Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

Admissible and attainable convergence behavior with stagnation mirroring in restarted (block) GMRES

Kirk M. Soodhalter

TL;DR

This work addresses the problem of prescribing restarted block GMRES convergence patterns by constructing matrices and block right-hand sides that yield a desired residual sequence, while decoupling eigenvalues and Ritz values from the convergence. The authors extend the scalar GMRES framework to Bl-Gmres using a right-vector-space viewpoint with $p\times p$ scalars and develop restarted Krylov-factorization constructions that realize arbitrary per-cycle eigenvalues and Ritz values under the mirroring of stagnation. Key contributions include a proof of stagnation mirroring for restarted Bl-Gmres, a detailed method to build restarted block factorizations, and a characterization of residual and Hessenberg structures for both no-end-of-cycle stagnation and end-of-cycle stagnation cases. These results enable precise design of block linear systems with prescribed convergence, providing a foundation for theoretical analysis and potential preconditioner-informed design, while noting current limitations such as block Arnoldi breakdown.

Abstract

In this work, we describe how to construct matrices and block right-hand sides the exhibit a specified restarted block \gmres convergence pattern, such that the eigenvalues and Ritz values at each iteration can be chosen independent of the specified convergence behavior. This work is a generalization of the work in [Meurant and Tebbens, Num. Alg. 2019] in which the authors do the same for restarted non-block \gmres. We use the same tools as were used in [Kubínová and Soodhalter, SIMAX 2020], namely to analyze block \gmres as an iteration over a right vector space with scalars from the $^\ast$-algebra of matrices. To facilitate our work, we also extend the work of Meurant and Tebbens and offer alternative proofs of some of their results, that can be more easily generalized to the block setting.

Admissible and attainable convergence behavior with stagnation mirroring in restarted (block) GMRES

TL;DR

This work addresses the problem of prescribing restarted block GMRES convergence patterns by constructing matrices and block right-hand sides that yield a desired residual sequence, while decoupling eigenvalues and Ritz values from the convergence. The authors extend the scalar GMRES framework to Bl-Gmres using a right-vector-space viewpoint with scalars and develop restarted Krylov-factorization constructions that realize arbitrary per-cycle eigenvalues and Ritz values under the mirroring of stagnation. Key contributions include a proof of stagnation mirroring for restarted Bl-Gmres, a detailed method to build restarted block factorizations, and a characterization of residual and Hessenberg structures for both no-end-of-cycle stagnation and end-of-cycle stagnation cases. These results enable precise design of block linear systems with prescribed convergence, providing a foundation for theoretical analysis and potential preconditioner-informed design, while noting current limitations such as block Arnoldi breakdown.

Abstract

In this work, we describe how to construct matrices and block right-hand sides the exhibit a specified restarted block \gmres convergence pattern, such that the eigenvalues and Ritz values at each iteration can be chosen independent of the specified convergence behavior. This work is a generalization of the work in [Meurant and Tebbens, Num. Alg. 2019] in which the authors do the same for restarted non-block \gmres. We use the same tools as were used in [Kubínová and Soodhalter, SIMAX 2020], namely to analyze block \gmres as an iteration over a right vector space with scalars from the -algebra of matrices. To facilitate our work, we also extend the work of Meurant and Tebbens and offer alternative proofs of some of their results, that can be more easily generalized to the block setting.

Paper Structure

This paper contains 19 sections, 26 theorems, 109 equations.

Table of Contents

  1. Introduction
  2. Gmres and admissible convergence
  3. Admissible and attainable convergence behavior
  4. Prescribing behavior of restarted Gmres
  5. Admissible convergence and the mirroring of stagnation
  6. Building a restarted Gmres factorization
  7. Residual components in the Arnoldi vectors
  8. Generating the prescribed Hessenberg matrices in restarted Arnoldi
  9. No stagnation at the end of any cycle
  10. Stagnation at the end of a cycle
  11. Bl-Gmres admissible convergence behavior
  12. Extension to restarted Bl-Gmres
  13. Mirroring of stagnation behavior
  14. Building a restarted Bl-Gmres factorization
  15. Residual components in the block Arnoldi vectors
  16. ...and 4 more sections

Key Result

proposition 1

For any admissible sequence $f_j _{j=0}^n$ defined as in definition:admissible-convergence, a matrix-right-hand-side pair $(A,{\boldsymbol{b}})$ can be constructed such that $A$ can be specified to have any set of eigenvalues; Gmres applied to eqn:Axb with ${\boldsymbol{x}}_0={\boldsymbol{0}}$ produ

Theorems & Definitions (57)

  • definition 1
  • definition 2
  • proposition 1: paraphrasing results from GreenbaumPtakStrakos:1996:1TebbensMeurant:2012:1
  • proposition 2
  • proof : Proof of \ref{['prop:simplified-stagnation-mirroring']}
  • remark 1
  • definition 3
  • proposition 3
  • proof
  • lemma 1
  • ...and 47 more