Iterative hard-thresholding applied to optimal control problems with $L^0(\Omega)$ control cost
Prof. Dr. Daniel Wachsmuth | University of Würzburg | Germany
We investigate the hard-thresholding method applied to optimal control problems with $L^0(\Omega)$ control
cost, which penalizes the measure of the support of the control. As the underlying measure space is non-atomic,
arguments of convergence proofs in $l^2$ or $\R^n$ cannot be applied.
Nevertheless, we prove the surprising property that the values of the objective functional are lower semicontinuous
along the iterates. That is, the function value in a weak limit point is less or equal than the lim-inf of the function
values along the iterates. Under a compactness assumption, we can prove that weak limit points are strong limit points,
which enables us to prove certain stationarity conditions for the limit points.
Numerical experiments are carried out, which show the performance of the method.
These indicates that the method is robust with respect to discretization.
In addition, we show that solutions obtained by the thresholding algorithm are superior to solutions of $L^1(\Omega)$-regularized problems.
A hybrid semismooth-/quasi-Newton method for structured nonsmooth operator equations in Banach spaces and its application to optimal control
Dr. Florian Mannel | University of Graz | Austria
We present an algorithm for the efficient solution of structured nonsmooth operator equations in Banach spaces. Here, the term structured indicates that we consider equations which are composed of a smooth and a semismooth mapping. Equations of this type occur, for instance, as optimality conditions of structured nonsmooth optimization problems. In particular, the algorithm can be applied to nonconvex PDE-constrained optimal control problems with sparsity.
The novel algorithm combines a semismooth Newton method with a quasi-Newton method. This hybrid approach retains the local superlinear convergence of both these methods under standard assumptions. Since it is known that, in general, quasi-Newton methods alone can not achieve superlinear convergence in semismooth settings, this is rather satisfying from a theoretical point of view. In addition, we observe in the infinite-dimensional setting that PDE-constrained optimal control problems are particularly well-suited for the application of the new method because of their inherent smoothing property.
The most striking feature of the new method, however, is its numerical performance. On nonsmooth PDE-constrained optimal control problems it is significantly faster than semismooth Newton methods, and these speedups persist when globalization techniques are added. Most notably, the hybrid approach can be embedded in a matrix-free limited-memory truncated trust-region framework to efficiently solve nonconvex and nonsmooth large-scale real-world optimization problems, as we will demonstrate by means of an example from magnetic resonance imaging. In this challenging environment it dramatically outperforms semismooth Newton methods, sometimes by a factor of fifty and more.
All of these topics are addressed in the talk.
Microscopic Derivation of Mean Field Game Models
Dr. Torsten Trimborn | RWTH Aachen | Germany
Mean field game theory studies the behavior of a large number of interacting individuals
in a game theoretic setting. In this talk, we derive the mean field game PDE system from deterministic microscopic agent dynamics. The dynamics are given by a general ODE which defines a large class of differential games. We use the concept of Nash equilibria and apply dynamic programming to derive the mean field limit equation. We extensively study the scaling
behavior of our system and observe new reasonable configurations. We show that well known mean field game limit systems are a subclass of our model. We motivate the novel scales with an example of an agent-based financial market model, inspired by the econophysical Levy-Levy-Solomon model.
On hysteresis-reaction-diffusion systems
Prof. Dr. Klemens Fellner | University of Graz | Austria
We report on recent progresses on models coupling a hysteresis operator to reaction-diffusion equations.
In a frist collaboration with M. Brokate, we prove weak-differentiability of the control-to-state mapping in a parabolic control problem with hysteresis. In a second paper with C. Münch, we present the derivation of hysteresis-reaction-diffusion systems
as singular fast-reaction limit of a suitable class of coupled ODE-PDE systems. Besides existence theory and numerical examples, we demonstrate in particular a hysteresis driven instability mechanism, where the shape of a generalised play operator may decide between large-time equilibration or large-time blow-up.