09:50
On convergence of solutions to optimal control problems governed by a regularized phase field
Prof. Dr. Ira Neitzel | University of Bonn | Germany
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Prof. Dr. Ira Neitzel | University of Bonn | Germany
Prof. Dr. Thomas Wick | Leibniz Universität Hannover
Prof. Winnifried Wollner | TU Darmstadt | Germany
We consider an optimal control problem of tracking type governed by a time-discrete regularized phase-field fracture or damage propagation model. The problem contains a regularization term that penalizes the violation of the irreversibility condition in the evolution of the fracture. We prove convergence of solutions of the regularized problem when taking the limit with respect to the penalty term, and obtain an estimate for the constraint violation in terms of the penalty Parameter.
10:10
Analysis and approximations of an optimal control problem related to the Allen-Cahn equation
Konstantinos Chrysafinos | National Technical University of Athens | Greece
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Konstantinos Chrysafinos | National Technical University of Athens | Greece
An optimal control problem related to the Allen-Cahn equation is considered. In particular, a tracking type functional is minimized subject to the Allen-Cahn equation, by using controls of distributed type satisfying point-wise control constraints.Various regularity and continuity results of the control to state mapping are presented. In addition, first and second order necessary and sufficient conditions are discussed. A fully-discrete scheme based on the lowest order (in time) discontinuous Galerkin scheme combined with conforming finite element subspaces is considered. Provided that the temporal and spatial discretization parameters are suitably chosen, we present error estimates for the difference between locally optimal controls and their discrete approximations. Further estimates for the corresponding state and adjoint variables are also presented.
10:30
Reduced Order Models for CVaR Estimation and Risk-Averse Optimization
Prof. Dr. Matthias Heinkenschloss | Rice University | Germany
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Prof. Dr. Matthias Heinkenschloss | Rice University | Germany
Dr. Boris Krämer | Massachusetts Institute of Technology | United States
M. Sc. Mae Markowski | Rice University | United States
Dr. Timur Takhtaganov | Lawrence Berkeley National Lab | United States
Prof. Dr. Karen Willcox | University of Texas at Austin | United States
This talk introduces new reduced order model (ROM) approaches for the efficient solution of risk averse PDE constrained optimization problems.
Risk-averse optimization arises in science and engineering applications where one needs to determine inputs (controls or design parameters) to maximize the performance of a system specified by a quantity of interest (QoI). The system is modeled by PDEs and some system parameters such as material parameters are uncertain. Instead of maximizing the expected performance of the system, risk-averse optimization also includes a so-called risk measure for the deviation of the actual performance below the expected performance. One example of such risk measures is the Conditional Value at Risk (CVaR). Estimation of the risk-measure at a single control/design requires many evaluations of the PDE, each corresponding to a realization of the uncertain system parameters. More importantly, only realizations in tail of the probability distribution of the QoI enter the estimation. Unfortunately, this distribution is not known explicitly, but is determined by the probability distributions of the uncertain inputs and the PDE solution through which these uncertain inputs enter the performance objective. Optimization of risk-measures requires even more PDE solves.
This talk discussed the rigorous use of ROM for the efficient estimation of CVaR. Efficiency is gained by using ROMs to replace the expensive full order model in the CVaR estimation, and also using the structure of CVaR estimation to guide the ROM construction. Improvements gained by using ROMs are quantified and illustrated numerically. These CVaR estimation approaches are then used to optimize CVAR subject to PDE constraints with uncertain parameters. In this context it is important that key quantities that need to be approximated for CVAR objective function evaluations are
also the key quantities arising in gradient computations. Numerical results are given to illustrate the performance gains due to ROM.