ACTA NUMERICA 是过去两年在 MATHSCINET 中被引用次数最多的期刊。其年度评论文章集包括数值分析和科学计算领域领先研究人员的调查论文。这些论文概述了最近的进展，并提供了最先进的技术和分析。涵盖数值分析的广度，文章以各级研究人员可以访问的风格编写，可以作为高级教具。广泛的学科领域包括线性​​代数、优化、常微分方程和偏微分方程、近似理论、随机分析和非线性动力系统中的计算方法，以及计算技术在科学和工程中的应用以及数值方法的数学理论。

Acta Numerica is the most cited journal in MATHSCINET over the past two years. Its annual collection of review articles features survey papers from leading researchers in the fields of numerical analysis and scientific computing. These papers outline recent advancements and provide state-of-the-art techniques and analyses. Covering a broad range of topics in numerical analysis, the articles are written in a style accessible to researchers at all levels and can serve as advanced teaching materials. The wide-ranging subjects include linear algebra, optimization, ordinary and partial differential equations, approximation theory, stochastic analysis, and computational methods in nonlinear dynamical systems, as well as the application of computational techniques in science and engineering and the mathematical theory of numerical methods.

《Acta Numerica》是过去两年在MATHSCINET中被引用次数最多的期刊。其年度评论文章集汇集了数值分析和科学计算领域的领先研究人员的调查论文。这些论文概述了最近的研究进展，并提供了最先进的技术和分析。涵盖数值分析的广泛主题，文章以各级研究人员都能理解的风格撰写，适合作为高级教学材料。相关学科包括线性代数、优化、常微分方程和偏微分方程、近似理论、随机分析以及非线性动力系统中的计算方法，同时也涉及计算技术在科学和工程中的应用及数值方法的数学理论。

Acta Numerica

Low-rank tensor representations can provide highly compressed approximations of functions. These concepts, which essentially amount to generalizations of classical techniques of separation of variables, have proved to be particularly fruitful for functions of many variables. We focus here on problems where the target function is given only implicitly as the solution of a partial differential equation. A first natural question is under which conditions we should expect such solutions to be efficiently approximated in low-rank form. Due to the highly nonlinear nature of the resulting low-rank approximations, a crucial second question is at what expense such approximations can be computed in practice. This article surveys basic construction principles of numerical methods based on low-rank representations as well as the analysis of their convergence and computational complexity.

Markus Bachmayr

Low-rank tensor methods for partial differential equations

偏微分方程的低秩张量方法

Physics-informed neural networks (PINNs) and their variants have been very popular in recent years as algorithms for the numerical simulation of both forward and inverse problems for partial differential equations. This article aims to provide a comprehensive review of currently available results on the numerical analysis of PINNs and related models that constitute the backbone of physics-informed machine learning. We provide a unified framework in which analysis of the various components of the error incurred by PINNs in approximating PDEs can be effectively carried out. We present a detailed review of available results on approximation, generalization and training errors and their behaviour with respect to the type of the PDE and the dimension of the underlying domain. In particular, we elucidate the role of the regularity of the solutions and their stability to perturbations in the error analysis. Numerical results are also presented to illustrate the theory. We identify training errors as a key bottleneck which can adversely affect the overall performance of various models in physics-informed machine learning.

Tim De Ryck

Siddhartha Mishra

Numerical analysis of physics-informed neural networks and related models in physics-informed machine learning

物理信息神经网络及相关模型在物理信息机器学习中的数值分析

In the past decade the mathematical theory of machine learning has lagged far behind the triumphs of deep neural networks on practical challenges. However, the gap between theory and practice is gradually starting to close. In this paper I will attempt to assemble some pieces of the remarkable and still incomplete mathematical mosaic emerging from the efforts to understand the foundations of deep learning. The two key themes will be interpolation and its sibling over-parametrization. Interpolation corresponds to fitting data, even noisy data, exactly. Over-parametrization enables interpolation and provides flexibility to select a suitable interpolating model. As we will see, just as a physical prism separates colours mixed within a ray of light, the figurative prism of interpolation helps to disentangle generalization and optimization properties within the complex picture of modern machine learning. This article is written in the belief and hope that clearer understanding of these issues will bring us a step closer towards a general theory of deep learning and machine learning.

Mikhail Belkin

Fit without fear: remarkable mathematical phenomena of deep learning through the prism of interpolation

无畏拟合：透过插值棱镜观察深度学习的显著数学现象

In this article we will focus on only a small part of financial mathematics, namely the use of partial differential equations for pricing futures. Even within this narrow range it is hard to be systematic and complete, or even to do better than existing books such as Wilmott, Howison and Dewynne (1995), Achdou and Pironneau (2005), or software manuals such as Lapeyre, Martini and Sulem (2010). So this article may be valuable only to the extent that it reflects ten years of teaching, conferences and interaction with the protagonists of financial mathematics. Also, because the theory of partial differential equations is not always well known, we have chosen a pragmatic approach and left out the details of the theory or the proofs of some results, and refer the reader to other books. The numerical algorithms, on the other hand, are given in detail.

Olivier Pironneau

Pricing futures by deterministic methods

用确定性方法为期货定价

This article describes the current state of the art of high-performance computing systems, and attempts to shed light on near-future developments that might prolong the steady growth in speed of such systems, which has been one of their most remarkable characteristics. We review the different ways devised to speed them up, both with regard to components and their architecture. In addition, we discuss the requirements for software that can take advantage of existing and future architectures.

J. J. Dongarra

A. J. van der Steen

High-performance computing systems: Status and outlook

高性能计算系统：现状与展望

This overview is devoted to splitting methods, a class of numerical integrators intended for differential equations that can be subdivided into different problems easier to solve than the original system. Closely connected with this class of integrators are composition methods, in which one or several low-order schemes are composed to construct higher-order numerical approximations to the exact solution. We analyse in detail the order conditions that have to be satisfied by these classes of methods to achieve a given order, and provide some insight about their qualitative properties in connection with geometric numerical integration and the treatment of highly oscillatory problems. Since splitting methods have received considerable attention in the realm of partial differential equations, we also cover this subject in the present survey, with special attention to parabolic equations and their problems. An exhaustive list of methods of different orders is collected and tested on simple examples. Finally, some applications of splitting methods in different areas, ranging from celestial mechanics to statistics, are also provided.

Sergio Blanes

Fernando Casas

Ander Murua

Splitting methods for differential equations

微分方程的分裂方法

We propose a geometric framework to describe and analyse a wide array of operator splitting methods for solving monotone inclusion problems. The initial inclusion problem, which typically involves several operators combined through monotonicity-preserving operations, is seldom solvable in its original form. We embed it in an auxiliary space, where it is associated with a surrogate monotone inclusion problem with a more tractable structure and which allows for easy recovery of solutions to the initial problem. The surrogate problem is solved by successive projections onto half-spaces containing its solution set. The outer approximation half-spaces are constructed by using the individual operators present in the model separately. This geometric framework is shown to encompass traditional methods as well as state-of-the-art asynchronous block-iterative algorithms, and its flexible structure provides a pattern to design new ones.

Patrick L. Combettes

The geometry of monotone operator splitting methods

单调算子分裂方法的几何学

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.

Huan Xun

Jagalur Jayanth

Marzouk Youssef

Optimal experimental design: Formulations and computations

最优实验设计：公式化与计算

The Moment-SOS hierarchy, first introduced in optimization in 2000, is based on the theory of the S -moment problem and its dual counterpart: polynomials that are positive on S . It turns out that this methodology can also be used to solve problems with positivity constraints ‘ f (x) ≥ 0 for all $\mathbf{x}\in S$ ’ or linear constraints on Borel measures. Such problems can be viewed as specific instances of the generalized moment problem (GMP), whose list of important applications in various domains of science and engineering is almost endless. We describe this methodology in optimization and also in two other applications for illustration. Finally we also introduce the Christoffel function and reveal its links with the Moment-SOS hierarchy and positive polynomials.

Abbreviation	Acta Numer.
Journal Impact	16.26
Quartiles(Global)	MATHEMATICS(Q1)

ISSN	0962-4929, 1474-0508
h-index	85
Top Journals	Yes

Publication Information	Publisher: Cambridge University Press，Publishing cycle: ，Journal Type: journal，Open Access Journals: No
Basic data	Year of publication: 1992，Proportion of original research papers: 100.00%，Self Citation Rate:1.20%， Gold OA Rate: 57.89%

Acta Numerica

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Low-rank tensor methods for partial differential equations

Numerical analysis of physics-informed neural networks and related models in physics-informed machine learning

Fit without fear: remarkable mathematical phenomena of deep learning through the prism of interpolation

Pricing futures by deterministic methods

High-performance computing systems: Status and outlook

Splitting methods for differential equations

Numerical analysis of physics-informed neural networks and related models in physics-informed machine learning

The geometry of monotone operator splitting methods

Optimal experimental design: Formulations and computations

The Moment-SOS hierarchy: Applications and related topics