Seminar in Numerical Analysis: Angela Kunoth (Universität Paderborn)
Optimization problems constrained by linear parabolic evolution PDEs are challenging from a computational point of view, as they require to solve a system of PDEs coupled globally in time and space. For their solution, conventional time-stepping methods quickly reach their limitations due to the enormous demand for storage. For such a coupled PDE system, adaptive methods which aim at distributing the available degrees of freedom in an a-posteriori-fashion to capture singularities in the data or domain, with respect to both space and time, appear to be most promising. Employing wavelet schemes for full weak space-time formulations of the parabolic PDEs, we can prove convergence and optimal complexity.
Yet another level of challenge are control problems constrained by evolution PDEs involving stochastic or countably many infinite parametric coefficients: for each instance of the parameters, this requires the solution of the complete control problem.
Our method of attack is based on the following new theoretical paradigm. It is first shown for control problems constrained by evolution PDEs, formulated in full weak space-time form, that state, costate and control are analytic as functions depending on these parameters. Moreover, we establish that these functions allow expansions in terms of sparse tensorized generalized polynomial chaos (gpc) bases. Their sparsity is quantified in terms of p-summability of the coefficient sequences for some 0 < p <= 1. Resulting a-priori estimates establish the existence of an index set, allowing for concurrent approximations of state, co-state and control for which the gpc approximations attain rates of best N-term approximation. These findings serve as the analytical foundation for the development of corresponding sparse realizations in terms of deterministic adaptive Galerkin approximations of state, co-state and control on the entire, possibly infinite-dimensional parameter space.
The results were obtained with Max Gunzburger (Florida State University) and with Christoph Schwab (ETH Zuerich).
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