European Numerical Mathematics and
Advanced Applications Conference 2019
30th sep - 4th okt 2019, Egmond aan Zee, The Netherlands
15:45   MS10: PDE constrained optimization and variational inequalities (Part 3)
Chair: Daniel Walter
25 mins
Learning nonlocal regularizers using bilevel optimization
Gernot Holler, Karl Kunisch
Abstract: In this talk we are concerned with how a particular class of nonlocal operators can be used for the regularization of ill-posed inverse problems. Prototypical examples for the considered class of nonlocal operators are fractional order Sobolev semi-norms. We address how some basic results from the theory of regularization with local operators can be extended to the nonlocal case. We then study a framework based on a bilevel optimization strategy which allows us to choose nonlocal regularization operators from a given class which are optimal with respect to a suitable performance measure on a given training set. Finally, we address some numerical difficulties attached to our approach and present results from numerical experiments.
25 mins
Optimal Control of the Relativistic Vlasov-Maxwell System with Boundary Conditions
Jörg Weber
Abstract: The time evolution of a collisionless plasma is modeled by the relativistic Vlasov-Maxwell system, which couples the Vlasov equation (the transport equation) with the Maxwell equations of electrodynamics. The plasma particles are located in a bounded domain, for example a fusion reactor. In the exterior, there are external currents that may serve as a control of the plasma if adjusted suitably. Also, we allow material parameters, that is to say permittivity and permeability, which may depend on the space coordinate. We discuss solution theory of the modeling nonlinear PDE system. Unfortunately, there is only a very weak solution concept and uniqueness of weak solutions to the initial value problem is not known. This causes many problems when treating an optimal control problem, on which the main focus of this talk lies. On the one hand, particles hitting the boundary of their container and thus causing damage, and, on the other hand, exhaustive control costs are penalized. We prove existence of minimizers of the arising minimizing problem and give an approach to derive first order optimality conditions.
25 mins
Approximation of Rate-Independent Evolution with Non-Convex Energies
Christian Meyer, Michael Sievers
Abstract: Rate-independent systems governed by non-convex energies provide a several mathematical challenges. Since solutions may in general show discontinuities in time, the design of a suitable, mathematically rigorous notion of solution is all but clear and several different solution concepts exist, such as weak, differential, and global energetic solutions. In the recent past a new promising solution concept was developed, the so-called parametrized solution. The principal idea is to introduce an artificial time, in which the solution is continuous, and to interpret the physical time as a function of the artificial time. A numerical scheme that allows to approximate this class of solutions is the so-called local time-incremental minimization scheme. We investigate this scheme (combined with a standard finite element discretization in space) in detail, provide convergence results in the general case, and prove convergence rates for problems with (locally) convex energies. Numerical tests confirm our theoretical findings.
25 mins
Linear convergence of accelerated conditional gradient algorithms in spaces of measures
Daniel Walter
Abstract: A class of generalized conditional gradient algorithms for the solution of optimization problem in spaces of Radon measures is presented. The method iteratively inserts additional Dirac-delta functions and optimizes the corresponding coefficients. Under general assumptions, a sub-linear $\mathcal{O}(1/k)$ rate in the objective functional is obtained, which is sharp in most cases. To improve efficiency, one can fully resolve the finite-dimensional subproblems occurring in each iteration of the method. We provide an analysis for the resulting procedure: under a structural assumption on the optimal solution, a linear $\mathcal{O}(\zeta^k)$ convergence rate is obtained locally. This is joint work with Konstantin Pieper.