Superfast solution of linear convolutional Volterra equations using QTT approximation
AffiliationUniversity of Chester ; Russian Academy of Sciences / University of Chester ; Russian Academy of Sciences / Lomonosov Moscow State University
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AbstractThis article address a linear fractional differential equation and develop effective solution methods using algorithms for the inversion of triangular Toeplitz matrices and the recently proposed QTT format. The inverses of such matrices can be computed by the divide and conquer and modified Bini’s algorithms, for which we present the versions with the QTT approximation. We also present an efficient formula for the shift of vectors given in QTT format, which is used in the divide and conquer algorithm. As a result, we reduce the complexity of inversion from the fast Fourier level O(nlogn) to the speed of superfast Fourier transform, i.e., O(log^2n). The results of the paper are illustrated by numerical examples.
CitationJournal of Computational and Applied Mathematics, 2014, 260, pp. 434-448
DescriptionNOTICE: this is the author’s version of a work that was accepted for publication in Journal of Computational and Applied Mathematics. Changes resulting from the publishing process, such as peer review, editing, corrections, structural formatting, and other quality control mechanisms may not be reflected in this document. Changes may have been made to this work since it was submitted for publication. A definitive version was subsequently published in Journal of Applied and Computational Mathematics, 260, 2014, doi: 10.1016/j.cam.2013.10.025
SponsorsDuring this work D. V. Savostyanov and E. E. Tyrtyshnikov were supported by the Leverhulme Trust to visit, stay and work at the University of Chester, as the Visiting Research Fellow and the Visiting Professor, respectively. Their work was also supported in part by RFBR grants 11-01-00549, 12-01-91333-nnio-a, 13-01-12061, and Russian Federation Government Contracts 14.740.11.0345, 14.740.11.1067, 16.740.12.0727.
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