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Risk-Averse PDE-Constrained Optimization using the Conditional Value-at-Risk

Please always quote using this URN:urn:nbn:de:0296-matheon-13652
  • Uncertainty is inevitable when solving science and engineering application problems. In the face of uncertainty, it is essential to determine robust and risk-averse solutions. In this work, we consider a class of PDE-constrained optimization problems in which the PDE coefficients and inputs may be uncertain. We introduce two approximations for minimizing the conditional value-at-risk for such PDE-constrained optimization problems. These approximations are based on the primal and dual formulations of the conditional value-at-risk. For the primal problem, we introduce a smooth approximation of the conditional value-at-risk in order to utilize derivative-based optimization algorithms and to take advantage of the convergence properties of quadrature-based discretizations. For this smoothed conditional value-at-risk, we prove differentiability as well as consistency of our approximation. For the dual problem, we regularize the inner maximization problem, rigorously derive optimality conditions, and demonstrate the consistency of our approximation. Furthermore, we propose a fixed-point iteration that takes advantage of the structure of the regularized optimality conditions and provides a means of calculating worst-case probability distributions based on the given probability level. We conclude with numerical results.

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Metadaten
Author:Thomas Surowiec, Drew Kouri
URN:urn:nbn:de:0296-matheon-13652
Referee:Dietmar Hömberg
Document Type:Preprint, Research Center Matheon
Language:English
Date of first Publication:2015/10/20
Release Date:2015/10/20
Tag:PDE Optimization, Conditional Value-At-Risk, Uncertainty Quantification
Institute:Humboldt-Universität zu Berlin
Project:D Optics and Electronics (Electronic and photonic devices) / D-OT1 Mathematical modeling, analysis, and optimization of strained Germanium-microbridges
MSC-Classification:49-XX CALCULUS OF VARIATIONS AND OPTIMAL CONTROL; OPTIMIZATION [See also 34H05, 34K35, 65Kxx, 90Cxx, 93-XX] / 49Mxx Numerical methods [See also 90Cxx, 65Kxx] / 49M15 Newton-type methods
65-XX NUMERICAL ANALYSIS / 65Kxx Mathematical programming, optimization and variational techniques / 65K05 Mathematical programming methods [See also 90Cxx]
65-XX NUMERICAL ANALYSIS / 65Nxx Partial differential equations, boundary value problems / 65N35 Spectral, collocation and related methods
90-XX OPERATIONS RESEARCH, MATHEMATICAL PROGRAMMING / 90Cxx Mathematical programming [See also 49Mxx, 65Kxx] / 90C15 Stochastic programming
Preprint Number:1082
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