A Spectral Preconditioner for the Conjugate Gradient Method with Iteration Budget

TL;DR

This paper introduces a spectral preconditioning strategy for the conjugate gradient method that optimizes eigenvalue clustering to improve early iteration accuracy in large symmetric positive-definite systems.

Contribution

It formulates spectral preconditioner design as a constrained optimization problem and proposes practical, low-cost strategies for selecting parameters to enhance initial convergence.

Findings

Significant error reduction in early iterations with optimized spectral preconditioning.

Optimal eigenvalue clustering improves the energy norm error during initial PCG steps.

Numerical experiments confirm the effectiveness of the proposed strategies.

Abstract

We study the solution of large symmetric positive-definite linear systems in a matrix-free setting with a limited iteration budget. We focus on the preconditioned conjugate gradient (PCG) method with spectral preconditioning. Spectral preconditioners map a subset of eigenvalues to a positive cluster via a scaling parameter, and leave the remainder of the spectrum unchanged, in hopes to reduce the number of iterations to convergence. We formulate the design of the spectral preconditioners as a constrained optimization problem. The optimal cluster placement is defined to minimize the error in energy norm at a fixed iteration. This optimality criterion provides new insight into the design of efficient spectral preconditioners when PCG is stopped short of convergence. We propose practical strategies for selecting the scaling parameter, hence the cluster position, that incur negligible computational cost. Numerical experiments highlight the importance of cluster placement and demonstrate significant improvements in terms of error in energy norm, particularly during the initial iterations.