Nonlinear Schwarz preconditioning for Quasi-Newton methods

TL;DR

The work targets speeding up nonlinear FE-discretized Quasi-Newton (QN) methods by introducing nonlinear restricted additive Schwarz (NRAS) preconditioning for quasi-Newton methods. Left and right preconditioning are formalized with residuals $F_L(x)= x - G(x)$ and $F_R(x)= F(G(x))$, and the secant equations become $B_L^{(k+1)} s_L^{(k)}= y_L^{(k)}$ and $B_R^{(k+1)} s_R^{(k)}= y_R^{(k)}$, with $s_L^{(k)}= x^{(k+1)} - x^{(k)}$ and $s_R^{(k)}= x^{(k+1)} - x^{(+)}$. A two-level NRAS scheme introduces a coarse correction to couple subdomains and achieve scalability, while memory-limited variants use L-BFGS/AA-I updates on the local problems. Numerical experiments demonstrate improved convergence speed and robustness over standard L-BFGS and Newton, especially for memory-constrained nonlinear PDE solves.

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.

Figures (1)

• Figure 1: Convergence history of the standard QN methods, preconditioned QN methods, Newton's method, and the TL-NRAS method for solving the minimal surface problem. The QN methods are configured to use the last $7$ secant pairs.