RB-Based PDE-Constrained Non-Smooth Optimization

Abstract

We investigate the Reduced Basis (RB) method for a semilinear, non-smooth, parameter dependent, elliptic PDE with a max-type response term, the so called max-PDE. Thereby, we first show the existence and uniqueness of so- lutions to the max-PDE with the help of monotone operator theory, as well as the Lipschitz continuity and compactness of the solution operator. Based on that we introduce a RB-greedy method, analyze its convergence behav- ior, introduce error estimates and an a posteriori error estimator. To solve the nonlinear equation on Finite Element (FE) and RB-level, a semismooth Newton method is used. The necessary theory on subdifferentials and semis- moothness is introduced and subsequently locally quadratic convergence of the semismooth Newton method for the max-PDE is shown. Additionally, the Discrete Empirical Interpolation method (DEIM) is used to approximate the nonlinearity in the RB-system. As application optimization problems con- strained by the parameter dependent max-PDE are analyzed. The theoretical results are verified through numerical examples for both the max-PDE itself and the optimization problems. It becomes apparent that RB can be a suitable approach to solve the max-PDE more efficient.