Session: Recent Advances in PDE-Constrained Optimizatin
Chair: Michael Ulbrich
Cluster: PDE-constrained Optimization
Talk 1: Probabilistic state constraints for optimal control problems under uncertainty
Speaker: Caroline Geiersbach
Abstract: In this talk, we discuss optimal control problems subject to random state constraints, where we distinguish between the chance-constrained case and the almost sure formulation. We highlight some of the difficulties in the infinite-dimensional setting, which is of interest in physics-based models where a control belonging to a Banach space acts on a system described by a partial differential equation (PDE) with random inputs or parameters. We study the setting in which the obtained state should be bounded uniformly over the physical domain with high probability, or even probability one. We apply our results to a model with a random elliptic PDE, where the randomness is induced by the right-hand side. For the chance-constrained setting, this structure allows us to obtain an explicit representation for the Clarke subdifferential of the probability function using the spherical radial decomposition of Gaussian random vectors. This representation is used for the numerical solution in a discretize-then-optimize approach. For the almost sure setting, we use a Moreau-Yosida regularization and solve a sequence of regularized problems in an optimize-then-discretize approach. The solutions are compared, providing insights for the development of further algorithms.
Talk 2: Image Registration in Non-Reflexive Banach Spaces
Speaker: Johannes Haubner
Abstract: We consider image registration as an optimal control problem using an optical flow formulation, i.e., we aim to solve an optimization problem that is governed by a linear hyperbolic transport equation. In order to be able to transform the image of a square into the image of a circle and ensure bi-Lipschitz continuity of the transformation, we aim for $W^{1,\infty}$-regularity of the velocity that parametrizes the transformation. This leads to an optimization problem in the non-reflexive Banach space $W^{1,\infty}$. We introduce relaxations of the optimization problem involving smoothed maximum and minimum functions and the Orlicz space $L_{\exp}$. To derive well-posedness results for the relaxed optimization problem, we revisit and establish new existence and uniqueness results for the linear hyperbolic transport equations. We present limit considerations and discuss differentiability.
Talk 3: Numerical methods for shape optimal shpe design of fluid-structure interaction problems
Speaker: Michael Ulbrich
Abstract: We consider the method of mappings for performing shape optimization for unsteady fluid–structure interaction (FSI) problems. The main focus is on the numerical implementation. Our modelling of the optimization problem is guided by several theoretical aspects, such as regularity requirements on the transformations and connections to differential geometric aspects. We discretize the problem in a way such that exact discrete gradients can be computed. This in combination with the way we represent shapes allows for an efficent application of general purpose optimization solvers. Our numerical tests focus on problems derived from an FSI benchmark to validate our implementation, which builds on FEniCS, dolfin-adjoint, and IPOPT. The method is used to optimize parts of the outer boundary and the interface.
