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

Acta Numerica

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。

Category	Quartile
数学	1
数学, 数学	1

Low-rank tensor methods for partial differential equations

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