Superfast solution of linear convolutional Volterra equations using QTT approximation

TL;DR

This paper introduces a superfast method for solving linear convolutional Volterra equations by combining QTT approximation with advanced matrix inversion algorithms, significantly reducing computational complexity.

Contribution

It develops a novel approach that integrates QTT format with divide and conquer algorithms for efficient inversion of triangular Toeplitz matrices, improving computational speed.

Findings

Achieves inversion complexity of O(log^2 n)

Demonstrates effectiveness through numerical examples

Reduces computational time compared to traditional methods

Abstract

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.