Speaker
Description
PDE constraint optimization problems arise in a wide field of application and therefore the numerical analysis of these problems is a highly relevant research topic. In this talk, we consider a time-dependent optimization problem with a tracking type objective function and standard regularization. The equality constraint is a parabolic linear time-dependent PDE. We also consider additional box constraints on the control variable.
There are several approaches to solve these optimization problems, one can first discretize then optimize, or first optimize and then discretize and solve the resulting system. This is usually done by using applying some time-stepping scheme and using a different discretization in space. We follow the approach to first optimize then discretize the problem. We utilize a space-time variational formulation of the problem and apply a semi-smooth Newton method in the occurring Lebesgue-Bochner spaces. For the discretization, a simultaneous finite element discretization in time and space is used.
At last year's annual GAMM conference, we presented the theory of the application of the semi-smooth Newton method in our setting. This year we want to take a closer look at the discretization and the implementation of this approach. The discretization of the Newton systems for each iteration leads to unsymmetric saddle point systems. If the discretization is done in a naive approach this leads to very large sparse systems which are quickly becoming infeasible to solve. To mitigate this we use a tensor discretization approach which greatly reduces the storage needed for the discretized systems.
In this presentation we want to show: - how to develop an iterative matrix-free solver for the arising Newton system, - the discretization leading to Kronecker structures, - how we can exploit the Kronecker structures that arise from the chosen discretization for an efficient implementation, - the possibilities for parallelization using this approach, - numerical results for large scale systems.
