Superfast solution of linear convolutional Volterra equations using QTT approximation
For researchers in numerical analysis and scientific computing, this provides a faster algorithm for a specific class of equations, though it is an incremental improvement on existing methods.
The paper develops methods for solving linear fractional differential equations by inverting triangular Toeplitz matrices using the QTT format, reducing complexity from O(n log n) to O(log^2 n).
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.