Weighted Inner Products for GMRES and GMRES-DR

TL;DR

This paper investigates how weighted inner products, including the discrete cosine transform, can enhance GMRES convergence, especially for problems with localized eigenvectors or when combined with deflation techniques.

Contribution

It introduces the use of weighted inner products, notably W-GMRES-DCT, and explores their effectiveness in improving GMRES convergence for specific problem classes.

Findings

Weighted inner products can improve convergence for localized eigenvector problems.

Incorporating DCT into the inner product significantly enhances GMRES performance.

Weighting combined with deflation techniques like GMRES-DR is effective.

Abstract

The convergence of the restarted GMRES method can be significantly improved, for some problems, by using a weighted inner product that changes at each restart. How does this weighting affect convergence, and when is it useful? We show that weighted inner products can help in two distinct ways: when the coefficient matrix has localized eigenvectors, weighting can allow restarted GMRES to focus on eigenvalues that otherwise slow convergence; for general problems, weighting can break the cyclic convergence pattern into which restarted GMRES often settles. The eigenvectors of matrices derived from differential equations are often not localized, thus limiting the impact of weighting. For such problems, incorporating the discrete cosine transform into the inner product can significantly improve GMRES convergence, giving a method we call W-GMRES-DCT. Integrating weighting with eigenvalue deflation via GMRES-DR also can give effective solutions.