Existence and Optimality Conditions for Risk-Averse PDE-Constrained Optimization

Journal Article · 04 June 2018 · SIAM/ASA Journal on Uncertainty Quantification

DOI: https://doi.org/10.1137/16M1086613 · OSTI ID:1441483

Kouri, Drew Philip ^[1]; Surowiec, Thomas M. ^[2]

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1. Sandia National Lab. (SNL-NM), Albuquerque, NM (United States)
2. Philipps-Universitat Marburg (Germany). FB12 Mathematik und Informatik

Uncertainty is ubiquitous in virtually all engineering applications, and, for such problems, it is inadequate to simulate the underlying physics without quantifying the uncertainty in unknown or random inputs, boundary and initial conditions, and modeling assumptions. Here in this paper, we introduce a general framework for analyzing risk-averse optimization problems constrained by partial differential equations (PDEs). In particular, we postulate conditions on the random variable objective function as well as the PDE solution that guarantee existence of minimizers. Furthermore, we derive optimality conditions and apply our results to the control of an environmental contaminant. Lastly, we introduce a new risk measure, called the conditional entropic risk, that fuses desirable properties from both the conditional value-at-risk and the entropic risk measures.

• Research Organization: Sandia National Laboratories (SNL-NM), Albuquerque, NM (United States)
• Sponsoring Organization: DARPA; USDOE National Nuclear Security Administration (NNSA)
• Grant/Contract Number: AC04-94AL85000; NA0003525
• OSTI ID: 1441483
• Report Number(s): SAND--2018-5345J; 663230
• Journal Information: SIAM/ASA Journal on Uncertainty Quantification, Journal Name: SIAM/ASA Journal on Uncertainty Quantification Journal Issue: 2 Vol. 6; ISSN 2166-2525
• Publisher: SIAMCopyright Statement
• Country of Publication: United States
• Language: English