Multigrid methods for Toeplitz linear systems with different size reduction

TL;DR

This paper introduces a new multigrid method for solving Toeplitz and circulant linear systems, with rigorous convergence analysis and demonstrated effectiveness through numerical experiments.

Contribution

It develops a multigrid approach based on g-circulant projectors, extending spectral analysis and proving optimality for Toeplitz systems with structure-preserving techniques.

Findings

Proves optimality of the multigrid method for certain recursive levels.

Extends convergence analysis from circulant to Toeplitz matrices.

Numerical results confirm the method's effectiveness.

Abstract

Starting from the spectral analysis of g-circulant matrices, we consider a new multigrid method for circulant and Toeplitz matrices with given generating function. We assume that the size n of the coefficient matrix is divisible by g \geq 2 such that at the lower level the system is reduced to one of size n/g by employing g-circulant based projectors. We perform a rigorous two-grid convergence analysis in the circulant case and we extend experimentally the results to the Toeplitz setting, by employing structure preserving projectors. The optimality of the proposed two-grid method and of the multigrid method is proved, when the number theta \in N of recursive calls is such that 1 < theta < g. The previous analysis is used also to overcome some pathological cases, in which the generating function has zeros located at "mirror points" and the standard two-grid method with g = 2 is not optimal. The numerical experiments show the correctness and applicability of the proposed ideas both for circulant and Toeplitz matrices.