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Computational Methods in Applied Mathematics

Editor-in-Chief: Carstensen, Carsten

Managing Editor: Matus, Piotr


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Volume 19, Issue 1

Issues

Quasi-Optimal Rank-Structured Approximation to Multidimensional Parabolic Problems by Cayley Transform and Chebyshev Interpolation

Ivan Gavrilyuk / Boris N. Khoromskij
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  • Max-Planck-Institute for Mathematics in the Sciences, Inselstr. 22–26, 04103 Leipzig, Germany
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Published Online: 2018-07-12 | DOI: https://doi.org/10.1515/cmam-2018-0021

Abstract

In the present paper we propose and analyze a class of tensor approaches for the efficient numerical solution of a first order differential equation ψ(t)+Aψ=f(t) with an unbounded operator coefficient A. These techniques are based on a Laguerre polynomial expansions with coefficients which are powers of the Cayley transform of the operator A. The Cayley transform under consideration is a useful tool to arrive at the following aims: (1) to separate time and spatial variables, (2) to switch from the continuous “time variable” to “the discrete time variable” and from the study of functions of an unbounded operator to the ones of a bounded operator, (3) to obtain exponentially accurate approximations. In the earlier papers of the authors some approximations on the basis of the Cayley transform and the N-term Laguerre expansions of the accuracy order 𝒪(e-N) were proposed and justified provided that the initial value is analytical for A. In the present paper we combine the Cayley transform and the Chebyshev–Gauss–Lobatto interpolation and arrive at an approximation of the accuracy order 𝒪(e-N) without restrictions on the input data. The use of the Laguerre expansion or the Chebyshev–Gauss–Lobatto interpolation allows to separate the time and space variables. The separation of the multidimensional spatial variable can be achieved by the use of low-rank approximation to the Cayley transform of the Laplace-like operator that is spectrally close to A. As a result a quasi-optimal numerical algorithm can be designed.

Keywords: High-Dimensional Problems; Rank Structured Tensor Approximation; Quantized Representation of Vectors; Matrix-Valued Functions; Cayley Transform from Unbounded to the Bounded Operators; Model Reduction; Exponentially Accurate Approximations; Variables Separation

MSC 2010: 65F30; 65F50; 65N35; 65F10

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About the article

Received: 2017-12-01

Revised: 2018-02-21

Accepted: 2018-05-02

Published Online: 2018-07-12

Published in Print: 2019-01-01


Citation Information: Computational Methods in Applied Mathematics, Volume 19, Issue 1, Pages 55–71, ISSN (Online) 1609-9389, ISSN (Print) 1609-4840, DOI: https://doi.org/10.1515/cmam-2018-0021.

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