A Measure Approximation for Distributionally Robust PDEConstrained Optimization Problems
Abstract
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 worstcase 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 minmax problem and conclude with numerical results.
 Authors:

 Sandia National Lab. (SNLNM), Albuquerque, NM (United States)
 Publication Date:
 Research Org.:
 Sandia National Lab. (SNLNM), Albuquerque, NM (United States)
 Sponsoring Org.:
 USDOE National Nuclear Security Administration (NNSA)
 OSTI Identifier:
 1429682
 Report Number(s):
 SAND201712112J
Journal ID: ISSN 00361429; 658537
 Grant/Contract Number:
 AC0494AL85000; NA0003525
 Resource Type:
 Accepted Manuscript
 Journal Name:
 SIAM Journal on Numerical Analysis
 Additional Journal Information:
 Journal Volume: 55; Journal Issue: 6; Journal ID: ISSN 00361429
 Publisher:
 Society for Industrial and Applied Mathematics
 Country of Publication:
 United States
 Language:
 English
 Subject:
 97 MATHEMATICS AND COMPUTING; 54 ENVIRONMENTAL SCIENCES; PDEconstrained optimization; uncertainty; distributionally robust; datadriven
Citation Formats
Kouri, Drew Philip. A Measure Approximation for Distributionally Robust PDEConstrained Optimization Problems. United States: N. p., 2017.
Web. doi:10.1137/15M1036944.
Kouri, Drew Philip. A Measure Approximation for Distributionally Robust PDEConstrained Optimization Problems. United States. doi:10.1137/15M1036944.
Kouri, Drew Philip. Tue .
"A Measure Approximation for Distributionally Robust PDEConstrained Optimization Problems". United States. doi:10.1137/15M1036944. https://www.osti.gov/servlets/purl/1429682.
@article{osti_1429682,
title = {A Measure Approximation for Distributionally Robust PDEConstrained Optimization Problems},
author = {Kouri, Drew Philip},
abstractNote = {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 worstcase 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 minmax problem and conclude with numerical results.},
doi = {10.1137/15M1036944},
journal = {SIAM Journal on Numerical Analysis},
number = 6,
volume = 55,
place = {United States},
year = {2017},
month = {12}
}
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