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International Journal for Uncertainty Quantification
IF: 4.911 5-Year IF: 3.179 SJR: 1.008 SNIP: 0.983 CiteScore™: 5.2

ISSN Print: 2152-5080
ISSN Online: 2152-5099

Open Access

International Journal for Uncertainty Quantification

DOI: 10.1615/Int.J.UncertaintyQuantification.2020032480
pages 225-248

HYPERDIFFERENTIAL SENSITIVITY ANALYSIS OF UNCERTAIN PARAMETERS IN PDE-CONSTRAINED OPTIMIZATION

Joseph Hart
Optimization and Uncertainty Quantification, Sandia National Laboratories, P.O. Box 5800, Albuquerque, New Mexico 87123-1320, USA
Bart van Bloemen Waanders
Optimization and Uncertainty Quantification, Sandia National Laboratories, P.O. Box 5800, Albuquerque, New Mexico 87123-1320, USA
Roland Herzog
Technical University Chemnitz, Faculty of Mathematics, 09107 Chemnitz, Germany

ABSTRACT

Many problems in engineering and sciences require the solution of large scale optimization constrained by partial differential equations (PDEs). Though PDE-constrained optimization is itself challenging, most applications pose additional complexity, namely, uncertain parameters in the PDEs. Uncertainty quantification (UQ) is necessary to characterize, prioritize, and study the influence of these uncertain parameters. Sensitivity analysis, a classical tool in UQ, is frequently used to study the sensitivity of a model to uncertain parameters. In this article, we introduce "hyperdifferential sensitivity analysis" which considers the sensitivity of the solution of a PDE-constrained optimization problem to uncertain parameters. Our approach is a goal-oriented analysis which may be viewed as a tool to complement other UQ methods in the service of decision making and robust design. We formally define hyperdifferential sensitivity indices and highlight their relationship to the existing optimization and sensitivity analysis literatures. Assuming the presence of low rank structure in the parameter space, computational efficiency is achieved by leveraging a generalized singular value decomposition in conjunction with a randomized solver which converts the computational bottleneck of the algorithm into an embarrassingly parallel loop. Two multiphysics examples, consisting of nonlinear steady state control and transient linear inversion, demonstrate efficient identification of the uncertain parameters which have the greatest influence on the optimal solution.

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