Nonlinear Schwarz preconditioning for Quasi-Newton methods

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.