An Efficient Scaled spectral preconditioner for sequences of symmetric positive definite linear systems

TL;DR

This paper introduces a scaled spectral preconditioner designed to accelerate the convergence of conjugate gradient methods for sequences of symmetric positive-definite linear systems, especially in early iterations, with minimal additional computational cost.

Contribution

The paper proposes a novel scaled spectral preconditioner with three strategies for selecting the scaling parameter, enhancing early convergence of CG in solving linear systems.

Findings

Significantly improves early CG convergence in numerical experiments.

Achieves this acceleration with negligible additional computational cost.

Effective in data assimilation applications.

Abstract

We explore a scaled spectral preconditioner for the efficient solution of sequences of symmetric and positive-definite linear systems. We design the scaled preconditioner not only as an approximation of the inverse of the linear system but also with consideration of its use within the conjugate gradient (CG) method. We propose three different strategies for selecting a scaling parameter, which aims to position the eigenvalues of the preconditioned matrix in a way that reduces the energy norm of the error, the quantity that CG monotonically decreases at each iteration. Our focus is on accelerating convergence especially in the early iterations, which is particularly important when CG is truncated due to computational cost constraints. Numerical experiments provide in data assimilation confirm that the scaled spectral preconditioner can significantly improve early CG convergence with negligible computational cost.