Stagnation of block GMRES and its relationship to block FOM

TL;DR

This paper investigates the convergence and stagnation phenomena of block GMRES, relating it to block FOM and principal angles, and introduces generalized methods to handle breakdowns, supported by numerical examples.

Contribution

It provides a theoretical analysis of block GMRES stagnation, generalizes block FOM to handle breakdowns, and links stagnation to principal angles and Arnoldi vectors.

Findings

Stagnation relates to the column space of Arnoldi vectors.

Generalized block FOM can generate solutions during breakdowns.

Numerical examples validate the theoretical analysis.

Abstract

We analyze the the convergence behavior of block GMRES and characterize the phenomenon of stagnation which is then related to the behavior of the block FOM method. We generalize the block FOM method to generate well-defined approximations in the case that block FOM would normally break down, and these generalized solutions are used in our analysis. This behavior is also related to the principal angles between the column-space of the previous block GMRES residual and the current minimum residual constraint space. At iteration $j$, it is shown that the proper generalization of GMRES stagnation to the block setting relates to the columnspace of the $j$th block Arnoldi vector. Our analysis covers both the cases of normal iterations as well as block Arnoldi breakdown wherein dependent basis vectors are replaced with random ones. Numerical examples are given to illustrate what we have proven, including a small application problem to demonstrate the validity of the analysis in a less pathological case.