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Superfast solution of linear convolutional Volterra equations using QTT approximation

Authors
Journal
Journal of Computational and Applied Mathematics
0377-0427
Publisher
Elsevier
Volume
260
Identifiers
DOI: 10.1016/j.cam.2013.10.025
Keywords
  • Fractional Calculus
  • Triangular Toeplitz Matrix
  • Divide And Conquer
  • Tensor Train Format
  • Fast Convolution
  • Superfast Fourier Transform
Disciplines
  • Computer Science
  • Mathematics

Abstract

Abstract We 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(log2n). The results of the paper are illustrated by numerical examples.

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