Trust Region POD for Optimal Boundary Control of a Semilinear Heat Equation

Abstract

In this diploma thesis an optimal control problem governed by a semilinear heat equation is considered. The problem is formulated as a reduced problem by including the semilinear heat equation in the formulation of the cost functional. This nonlinear reduced problem is numerically solved by a globalized inexact Newton method. The inexact Newton steps are computed with a conjugate gradient (CG) algorithm. In a first approach, an Armijo backtracking strategy is chosen for globalization of the Newton-CG method. A classical Finite Element Galerkin technique is used for spatial discretization. To reduce the computational effort a model reduction approach based on proper orthogonal decomposition (POD) is applied. A control which is utilized to set up the POD basis has to be chosen at the beginning and the reduced-order models (ROMs) are fixed during optimization. If the required control is chosen badly, few POD basis functions do not suffice to obtain good POD suboptimal controls. To overcome this problem the reduced-order Newton-CG strategy is embedded in a trust region framework, where the POD basis and hence the ROMs are improved successively by utilizing the updated control values. The proposed methods are tested by numerical examples. In particular, the adaptation of the POD basis when applying the trust region POD strategy is analyzed.