Optimal Uncertainty Quantification
We propose a rigorous framework for uncertainty quantification (UQ) in which the UQ objectives and its assumptions/information set are brought to the forefront. This framework, which we call optimal uncertainty quantification (OUQ), is based on the observation that, given a set of assumptions and information about the problem, there exist optimal bounds on uncertainties: these are obtained as values of well-defined optimization problems corresponding to extremizing probabilities of failure, or of deviations, subject to the constraints imposed by the scenarios compatible with the assumptions and information. In particular, this framework does not implicitly impose inappropriate assumptions, nor does it repudiate relevant information. Although OUQ optimization problems are extremely large, we show that under general conditions they have finite-dimensional reductions. As an application, we develop optimal concentration inequalities (OCI) of Hoeffding and McDiarmid type. Surprisingly, these results show that uncertainties in input parameters, which propagate to output uncertainties in the classical sensitivity analysis paradigm, may fail to do so if the transfer functions (or probability distributions) are imperfectly known. We show how, for hierarchical structures, this phenomenon may lead to the nonpropagation of uncertainties or information across scales. In addition, a general algorithmic framework is developed for OUQ and is tested on the Caltech surrogate model for hypervelocity impact and on the seismic safety assessment of truss structures, suggesting the feasibility of the framework for important complex systems. The introduction of this paper provides both an overview of the paper and a self-contained minitutorial on the basic concepts and issues of UQ.
© 2013, Society for Industrial and Applied Mathematics. Received by the editors September 7, 2010; accepted for publication (in revised form) May 22, 2012; published electronically May 8, 2013. This work was partially supported by the Department of Energy National Nuclear Security Administration under award DE-FC52-08NA28613 through Caltech's ASC/PSAAP Center for the Predictive Modeling and Simulation of High Energy Density Dynamic Response of Materials. Calculations for this paper were performed using the mystic optimization framework . We thank the Caltech PSAAP Experimental Science Group—Marc Adams, Leslie Lamberson, Jonathan Mihaly, Laurence Bodelot, Justin Brown, Addis Kidane, Anna Pandolfi, Guruswami Ravichandran, and Ares Rosakis— for formula (1.5) and Figure 1.2. We thank Sydney Garstang and Carmen Sirois for proofreading the manuscript. We thank Ilse Ipsen and four anonymous referees for detailed comments and suggestions.
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