Toward Efficient Polynomial Preconditioning for GMRES

TL;DR

This paper introduces a stable, efficient polynomial preconditioner derived from the GMRES polynomial, which enhances convergence of GMRES and potentially other Krylov methods for large linear systems.

Contribution

The paper proposes a new polynomial preconditioning method based on the GMRES polynomial that is easier to compute and more stable than previous approaches.

Findings

Improved convergence for difficult linear systems.

Enhanced stability through root-adding and stability checks.

Potential for significant reduction in dot products during iterations.

Abstract

We present a polynomial preconditioner for solving large systems of linear equations. The polynomial is derived from the minimum residual polynomial (the GMRES polynomial) and is more straightforward to compute and implement than many previous polynomial preconditioners. Our current implementation of this polynomial using its roots is naturally more stable than previous methods of computing the same polynomial. We implement further stability control using added roots, and this allows for high degree polynomials. We discuss the effectiveness and challenges of root-adding and give an additional check for stability. In this paper, we study the polynomial preconditioner applied to GMRES; however it could be used with any Krylov solver. This polynomial preconditioning algorithm can dramatically improve convergence for some problems, especially for difficult problems, and can reduce dot products by an even greater margin.