scholarly journals Risk-averse optimal control of semilinear elliptic PDEs

2020 ◽  
Vol 26 ◽  
pp. 53 ◽  
Author(s):  
D.P. Kouri ◽  
T.M. Surowiec
Keyword(s):  
Optimal Control ◽  
Stochastic Optimization ◽  
Optimization Problems ◽  
Risk Measures ◽  
Elliptic Pdes ◽  
Infinite Dimensional ◽  
Continuity Properties ◽  
Semilinear Elliptic ◽  
Solution Regularity ◽  
Semilinear Elliptic Pdes

In this paper, we consider the optimal control of semilinear elliptic PDEs with random inputs. These problems are often nonconvex, infinite-dimensional stochastic optimization problems for which we employ risk measures to quantify the implicit uncertainty in the objective function. In contrast to previous works in uncertainty quantification and stochastic optimization, we provide a rigorous mathematical analysis demonstrating higher solution regularity (in stochastic state space), continuity and differentiability of the control-to-state map, and existence, regularity and continuity properties of the control-to-adjoint map. Our proofs make use of existing techniques from PDE-constrained optimization as well as concepts from the theory of measurable multifunctions. We illustrate our theoretical results with two numerical examples motivated by the optimal doping of semiconductor devices.

Combinatorial optimal control of semilinear elliptic PDEs

2018 ◽  
Vol 70 (3) ◽  
pp. 641-675 ◽  
Author(s):  
Christoph Buchheim ◽  
Renke Kuhlmann ◽  
Christian Meyer
Keyword(s):  
Optimal Control ◽  
Elliptic Pdes ◽  
Semilinear Elliptic ◽  
Semilinear Elliptic Pdes

scholarly journals On quantitative stability in infinite-dimensional optimization under uncertainty

2021 ◽  
Author(s):  
M. Hoffhues ◽  
W. Römisch ◽  
T. M. Surowiec
Keyword(s):  
Stochastic Optimization ◽  
Constrained Optimization ◽  
Optimization Problems ◽  
Optimization Under Uncertainty ◽  
Optimal Solutions ◽  
Probability Metrics ◽  
Pde Constrained Optimization ◽  
Quantitative Stability ◽  
Infinite Dimensional ◽  
Optimal Value

AbstractThe vast majority of stochastic optimization problems require the approximation of the underlying probability measure, e.g., by sampling or using observations. It is therefore crucial to understand the dependence of the optimal value and optimal solutions on these approximations as the sample size increases or more data becomes available. Due to the weak convergence properties of sequences of probability measures, there is no guarantee that these quantities will exhibit favorable asymptotic properties. We consider a class of infinite-dimensional stochastic optimization problems inspired by recent work on PDE-constrained optimization as well as functional data analysis. For this class of problems, we provide both qualitative and quantitative stability results on the optimal value and optimal solutions. In both cases, we make use of the method of probability metrics. The optimal values are shown to be Lipschitz continuous with respect to a minimal information metric and consequently, under further regularity assumptions, with respect to certain Fortet-Mourier and Wasserstein metrics. We prove that even in the most favorable setting, the solutions are at best Hölder continuous with respect to changes in the underlying measure. The theoretical results are tested in the context of Monte Carlo approximation for a numerical example involving PDE-constrained optimization under uncertainty.

scholarly journals A certified RB method for PDE-constrained parametric optimization problems

2019 ◽  
Vol 10 (1) ◽  
pp. 123-152
Author(s):  
Andrea Manzoni ◽  
Stefano Pagani
Keyword(s):  
Optimal Control ◽  
Efficient Solution ◽  
Parametric Optimization ◽  
Optimization Problems ◽  
A Posteriori Error Estimates ◽  
Optimal Solution ◽  
Posteriori Error ◽  
Elliptic Pdes ◽  
Reduced Basis ◽  
Cost Functionals

Abstract We present a certified reduced basis (RB) framework for the efficient solution of PDE-constrained parametric optimization problems. We consider optimization problems (such as optimal control and optimal design) governed by elliptic PDEs and involving possibly non-convex cost functionals, assuming that the control functions are described in terms of a parameter vector. At each optimization step, the high-fidelity approximation of state and adjoint problems is replaced by a certified RB approximation, thus yielding a very efficient solution through an “optimize-then-reduce” approach. We develop a posteriori error estimates for the solutions of state and adjoint problems, the cost functional, its gradient and the optimal solution. We confirm our theoretical results in the case of optimal control/design problems dealing with potential and thermal flows.

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