分数阶微分方程的低秩解

IF 0.8 4区 数学 Q3 MATHEMATICS, APPLIED
T. Breiten, V. Simoncini, M. Stoll
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引用次数: 47

摘要

许多科学技术问题都可以用具有分数阶时间导数和空间导数的微分方程来解决。为了使用该技术精确地模拟自然现象,需要精细的空间和时间离散化,从而导致大规模的线性系统或矩阵方程,特别是当考虑多个空间维度时。分数阶微分方程的离散化通常涉及具有Toeplitz结构或在变系数情况下具有Toeplitz结构的密集矩阵。我们将Toeplitz矩阵及其循环前置条件的快速评估与最先进的线性矩阵方程方法相结合,以有效地解决这些问题,无论是在CPU时间和内存需求方面。此外,我们还说明了在存在可变系数时如何调整这些技术。对典型的含分数阶导数的微分问题在空间和时间上的数值实验表明了该方法的有效性。
本文章由计算机程序翻译,如有差异,请以英文原文为准。
Low-Rank Solvers for Fractional Differential Equations
Many problems in science and technology can be cast using differential equations with both fractional time and spatial derivatives. To accurately simulate natural phenomena using this technology fine spatial and temporal discretizations are required, leading to large-scale linear systems or matrix equations, especially whenever more than one space dimension is considered. The discretization of fractional differential equations typically involves dense matrices with a Toeplitz or in the variable coefficient case Toeplitz-like structure . We combine the fast evaluation of Toeplitz matrices and their circulant preconditioners with state-of-the-art linear matrix equation methods to efficiently solve these problems, both in terms of CPU time and memory requirements. Additionally, we illustrate how these techniqes can be adapted when variable coefficients are present. Numerical experiments on typical differential problems with fractional derivatives in both space and time showing the effectiveness of the approaches are reported.
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来源期刊
CiteScore
2.10
自引率
7.70%
发文量
36
审稿时长
6 months
期刊介绍: Electronic Transactions on Numerical Analysis (ETNA) is an electronic journal for the publication of significant new developments in numerical analysis and scientific computing. Papers of the highest quality that deal with the analysis of algorithms for the solution of continuous models and numerical linear algebra are appropriate for ETNA, as are papers of similar quality that discuss implementation and performance of such algorithms. New algorithms for current or new computer architectures are appropriate provided that they are numerically sound. However, the focus of the publication should be on the algorithm rather than on the architecture. The journal is published by the Kent State University Library in conjunction with the Institute of Computational Mathematics at Kent State University, and in cooperation with the Johann Radon Institute for Computational and Applied Mathematics of the Austrian Academy of Sciences (RICAM).
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