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Analysis on aggregation and block smoothers in multigrid methods for block Toeplitz linear systems

Matthias Bolten, Marco Donatelli, Paola Ferrari, Isabella Furci

TL;DR

The convergence analysis of the Two-Grid Method (TGM) reveals the connection between the features of the scalar-valued symbol at the coarser level and the properties of the original matrix-valued one, allowing us to prove the convergence of a V-cycle multigrid with standard grid transfer operators for scalar Toeplitz systems at the coarser levels.

Abstract

We present novel improvements in the context of symbol-based multigrid procedures for solving large block structured linear systems. We study the application of an aggregation-based grid transfer operator that transforms the symbol of a block Toeplitz matrix from matrix-valued to scalar-valued at the coarser level. Our convergence analysis of the Two-Grid Method (TGM) reveals the connection between the features of the scalar-valued symbol at the coarser level and the properties of the original matrix-valued one. This allows us to prove the convergence of a V-cycle multigrid with standard grid transfer operators for scalar Toeplitz systems at the coarser levels. Consequently, we extend the class of suitable smoothers for block Toeplitz matrices, focusing on the efficiency of block strategies, particularly the relaxed block Jacobi method. General conditions on smoothing parameters are derived, with emphasis on practical applications where these parameters can be calculated with negligible computational cost. We test the proposed strategies on linear systems stemming from the discretization of differential problems with $\mathbb{Q}_{d} $ Lagrangian FEM or B-spline with non-maximal regularity. The numerical results show in both cases computational advantages compared to existing methods for block structured linear systems.

Analysis on aggregation and block smoothers in multigrid methods for block Toeplitz linear systems

TL;DR

The convergence analysis of the Two-Grid Method (TGM) reveals the connection between the features of the scalar-valued symbol at the coarser level and the properties of the original matrix-valued one, allowing us to prove the convergence of a V-cycle multigrid with standard grid transfer operators for scalar Toeplitz systems at the coarser levels.

Abstract

We present novel improvements in the context of symbol-based multigrid procedures for solving large block structured linear systems. We study the application of an aggregation-based grid transfer operator that transforms the symbol of a block Toeplitz matrix from matrix-valued to scalar-valued at the coarser level. Our convergence analysis of the Two-Grid Method (TGM) reveals the connection between the features of the scalar-valued symbol at the coarser level and the properties of the original matrix-valued one. This allows us to prove the convergence of a V-cycle multigrid with standard grid transfer operators for scalar Toeplitz systems at the coarser levels. Consequently, we extend the class of suitable smoothers for block Toeplitz matrices, focusing on the efficiency of block strategies, particularly the relaxed block Jacobi method. General conditions on smoothing parameters are derived, with emphasis on practical applications where these parameters can be calculated with negligible computational cost. We test the proposed strategies on linear systems stemming from the discretization of differential problems with Lagrangian FEM or B-spline with non-maximal regularity. The numerical results show in both cases computational advantages compared to existing methods for block structured linear systems.
Paper Structure (19 sections, 6 theorems, 66 equations, 3 figures, 12 tables, 1 algorithm)

This paper contains 19 sections, 6 theorems, 66 equations, 3 figures, 12 tables, 1 algorithm.

Table of Contents

  1. Introduction
  2. Multigrid methods for structured linear systems
  3. Multigrid methods for block-circulant and block-Toeplitz matrices
  4. Aggregation and Block Smoothers: MGM Convergence analysis
  5. Grid Transfer Operator and Symbol at the Coarser Level
  6. Smoothing Property for Block Jacobi
  7. Approximation Property
  8. Multigrid Convergence and Optimality
  9. Choice of Optimal Parameters
  10. Numerical Experiments
  11. Examples
  12. Scalar Toeplitz Matrices Interpreted as Block Toeplitz Matrices
  13. Stiffness Matrices Using $\mathbb{Q}_{\textcolor{black}{d}}$ Lagrangian FEM
  14. Stiffness matrices Using B-Splines
  15. Scalar and Block smoothers in matrix-valued multigrid approach
  16. ...and 4 more sections

Key Result

Theorem 1

(RStub) Let $A_{n}$ be a positive definite matrix of size $n$ and let $V_{n,{\rm post}},$$V_{n,{\rm pre}}$ be defined as in the TGM algorithm. Assume Then $\gamma\geq a_{\rm{post}}$ and

Figures (3)

  • Figure 1: Plot of the magnitude of the spectral radius of the TGM iteration matrix for the system $C_n(\mathbf{f}^{[2]})$ computed over equispaced values of the pair $(\alpha,\omega)$.
  • Figure 2: Plot of the magnitude of the spectral radius of the TGM iteration matrix for the system $C_n(\mathbf{\textbf{f}}_{\mathbb{Q}_2})$ computed over 15 equispaced values of the pair $(\alpha,\omega)$.
  • Figure 3: Plot of the magnitude of the spectral radius of the TGM iteration matrix for the system $A_{(2,0)}$ computed over equispaced values of the pair $(\alpha,\omega)$.

Theorems & Definitions (13)

  • Theorem 1
  • Lemma 1
  • proof
  • Theorem 2
  • proof
  • Remark 1
  • Theorem 3
  • proof
  • Theorem 4
  • proof
  • ...and 3 more