A Measure Approximation for Distributionally Robust PDE-Constrained Optimization Problems

Journal Article · Tue Dec 19 00:00:00 EST 2017 · SIAM Journal on Numerical Analysis

DOI:https://doi.org/10.1137/15M1036944· OSTI ID:1429682

Kouri, Drew Philip ^[1]

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1. Sandia National Lab. (SNL-NM), Albuquerque, NM (United States)

In numerous applications, scientists and engineers acquire varied forms of data that partially characterize the inputs to an underlying physical system. This data is then used to inform decisions such as controls and designs. Consequently, it is critical that the resulting control or design is robust to the inherent uncertainties associated with the unknown probabilistic characterization of the model inputs. Here in this work, we consider optimal control and design problems constrained by partial differential equations with uncertain inputs. We do not assume a known probabilistic model for the inputs, but rather we formulate the problem as a distributionally robust optimization problem where the outer minimization problem determines the control or design, while the inner maximization problem determines the worst-case probability measure that matches desired characteristics of the data. We analyze the inner maximization problem in the space of measures and introduce a novel measure approximation technique, based on the approximation of continuous functions, to discretize the unknown probability measure. Finally, we prove consistency of our approximated min-max problem and conclude with numerical results.

• Research Organization: Sandia National Laboratories (SNL-NM), Albuquerque, NM (United States)
• Sponsoring Organization: USDOE National Nuclear Security Administration (NNSA)
• Grant/Contract Number: AC04-94AL85000; NA0003525
• OSTI ID: 1429682
• Report Number(s): SAND--2017-12112J; 658537
• Journal Information: SIAM Journal on Numerical Analysis, Journal Name: SIAM Journal on Numerical Analysis Journal Issue: 6 Vol. 55; ISSN 0036-1429
• Publisher: Society for Industrial and Applied Mathematics
• Country of Publication: United States
• Language: English

Related Subjects

54 ENVIRONMENTAL SCIENCES
97 MATHEMATICS AND COMPUTING
PDE-constrained optimization
data-driven
distributionally robust
uncertainty