A symbol based analysis for multigrid methods for Block-Circulant and Block-Toeplitz Systems

by   Matthias Bolten, et al.

In the literature, there exist several studies on symbol-based multigrid methods for the solution of linear systems having structured coefficient matrices. In particular, the convergence analysis for such methods has been obtained in an elegant form in the case of Toeplitz matrices generated by a scalar-valued function. In the block-Toeplitz setting, that is, in the case where the matrix entries are small generic matrices instead of scalars, some algorithms have already been proposed regarding specific applications and a first rigorous convergence analysis has been performed in [7]. However, with the existent symbol-based theoretical tools, it is still not possible to prove the convergence of many multigrid methods known in the literature. This paper aims to generalize the previous results giving more general sufficient conditions on the symbol of the grid transfer operators.In particular, we treat matrix-valued trigonometric polynomials which can be non-diagonalizable and singular at all points and we express the new conditions in terms of the eigenvectors associated with the ill-conditioned subspace. Moreover, we extend the analysis to the V-cycle method proving a linear convergence rate under stronger conditions, which resemble those given in the scalar case. In order to validate our theoretical findings, we present a classical block structured problem stemming from a FEM approximation of a second order differential problem. We focus on two multigrid strategies that use the geometric and the standard bisection grid transfer operators and we prove that both fall into the category of projectors satisfying the proposed conditions. In addition, using a tensor product argument, we provide a strategy to construct efficient V-cycle procedures in the block multilevel setting.


page 1

page 2

page 3

page 4


Multigrid methods for block-Toeplitz linear systems: convergence analysis and applications

In the past decades, multigrid methods for linear systems having multile...

Symbol Based Convergence Analysis in Block Multigrid Methods with applications for Stokes problems

The main focus of this paper is the study of efficient multigrid methods...

On the matrices in B-spline collocation methods for Riesz fractional equations and their spectral properties

In this work, we focus on a fractional differential equation in Riesz fo...

Symbol based convergence analysis in multigrid methods for saddle point problems

Saddle point problems arise in a variety of applications, e.g., when sol...

Krylov subspace methods for the solution of linear Toeplitz systems

In this thesis we study the preconditioning of square, non-symmetric and...

An effcient block rational Krylov solver for Sylvester equations with adaptive pole selection

We present an algorithm for the solution of Sylvester equations with rig...

Approximations of inverse block Toeplitz matrices and Baxter-type theorems for long-memory processes

We derive sharp approximation error bounds for inverse block Toeplitz ma...

Please sign up or login with your details

Forgot password? Click here to reset