Numerical tensor calculus*

IF 16.3 1区数学 Q1 MATHEMATICS

Acta Numerica Pub Date : 2014-05-01 DOI:10.1017/S0962492914000087

W. Hackbusch

引用次数: 59

Abstract

The usual large-scale discretizations are applied to two or three spatial dimensions. The standard methods fail for higher dimensions because the data size increases exponentially with the dimension. In the case of a regular grid with n grid points per direction, a spatial dimension d yields nd grid points. A grid function defined on such a grid is an example of a tensor of order d. Here, suitable tensor formats help, since they try to approximate these huge objects by a much smaller number of parameters, which increases only linearly in d. In this way, data of size nd = 10001000 can also be treated. This paper introduces the algebraic and analytical aspects of tensor spaces. The main part concerns the numerical representation of tensors and the numerical performance of tensor operations.

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数值张量微积分*

通常的大规模离散化应用于二维或三维空间。标准方法在高维情况下失效，因为数据大小随着维度的增加呈指数增长。在每个方向有n个网格点的规则网格的情况下，一个空间维d产生两个网格点。在这样的网格上定义的网格函数是d阶张量的一个例子。在这里，合适的张量格式会有所帮助，因为它们试图用更少的参数来近似这些巨大的对象，这些参数只在d中线性增加。这样，大小为nd = 10001000的数据也可以处理。本文介绍了张量空间的代数和解析方面。主要讨论张量的数值表示和张量运算的数值性能。

本文章由计算机程序翻译，如有差异，请以英文原文为准。

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来源期刊

Acta Numerica MATHEMATICS-

CiteScore

26.00

自引率

0.70%

发文量

期刊介绍： Acta Numerica stands as the preeminent mathematics journal, ranking highest in both Impact Factor and MCQ metrics. This annual journal features a collection of review articles that showcase survey papers authored by prominent researchers in numerical analysis, scientific computing, and computational mathematics. These papers deliver comprehensive overviews of recent advances, offering state-of-the-art techniques and analyses. Encompassing the entirety of numerical analysis, the articles are crafted in an accessible style, catering to researchers at all levels and serving as valuable teaching aids for advanced instruction. The broad subject areas covered include computational methods in linear algebra, optimization, ordinary and partial differential equations, approximation theory, stochastic analysis, nonlinear dynamical systems, as well as the application of computational techniques in science and engineering. Acta Numerica also delves into the mathematical theory underpinning numerical methods, making it a versatile and authoritative resource in the field of mathematics.