Group Seminar: Computational Methods for PDEs
Date: Tuesday, November 7, 2023
Time: 3:30 pm, S2 416-1
Abstract: We consider a distributed optimal control problem constrained by the partial differential equation (PDE) $By=u$, with the goal, to reach a given target $y_d$ with some given accuracy under minimal costs. Assuming that the operator $B:Y\to X^\ast$ defines an isomorphism, this problem will be analyzed on an abstract level. For a conforming trial space $Y_h\subset Y$ this will lead to quasi optimal estimates for the distance between the computable state $y_h\in Y_h$ and the desired target $y_d$, depending on the regularity of the target and the cost parameter. The control $u_H\in U_H\subset X^\ast$ is then reconstructed in a post processing step, for which a rigorous analysis will be given. Moreover, the choice of different, but equivalent, norms for the control and the incorporation of state constraints will be discussed. While this framework applies to a wide class of PDEs, we will consider the Poisson equation as a model problem to apply the proposed methods. Numerical examples, including an adaptive refinement scheme and a discontinuous target, will complement the theory.
This talk is based on joint work with Ulrich Langer (Linz) and Huidong Yang (Wien).