How to optimize preconditioners for the conjugate gradient method: a stochastic approach

TL;DR

This paper introduces a stochastic method for optimizing preconditioner parameters in the conjugate gradient method by using trial runs with random initial guesses, leading to improved convergence performance.

Contribution

It proposes a novel stochastic functional for preconditioner optimization in CG, focusing on mean convergence rather than worst-case estimates.

Findings

Optimization of the new functional outperforms spectral condition number-based methods

Numerical experiments demonstrate improved convergence rates

The approach adapts preconditioner parameters more effectively

Abstract

The conjugate gradient method (CG) is typically used with a preconditioner which improves efficiency and robustness of the method. Many preconditioners include parameters and a proper choice of a preconditioner and its parameters is often not a trivial task. Although many convergence estimates exist which can be used for optimizing preconditioners, these estimates typically hold for all initial guess vectors, in other words, they reflect the worst convergence rate. To account for the mean convergence rate instead, in this paper, we follow a stochastic approach. It is based on trial runs with random initial guess vectors and leads to a functional which can be used to monitor convergence and to optimize preconditioner parameters in CG. Presented numerical experiments show that optimization of this new functional with respect to preconditioner parameters usually yields a better parameter value than optimization of the functional based on the spectral condition number.