BTU overview
Faculty 1
Chair of Optimal Control Prof. Dr. rer. nat. habil. Gerd Wachsmuth

Wasserstein Regularization in Inverse Problems and Optimal Control

General information

Project description

In the recent past optimal control problems and inverse problems in the space of regular Borel measures have attracted considerable attention. This increasing interest is due to the special structure of optimal solutions of these kind of problems that frequently provides a sparsity pattern, which is favorable in many applications. In the vast majority of contributions in this area, the regularity of optimal solutions is ensured by adding the Radon norm to the objective functional. However, this norm is not continuous with respect to the "natural" form of convergence in the space of regular Borel measures, i.e., weak-* convergence. This in turn has a negative impact on the performance of algorithms and approximation methods. The basic idea of this project is to replace the Radon norm by the Wasserstein distance of the control to a given prior. In contrast to the Radon norm this distance is continuous with respect to weak-* convergence.

The project starts with a thorough mathematical analysis of optimal control and inverse problems with Wasserstein regularizer. Based on the theoretical findings, three different algorithmic concepts will be developed to solve optimization problems with Wasserstein regularizer. These concepts are based on the identification of the Wasserstein distance as optimal value of optimal transportation problems. While two algorithmic concepts employ tailored regularization methods for optimal transport problems, another concept uses the special structure of optimal transport plans in case of strictly convex transport costs. By means of these algorithms we will finally solve a prototypical optimal control problem with Wasserstein regularizer subject to a semilinear elliptic partial differential equation.