Project C9 – M. Hinze: Sparse discretization of sparse control problems
Scientific Background and Motivation
Optimal control theory is an active field of research in applied mathematics. Sparse control problems in particular are interesting in the context of source identification and optimal actuator placement. Especially problems with measure-valued controls have nice sparsity properties, which can be expressed via optimality conditions. To numerically solve these problems a discretization of the problem is necessary. Obviously the choice of the discretization strategy has an immediate impact on the sparsity of the discrete problem. The occurring question is: How can we discretize sparse control problems to retain their sparsity structure?
Aims and Objectives
We aim at comparing different discretization strategies and their application to sparse control problems. To this end we intend to analyze problems with the mentioned properties, for example optimal control problems governed by parabolic partial differential equations with space-time measure controls. By derivation of optimality conditions the inherited sparsity structure of the problem can be observed. Adjusted to these findings we aim at determining the discretization strategy that retains as much as possible of the problems sparsity structure. Of special interest will be the comparison of full discretization to the variational discretization concept. By not discretizing the control space we may be able to adapt to the sparsity structure and derive a naturally occurring implicitly discrete space of controls. Another benefit of the variational discretization worth examining is the impact the choice of the test space has on the discrete sparsity properties of the control. The comparison of different discretization approaches could be quantified via error estimates. To support theoretical findings we intend to provide a numerical solution and visualization.
PhD student: Evelyn Christin Herberg