| Issue |
ESAIM: COCV
Volume 26, 2020
|
|
|---|---|---|
| Article Number | 53 | |
| Number of page(s) | 19 | |
| DOI | https://doi.org/10.1051/cocv/2019061 | |
| Published online | 04 September 2020 | |
Risk-averse optimal control of semilinear elliptic PDEs*,**,***
1
Optimization and Uncertainty Quantification, MS-1320, Sandia National Laboratories,
Albuquerque,
NM, USA.
2
FB12 Mathematik und Informatik, Philipps-Universität Marburg,
Marburg, Germany.
**** Corresponding author: dpkouri@sandia.gov
Received:
22
May
2018
Accepted:
27
September
2019
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.
Mathematics Subject Classification: 49J20 / 49J55 / 49K20 / 49K45 / 90C15
Key words: Risk-averse / PDE-constrained optimization / semilinear PDEs / uncertainty quantification / stochastic optimization / measurable multifunctions
TMS’s research was sponsored by DFG grant no. SU 963/1-1 “Generalized Nash Equilibrium Problems with Partial Differential Operators: Theory, Algorithms, and Risk Aversion”.
Sandia National Laboratories is a multimission laboratory managed and operated by National Technology and Engineering Solutions of Sandia, LLC., a wholly owned subsidiary of Honeywell International, Inc., for the U.S. Department of Energy’s National Nuclear Security Administration under contract DE-NA0003525. This paper describes objective technical results and analysis. Any subjective views or opinions that might be expressed in the paper do not necessarily represent the views of the U.S. Department of Energy or the United States Government.
© EDP Sciences, SMAI 2020
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