POD-Based A-posteriori Error Estimation for Control Problems Governed by Nonlinear PDEs

Abstract

In this thesis a reliable and (numerical) efficient a-posteriori error estimation for reduced order nonlinear control constrained optimization problems, governed by semilinear partial differential equations (PDEs) of parabolic type, is approached. For optimization the general concept of line search methods in form of a projected (Quasi-)Newton approach is applied. The reduced order model (ROM) is realised by the method of proper orthogonal decomposition (POD) in combination with an empirical interpolation method (EIM) to deal with the nonlinear term. The computation of the smallest eigenvalue of the Hessian, which is a substantial ingredient for the numerical estimation of the a-posteriori error, is investigated by utilizing four diverging concepts of providing second-order derivative information. In course of the work, this task is specified for parametrized optimal control problems, which are subject to parameter-invariant PDE constraints. A grid-based eigenvalue approximation approach is presented, that is used to formulate a weak form of the a-posteriori error estimate to be utilized in a POD-greedy algorithm for reduced basis (RB) generation. This then findsnumerical realisation and application in the context of a multiobjective optimal control problem (MOCP). Numerical examples and results are presented and discussed at relevant positions.