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Symbol-based multilevel block $τ$ preconditioners for multilevel block Toeplitz systems: GLT-based analysis and applications

Sean Y. Hon, Congcong Li, Rosita L. Sormani, Rolf Krause, Stefano Serra-Capizzano

Abstract

In recent years, there has been a renewed interest in preconditioning for multilevel Toeplitz systems, a research field that has been extensively explored over the past several decades. This work introduces novel preconditioning strategies using multilevel $τ$ matrices for both symmetric and nonsymmetric multilevel Toeplitz systems. Our proposals constitute a general framework, as they are constructed solely based on the generating function of the multilevel Toeplitz coefficient matrix, when it can be defined. We begin with nonsymmetric systems, where we employ a symmetrization technique by permuting the coefficient matrix to produce a real symmetric multilevel Hankel structure. We propose a multilevel $τ$ preconditioner tailored to the symmetrized system and prove that the eigenvalues of the preconditioned matrix sequence cluster at $\pm 1$, leading to rapid convergence when using the preconditioned minimal residual method. The high effectiveness of this approach is demonstrated through its application in solving space fractional diffusion equations. Next, for symmetric systems we introduce another multilevel $τ$ preconditioner and show that the preconditioned conjugate gradient method can achieve an optimal convergence rate, namely a rate that is independent of the matrix size, when employed for a class of ill-conditioned multilevel Toeplitz systems. Numerical examples are provided to critically assess the effectiveness of our proposed preconditioners compared to several leading existing preconditioned solvers, highlighting their superior performance.

Symbol-based multilevel block $τ$ preconditioners for multilevel block Toeplitz systems: GLT-based analysis and applications

Abstract

In recent years, there has been a renewed interest in preconditioning for multilevel Toeplitz systems, a research field that has been extensively explored over the past several decades. This work introduces novel preconditioning strategies using multilevel matrices for both symmetric and nonsymmetric multilevel Toeplitz systems. Our proposals constitute a general framework, as they are constructed solely based on the generating function of the multilevel Toeplitz coefficient matrix, when it can be defined. We begin with nonsymmetric systems, where we employ a symmetrization technique by permuting the coefficient matrix to produce a real symmetric multilevel Hankel structure. We propose a multilevel preconditioner tailored to the symmetrized system and prove that the eigenvalues of the preconditioned matrix sequence cluster at , leading to rapid convergence when using the preconditioned minimal residual method. The high effectiveness of this approach is demonstrated through its application in solving space fractional diffusion equations. Next, for symmetric systems we introduce another multilevel preconditioner and show that the preconditioned conjugate gradient method can achieve an optimal convergence rate, namely a rate that is independent of the matrix size, when employed for a class of ill-conditioned multilevel Toeplitz systems. Numerical examples are provided to critically assess the effectiveness of our proposed preconditioners compared to several leading existing preconditioned solvers, highlighting their superior performance.
Paper Structure (14 sections, 17 theorems, 96 equations, 4 tables)

This paper contains 14 sections, 17 theorems, 96 equations, 4 tables.

Table of Contents

  1. Introduction
  2. Preliminaries on spectral tools
  3. Notation
  4. Distribution and clustering
  5. Multilevel block Toeplitz matrices
  6. Multilevel block GLT sequences
  7. Multilevel m matrices
  8. Spectral properties of symmetrized multilevel Toeplitz matrices
  9. The block setting
  10. Spectral analysis and preconditioning proposal for nonsymmetric multilevel Toeplitz systems
  11. Extension to the multilevel case
  12. Spectral analysis and preconditioning proposal for ill-conditioned symmetric multilevel Toeplitz systems
  13. Numerical examples
  14. Conclusions

Key Result

Proposition 1

If $\{A_n\}_n\sim_\lambda\psi$, then $\{A_n\}_n$ is weakly clustered at the essential range of $\psi$ in the sense of the eigenvalues. Furthermore, if the essential range of $\psi$ is the singleton $\{s\}$ with $s\in\mathbb{C}$ fixed, then $\{A_n\}_n\sim_\lambda\psi$ if and only if $\{A_n\}_n$ is we

Theorems & Definitions (39)

  • Definition 1
  • Definition 2
  • Proposition 1
  • Definition 3
  • Lemma 1
  • Proposition 2
  • Proposition 3
  • proof
  • Theorem 4
  • Theorem 5
  • ...and 29 more