Multigrid methods for Toeplitz linear systems with different size reduction

Abstract

Starting from the spectral analysis of g-circulant matrices, we study the convergence of a multigrid method for circulant and Toeplitz matrices with various size reductions. We assume that the size n of the coefficient matrix is divisible by g ≥ 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 two-grid method and of the multigrid method is proved, when the number θ ∈ N of recursive calls is such that 1 < θ < 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.

Multigrid methods: Grid transfer operators and subdivision schemes

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“…The operator K nj,g allows to express F nj+1 in terms of F nj , see [18], i.e F nj and K nj,g satisfy the following packaging property…”

“…The proof consists of three steps. The first and second steps are borrowed from [18] and [28], thus we only state them shortly. We present in detail the proof of the main step, 3. step.…”

Symbol approach in a signal-restoration problem involving block Toeplitz matrices

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Mixed two‐grid finite difference methods for solving one‐dimensional and two‐dimensional Fitzhugh–Nagumo equations

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