492923000077

Sergio Blanes, Fernando Casas, Ander Murua

引用次数: 0

The geometry of monotone operator splitting methods 单调算子拆分法的几何原理

IF 14.2 1区数学

Acta Numerica Pub Date : 2024-09-04 DOI: 10.1017/s0962492923000065

Patrick L. Combettes

引用次数: 0

Numerical analysis of physics-informed neural networks and related models in physics-informed machine learning 物理信息机器学习中的物理信息神经网络和相关模型的数值分析

IF 14.2 1区数学

Acta Numerica Pub Date : 2024-09-04 DOI: 10.1017/s0962492923000089

Tim De Ryck, Siddhartha Mishra

引用次数: 0

Adaptive finite element methods 自适应有限元方法

IF 14.2 1区数学

Acta Numerica Pub Date : 2024-09-04 DOI: 10.1017/s0962492924000011

Andrea Bonito, Claudio Canuto, Ricardo H. Nochetto, Andreas Veeser

{"title":"Adaptive finite element methods","authors":"Andrea Bonito, Claudio Canuto, Ricardo H. Nochetto, Andreas Veeser","doi":"10.1017/s0962492924000011","DOIUrl":"https://doi.org/10.1017/s0962492924000011","url":null,"abstract":"This is a survey of the theory of adaptive finite element methods (AFEMs), which are fundamental to modern computational science and engineering but whose mathematical assessment is a formidable challenge. We present a self-contained and up-to-date discussion of AFEMs for linear second-order elliptic PDEs and dimension d > 1, with emphasis on foundational issues. After a brief review of functional analysis and basic finite element theory, including piecewise polynomial approximation in graded meshes, we present the core material for coercive problems. We start with a novel a posteriori error analysis applicable to rough data, which delivers estimators fully equivalent to the solution error. They are used in the design and study of three AFEMs depending on the structure of data. We prove linear convergence of these algorithms and rate-optimality provided the solution and data belong to suitable approximation classes. We also address the relation between approximation and regularity classes. We finally extend this theory to discontinuous Galerkin methods as prototypes of non-conforming AFEMs, and beyond coercive problems to inf-sup stable AFEMs.","PeriodicalId":48863,"journal":{"name":"Acta Numerica","volume":"59 1","pages":""},"PeriodicalIF":14.2,"publicationDate":"2024-09-04","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142130833","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}

引用次数: 0

The Moment-SOS hierarchy: Applications and related topics Moment-SOS 层次结构：应用和相关主题

IF 14.2 1区数学

Acta Numerica Pub Date : 2024-09-04 DOI: 10.1017/s0962492923000053

Jean B. Lasserre

引用次数: 0

Optimal experimental design: Formulations and computations 最佳实验设计：公式与计算

IF 14.2 1区数学

Acta Numerica Pub Date : 2024-09-04 DOI: 10.1017/s0962492924000023

Xun Huan, Jayanth Jagalur, Youssef Marzouk

{"title":"Optimal experimental design: Formulations and computations","authors":"Xun Huan, Jayanth Jagalur, Youssef Marzouk","doi":"10.1017/s0962492924000023","DOIUrl":"https://doi.org/10.1017/s0962492924000023","url":null,"abstract":"Questions of ‘how best to acquire data’ are essential to modelling and prediction in the natural and social sciences, engineering applications, and beyond. Optimal experimental design (OED) formalizes these questions and creates computational methods to answer them. This article presents a systematic survey of modern OED, from its foundations in classical design theory to current research involving OED for complex models. We begin by reviewing criteria used to formulate an OED problem and thus to encode the goal of performing an experiment. We emphasize the flexibility of the Bayesian and decision-theoretic approach, which encompasses information-based criteria that are well-suited to nonlinear and non-Gaussian statistical models. We then discuss methods for estimating or bounding the values of these design criteria; this endeavour can be quite challenging due to strong nonlinearities, high parameter dimension, large per-sample costs, or settings where the model is implicit. A complementary set of computational issues involves optimization methods used to find a design; we discuss such methods in the discrete (combinatorial) setting of observation selection and in settings where an exact design can be continuously parametrized. Finally we present emerging methods for sequential OED that build non-myopic design policies, rather than explicit designs; these methods naturally adapt to the outcomes of past experiments in proposing new experiments, while seeking coordination among all experiments to be performed. Throughout, we highlight important open questions and challenges.","PeriodicalId":48863,"journal":{"name":"Acta Numerica","volume":"6 1","pages":""},"PeriodicalIF":14.2,"publicationDate":"2024-09-04","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142130836","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}

引用次数: 0

Floating-point arithmetic 浮点算术

IF 14.2 1区数学

Acta Numerica Pub Date : 2023-05-01 DOI: 10.1017/S0962492922000101

S. Boldo, C. Jeannerod, G. Melquiond, Jean-Michel Muller

{"title":"Floating-point arithmetic","authors":"S. Boldo, C. Jeannerod, G. Melquiond, Jean-Michel Muller","doi":"10.1017/S0962492922000101","DOIUrl":"https://doi.org/10.1017/S0962492922000101","url":null,"abstract":"Floating-point numbers have an intuitive meaning when it comes to physics-based numerical computations, and they have thus become the most common way of approximating real numbers in computers. The IEEE-754 Standard has played a large part in making floating-point arithmetic ubiquitous today, by specifying its semantics in a strict yet useful way as early as 1985. In particular, floating-point operations should be performed as if their results were first computed with an infinite precision and then rounded to the target format. A consequence is that floating-point arithmetic satisfies the ‘standard model’ that is often used for analysing the accuracy of floating-point algorithms. But that is only scraping the surface, and floating-point arithmetic offers much more. In this survey we recall the history of floating-point arithmetic as well as its specification mandated by the IEEE-754 Standard. We also recall what properties it entails and what every programmer should know when designing a floating-point algorithm. We provide various basic blocks that can be implemented with floating-point arithmetic. In particular, one can actually compute the rounding error caused by some floating-point operations, which paves the way to designing more accurate algorithms. More generally, properties of floating-point arithmetic make it possible to extend the accuracy of computations beyond working precision.","PeriodicalId":48863,"journal":{"name":"Acta Numerica","volume":"32 1","pages":"203 - 290"},"PeriodicalIF":14.2,"publicationDate":"2023-05-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"44272147","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}

引用次数: 14

ANU volume 32 Cover and Back matter 澳大利亚国立大学第32卷封面和封底

IF 14.2 1区数学

Acta Numerica Pub Date : 2023-05-01 DOI: 10.1017/s0962492923000041

引用次数: 0

The virtual element method 虚元法

IF 14.2 1区数学

Acta Numerica Pub Date : 2023-05-01 DOI: 10.1017/S0962492922000095

L. Beirão Da Veiga, F. Brezzi, L. D. Marini, A. Russo

引用次数: 0

Compatible finite element methods for geophysical fluid dynamics 地球物理流体动力学的兼容有限元方法

1区数学

Acta Numerica Pub Date : 2023-05-01 DOI: 10.1017/s0962492923000028

Colin J. Cotter

{"title":"Compatible finite element methods for geophysical fluid dynamics","authors":"Colin J. Cotter","doi":"10.1017/s0962492923000028","DOIUrl":"https://doi.org/10.1017/s0962492923000028","url":null,"abstract":"This article surveys research on the application of compatible finite element methods to large-scale atmosphere and ocean simulation. Compatible finite element methods extend Arakawa’s C-grid finite difference scheme to the finite element world. They are constructed from a discrete de Rham complex, which is a sequence of finite element spaces linked by the operators of differential calculus. The use of discrete de Rham complexes to solve partial differential equations is well established, but in this article we focus on the specifics of dynamical cores for simulating weather, oceans and climate. The most important consequence of the discrete de Rham complex is the Hodge–Helmholtz decomposition, which has been used to exclude the possibility of several types of spurious oscillations from linear equations of geophysical flow. This means that compatible finite element spaces provide a useful framework for building dynamical cores. In this article we introduce the main concepts of compatible finite element spaces, and discuss their wave propagation properties. We survey some methods for discretizing the transport terms that arise in dynamical core equation systems, and provide some example discretizations, briefly discussing their iterative solution. Then we focus on the recent use of compatible finite element spaces in designing structure preserving methods, surveying variational discretizations, Poisson bracket discretizations and consistent vorticity transport.","PeriodicalId":48863,"journal":{"name":"Acta Numerica","volume":"21 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2023-05-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"135504227","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}

引用次数: 2