Superfast solution of linear convolutional Volterra equations using QTT approximation

• Published in: arXiv.org
• Main Authors: Roberts, Jason A, Savostyanov, Dmitry V, Tyrtyshnikov, Eugene E
• Format: Paper Journal Article
• Language: English
• Published: Ithaca Cornell University Library, arXiv.org 22.11.2012
• Subjects: Algorithms; Approximation; Differential equations; Format; Fourier transforms; Mathematical analysis; Mathematics - Numerical Analysis; Volterra integral equations
• ISSN: 2331-8422
• DOI: 10.48550/arxiv.1211.5384

We address a linear fractional differential equation and develop effective solution methods using algorithms for 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 the result, we reduce the complexity of inversion from the fast Fourier level \(O(n\log n)\) to the speed of superfast Fourier transform, i.e., \(O(\log^2 n).\) The results of the paper are illustrated by numerical examples.