Volume 45, pp. 107-132, 2016.

Low-rank solvers for fractional differential equations

Tobias Breiten, Valeria Simoncini, and Martin Stoll

Abstract

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 structure in the constant coefficient case. 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 techniques 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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Key words

fractional calculus, fast solvers, Sylvester equations, preconditioning, low-rank methods, tensor equations

AMS subject classifications

65F08, 65F10, 65F50, 92E20, 93C20

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