11:20
On instationary mean field games with diffusion
Dr. Roman Andreev | UP7D/LJLL | France
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Dr. Roman Andreev | UP7D/LJLL | France
We apply the augmented Lagrangian method to the convex optimization problem of the instationary variational mean field games with diffusion. The system is first discretized with space-time tensor product piecewise polynomial bases. This leads to a sequence of linear problems posed on the space-time cylinder that are 2nd order in the temporal variable and 4th order in the spatial variable. To solve these (large) linear problems with the preconditioned conjugate gradients method we propose a parameter-robust preconditioner that is based on a temporal transformation coupled with a spatial multigrid. Numerical examples illustrate the method.
11:40
A novel numerical method for a class of Liouville control problems
Dr. Souvik Roy | University of Würzburg | Germany
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Dr. Souvik Roy | University of Würzburg | Germany
Prof. Dr. Alfio Borzì | University of Würzburg | Germany
An accurate and efficient numerical scheme for solving a Liouville optimal control problem in the framework of the Pontryagin's maximum principle (PMP) is presented. The Liouville equation models the time-evolution of a density function that may represent a distribution of non-interacting particles or a probability density. In this work, the purpose of the control is to maximize the measure of a target set at a given final time. In order to solve this problem, a high-order accurate conservative and positive preserving discretization scheme is investigated and a novel iterative optimization method is formulated that solves the PMP optimality condition without requiring differentiability with respect to the control variable. Results of numerical experiments are presented that demonstrate the effectiveness of the proposed solution procedure.
12:00
Optimal control of PDEs in a complex space setting; application to the Schrödinger equation
Axel Kröner | INRIA Saclay and CMAP, Ecole polytechnique | France
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Axel Kröner | INRIA Saclay and CMAP, Ecole polytechnique | France
Dr. M. Soledad Aronna | FGV - Fundacao Getulio Vargas | Brazil
Prof. Dr. J. Frédéric Bonnans | INRIA Saclay and CMAP, Ecole polytechnique | France
In this paper we discuss optimality conditions for abstract optimization problems over complex spaces. We then apply these results to optimal control problems with a semigroup structure. As an application we detail the case when the state equation is the Schrödinger one, with pointwise constraints on the bilinear control. We derive first and second order optimality conditions and address in particular the case that the control enters the state equation and cost function linearly.
12:20
An iterative Bregman regularization method for optimal control problems with inequality constraints
Frank Pörner | Julius-Maximilians Universität Würzburg | Germany
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Frank Pörner | Julius-Maximilians Universität Würzburg | Germany
Prof. Dr. Daniel Wachsmuth | Germany
This contribution presents an iterative Bregman regularization method for solving optimal control problems of linear elliptic partial differential equations with control constraints. The regularization method is based on generalized Bregman distances. We provide convergence results under a combination of a source condition and a regularity condition on the active sets. We do not assume attainability of the desired state. Furthermore, a priori regularization error estimates for the control, state and adjoint state are presented. Numerical estimates indicate that the a priori estimate is sharp.