Parameter Estimation in the Hermitian and Skew-Hermitian Splitting Method Using Gradient Iterations

TL;DR

This paper improves the Hermitian and skew-Hermitian splitting method by using gradient iterations for better parameter estimation, leading to enhanced convergence and stability, especially with early stopping and lagged gradient methods.

Contribution

It introduces gradient-based strategies for parameter estimation in the splitting method, improving convergence and stability over arbitrary choices.

Findings

Early stopping with steepest descent provides good parameter estimates.

Lagged gradient methods are competitive with conjugate gradient in low precision.

Enhanced parameter estimation improves convergence and stability.

Abstract

This paper presents enhancement strategies for the Hermitian and skew-Hermitian splitting method based on gradient iterations. The spectral properties are exploited for the parameter estimation, often resulting in a better convergence. In particular, steepest descent with early stopping can generate a rough estimate of the optimal parameter. This is better than an arbitrary choice since the latter often causes stability problems or slow convergence. Additionally, lagged gradient methods are considered as inner solvers for the splitting method. Experiments show that they are competitive with conjugate gradient in low precision.