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Optimal Uncertainty Quantification for Legacy Data Observations of Lipschitz Functions

Sullivan, T. J. and McKerns, M. and Meyer, D. and Theil, F. and Owhadi, H. and Ortiz, M. (2013) Optimal Uncertainty Quantification for Legacy Data Observations of Lipschitz Functions. ESAIM-Mathematical Modelling and Numerical Analysis, 47 (6). pp. 1657-1689. ISSN 0764-583X.

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We consider the problem of providing optimal uncertainty quantification (UQ) – and hence rigorous certification – for partially-observed functions. We present a UQ framework within which the observations may be small or large in number, and need not carry information about the probability distribution of the system in operation. The UQ objectives are posed as optimization problems, the solutions of which are optimal bounds on the quantities of interest; we consider two typical settings, namely parameter sensitivities (McDiarmid diameters) and output deviation (or failure) probabilities. The solutions of these optimization problems depend non-trivially (even non-monotonically and discontinuously) upon the specified legacy data. Furthermore, the extreme values are often determined by only a few members of the data set; in our principal physically-motivated example, the bounds are determined by just 2 out of 32 data points, and the remainder carry no information and could be neglected without changing the final answer. We propose an analogue of the simplex algorithm from linear programming that uses these observations to offer efficient and rigorous UQ for high-dimensional systems with high-cardinality legacy data. These findings suggest natural methods for selecting optimal (maximally informative) next experiments.

Item Type:Article
Related URLs:
URLURL TypeDescription Paper DOIArticle
Owhadi, H.0000-0002-5677-1600
Ortiz, M.0000-0001-5877-4824
Additional Information:© 2013 EDP Sciences, SMAI. Article published by EDP Sciences. Received February 9, 2012. Published online by Cambridge University Press: 30 August 2013. Portions of this work were supported by the US Department of Energy NNSA under award DEFC52-08NA28613 through the California Institute of Technology’s ASC/PSAAP Center for the Predictive Modeling and Simulation of High Energy Density Dynamic Response of Materials. We thank the California Institute of Technology PSAAP Center’s Experimental Science Group – in particular, M. Adams, J.M. Mihaly and A. Rosakis – for the data set in Table 1. Finally, we thank three anonymous referees for their helpful comments.
Funding AgencyGrant Number
Department of Energy (DOE) National Nuclear Security AdministrationDEFC52-08NA28613
Subject Keywords:Uncertainty quantification; probability inequalities; non-convex optimization; Lipschitz functions; legacy data; point observations
Issue or Number:6
Classification Code:Mathematics Subject Classification: 60E15, 62G99, 65C50, 90C26
Record Number:CaltechAUTHORS:20131119-093417975
Persistent URL:
Official Citation: T.J. Sullivan, M. McKerns, D. Meyer, F. Theil, H. Owhadi and M. Ortiz (2013). Optimal uncertainty quantification for legacy data observations of Lipschitz functions. ESAIM: Mathematical Modelling and Numerical Analysis, 47, pp 1657-1689. doi:10.1051/m2an/2013083.
Usage Policy:No commercial reproduction, distribution, display or performance rights in this work are provided.
ID Code:42555
Deposited By: Tony Diaz
Deposited On:19 Nov 2013 22:27
Last Modified:24 Feb 2020 10:30

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