We are interested in solving mathematical programs that feature distributed and discrete-valued decision variables that allow to control a dynamical system that is modeled by means of a PDE constraint. The distributed decision variables are measurable functions and regularized with a total variation penalty, that is the value of the regularizer is the sum of the interface areas of the level sets of decision variables weighted by the jump heights over the respective interfaces. In particular, the level sets have finite perimeters. In the multi-dimensional case, current discrete optimization-based techniques enable us to solve medium-sized problem instances to optimality in an acceptable amount of time in the context of descent algorithms. Realistic problem sizes may currently be deemed intractable, however. In order to overcome this issue, we propose and analyze a domain decomposition technique that allows to replace the solution of large-scale problem instances to many medium-sized problems that are coupled through overlaps of the subdomains on which the smaller problems are defined. To this end, we employ covering arguments inside of an algorithmic framework to localize the optimality and sufficient decrease conditions for a superordinate trust-region strategy. We employ an algorithm leans on greedy approaches in coordinate descent algorithms in order to obtain convergence to stationary points.

Bio: Paul Manns did his Bachelor studies in Computer Science (diploma 2013) and Master studies in Computational Engineering (diploma 2015) at Technical University of Darmstadt. Afterwards, he was a research associate at the research group Optimization in Robotics and Biomechanics at Heidelberg University for a year (2016-2017). In April 2017, he joined the Institute for Mathematical Optimization at Technical University of Braunschweig as a research associate, where he finished his PhD in 2019. From August 2019 to November 2019, he was on a research stay at Argonne National Laboratory. In September 2020, Paul left Technical University of Braunschweig and joined the Mathematics and Computer Science Division of Argonne National Laboratory as James H Wilkinson Fellow in Scientific Computing. In September 2021, Paul assumed the position as assistant professor at TU Dortmund University.  

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