A modified implementation of MINRES to monitor residual subvector norms for block systems

TL;DR

This paper introduces a modified MINRES algorithm tailored for saddle-point systems that enables separate monitoring of residual subvector norms without significant additional computational cost.

Contribution

The paper presents a novel modification of the MINRES method allowing individual residual subvector norm monitoring with minimal extra storage and no additional preconditioner applications.

Findings

Effective monitoring of residual subvectors demonstrated in numerical experiments.

Minimal additional storage required for the modified algorithm.

No extra preconditioner applications needed compared to standard MINRES.

Abstract

Saddle-point systems, i.e., structured linear systems with symmetric matrices are considered. A modified implementation of (preconditioned) MINRES is derived which allows to monitor the norms of the subvectors individually. Compared to the implementation from the textbook of [Elman, Sylvester and Wathen, Oxford University Press, 2014], our method requires one extra vector of storage and no additional applications of the preconditioner. Numerical experiments are included.