Optimization Problems for PDEs in Weak Space-Time Form

Abstract

Optimization problems constrained by time-dependent Partial Differential Equations (PDEs) are challenging from a computational point of view: even in the simplest case, one needs to solve a system of PDEs coupled globally in time and space for the unknown solutions (the state, the costate and the control of the system). Typical and practically relevant examples are the control of nonlinear heat equations as they appear in laser hardening or the thermic control of flow problems (Boussinesq equations). Specifically for PDEs with a long time horizon, conventional time-stepping methods require an enormous amount of computer memory allocations for the respective other variables. In contrast, adaptive-in-time-and-space methods aim at distributing the available degrees of freedom in an a-posteriori fashion to capture singularities and are, therefore, most promising. Recently, well-posed weak variational formulations have been introduced for time-dependent PDEs such as the heat equation, linear transport and the wave equation. Those formulations also allow for a sharp relation between the approximation error and the residual, which is particularly relevant for model reduction. Moreover, for those tensor-basis formulations, advanced algebraic solvers designed to take into account these multiarray (tensorial) formulations appear to be particularly competitive with respect to time-marching schemes, especially in higher dimensions. We plan to discuss whether these techniques can be extended to nonlinear PDEs like Hamilton-Jacobi-Bellman equations, or stochastic PDEs and variational inequalities. Another topic will be adaptive schemes which, when properly designed, inherit the stability of the continuous formulation.

The central goals of the workshop are the analysis, fast solvers and model reduction for PDE-constrained control and optimization problems based on weak formulations of the underlying PDE(s).