Dual Regularization and Outer Approximation of Optimal Control Problems with Controls of Bounded Variation

• Applicant: Professor Dr. Christian Meyer
• Subject Area: Mathematics
• Term: since 2024
• Project identifier: Deutsche Forschungsgemeinschaft (DFG) - Project number 552141670

While it is meanwhile standard to use the TV-seminorm in imaging and for the regularization of inverse problems, optimal control problems with controls in the space of functions of bounded variations have only recently come to the focus of applied mathematicians. This concerns theoretical aspects such as the derivation of necessary and sufficient optimality conditions as well as numerical and algorithmic issues. Most of the algorithmic approaches so far are based on a regularization of the control and/or a discretization of the control by means of continuous ansatz functions. Both approaches imply that the numerical solution is continuous, although one of the main motivations for taking the space of functions of bounded variation as control space is to allow for discontinuous solutions. The main goal of the project is therefore to develop an algorithmic concept that on the one hand allows for discontinuous numerical solutions and on the other hand admits a rigorous convergence analysis in function space, which forms the basis for mesh independence of the algorithm. The basic idea of the algorithmic approach is a tailored regularization of the dual formulation of the TV-seminorm, which leads to an enlarged control space. The controls are therefore approximated "from the outside" such that discontinuous controls are feasible as desired. The second essential idea of the algorithmic concept is to solve the dual regularized problems by means of an outer approximation algorithm, where infeasible controls are cut off by suitable cutting planes. This leads to an algorithm that can be formulated in function space and just contains subproblems that can be solved by standard methods. We aim to develop the algorithmic concept by means of a model hierarchy with three model problems of different level of difficulty. We start with a strictly convex problem with a linear elliptic partial differential equation (PDE) and a quadratic objective, followed by a problem with a semi-linear elliptic PDE and finally we treat a model problem, where the control appears as coefficient in the main part of the differential operator. The development of an efficient and mathematically sound algorithm for this third, most complex model problem is the ultimate goal of the project.

• DFG Programme: Research Grants