A perturbed preconditioned gradient descent method for the unconstrained minimization of composite objectives

Abstract

We introduce a perturbed preconditioned gradient descent (PPGD) method for the unconstrained minimization of a strongly convex objective $G$ with a locally Lipschitz continuous gradient. We assume that $G(v)=E(v)+F(v)$ and that the gradient of $F$ is only known approximately. Our analysis is conducted in infinite dimensions with a preconditioner built into the framework. We prove a linear rate of convergence, up to an error term dependent on the gradient approximation. We apply the PPGD to the stationary Cahn-Hilliard equations with variable mobility under periodic boundary conditions. Numerical experiments are presented to validate the theoretical convergence rates and explore how the mobility affects the computation.