A restarted GMRES-based implementation of IDR(s)stab(L) to yield higher robustness

TL;DR

This paper introduces a new IDRstab implementation based on restarted GMRES that enhances robustness against breakdowns and can leverage lucky breakdowns for exact solutions, improving convergence and accuracy.

Contribution

A novel IDRstab implementation using restarted GMRES that avoids breakdowns and exploits lucky breakdowns for improved robustness and efficiency.

Findings

Demonstrates superior robustness in convergence

Achieves higher numerical accuracy

Effectively exploits lucky breakdown scenarios

Abstract

In this thesis we propose a novel implementation of IDRstab that avoids several unlucky breakdowns of current IDRstab implementations and is further capable of benefiting from a particular lucky breakdown scenario. IDRstab is a very efficient short-recurrence Krylov subspace method for the numerical solution of linear systems. Current IDRstab implementations suffer from slowdowns in the rate of convergence when the basis vectors of their oblique projectors become linearly dependent. We propose a novel implementation of IDRstab that is based on a successively restarted GMRES method. Whereas the collinearity of basis vectors in current IDRstab implementations would lead to an unlucky breakdown, our novel IDRstab implementation can strike a benefit from it in that it terminates with the exact solution whenever a new basis vector lives in the span of the formerly computed basis vectors. Numerical experiments demonstrate the superior robustness of our novel implementation with regards to convergence maintenance and the achievable accuracy of the numerical solution.