Nonlinear Schwarz preconditioning for Quasi-Newton methods

TL;DR

This paper introduces a nonlinear restricted additive Schwarz preconditioning strategy to enhance the convergence of limited memory quasi-Newton methods, addressing the challenge of constructing secant pairs in the preconditioned framework.

Contribution

It presents a novel nonlinear preconditioning approach for quasi-Newton methods, including strategies for secant pair construction and demonstrates improved robustness and efficiency.

Findings

Preconditioned QN methods show faster convergence in experiments.

The proposed approach is robust across different problem instances.

Numerical results validate the effectiveness of the nonlinear Schwarz preconditioning.

Abstract

We propose the nonlinear restricted additive Schwarz (RAS) preconditioning strategy to improve the convergence speed of limited memory quasi-Newton (QN) methods. We consider both "left-preconditioning" and "right-preconditioning" strategies. As the application of the nonlinear preconditioning changes the standard gradients and Hessians to their preconditioned counterparts, the standard secant pairs cannot be used to approximate the preconditioned Hessians. We discuss how to construct the secant pairs in the preconditioned QN framework. Finally, we demonstrate the robustness and efficiency of the preconditioned QN methods using numerical experiments.