10:40
MS10: PDE constrained optimization and variational inequalities (Part 2)
Chair: Daniel Walter
10:40
25 mins
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A Priori Error Estimates in Non-Energy Norms and the Aubin-Nitsche Trick for the Two-Dimensional Scalar Signorini Problem
Constantin Christof, Christof Haubner
Abstract: This talk is concerned with error estimates for the piecewise linear finite element approximation of the two-dimensional scalar Signorini problem on a convex polygonal domain. Using a Céa-type lemma, a supercloseness result and a non-standard duality argument, we prove various Lp- and W1p-error estimates under mild assumptions on the contact sets of the continuous and the discrete solution. The obtained orders of convergence turn out to be optimal for problems with bounded right-hand sides.
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11:05
25 mins
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A Priori Error Estimates for the Optimal Control of a Simplified Signorini Problem
Christof Haubner
Abstract: In the context of distributed control we consider a simplified Signorini problem, an elliptic variational inequality of first kind with unilateral constraints on the boundary. The state is discretized using linear finite elements while a variational discretization is applied to the control. We derive a priori error estimates for the control and state based on strong stationarity and a quadratic growth condition. The convergence rates depend on $H^1$ and $L^2$ error estimates of the simplified Signorini problem. Furthermore, we discuss under what conditions quadratic growth can be expected. Numerical experiments are presented that confirm the results.
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11:30
25 mins
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Shape and Topology Optimization subject to 3D Nonlinear Magnetostatics - Sensitivity Analysis
Peter Gangl, Kevin Sturm
Abstract: We present shape and topological sensitivities for a 3D nonlinear magnetostatic model of an electric motor. In order to derive the sensitivities,
we use a Lagrangian approach, which allows us to simplify the derivation
under realistic physical assumptions. The topological derivative for this
quasilinear problem involves the solution of two transmission problems on
an unbounded domain for each point of evaluation.
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11:55
25 mins
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Optimal control of pedestrian dynamics
Max Winkler, Roland Herzog, Jan Pietschmann, Ailyn Stötzner
Abstract: This talk is devoted to optimal control problems for the Hughes model which is a mathematical description of pedestrian dynamics. As an application, we consider the optimal evacuation of a crowd in e.,g. a burning building. The typical observation is that all people run to the closest exit and depending on the initial distribution of the crowd some exits slightly further away are not minded. Thus, we modified the model and introduced so-called agents which may also attract the crowd and the aim is to control the movement of these agents such that the evacuation is optimized by evenly spreading the crowd to all exits.
The model is based on a coupled system of a transport equation for the crowd, an Eikonal equation for the potential and an ODE for the agent dynamics. The problem is discretized with a discontinuous Galerkin scheme. Moreover, we investigate first-order necessary optimality conditions and gradient based optimization methods for the optimal control problem.
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