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Optimal uncertainty quantification for legacy data observationsof Lipschitz functions

Published online by Cambridge University Press:  30 August 2013

T.J. Sullivan ,
M. McKerns ,
D. Meyer ,
F. Theil ,
H. Owhadi  and
M. Ortiz
T.J. Sullivan
Affiliation:
Mathematics Institute, University of Warwick, Coventry, CV4 7AL, UK.. [email protected]
M. McKerns
Affiliation:
Center for Advanced Computing Research, California Institute of Technology, 1200 East California Boulevard, Mail Code 158-79, Pasadena, CA 91125, USA.; [email protected]
D. Meyer
Affiliation:
Mathematics Institute, University of Warwick, Coventry, CV4 7AL, UK.; [email protected]
F. Theil
Affiliation:
Mathematics Institute, University of Warwick, Coventry, CV4 7AL, UK.; [email protected]
H. Owhadi
Affiliation:
Applied & Computational Mathematics and Control & Dynamical Systems, California Institute of Technology, Mail Code 9-94, 1200 East California Boulevard, Pasadena, CA 91125, USA.; [email protected]
M. Ortiz
Affiliation:
Division of Engineering and Applied Science, California Institute of Technology, Mail Code 105-50, 1200 East California Boulevard, Pasadena, CA 91125, USA.; [email protected]
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Abstract

We consider the problem of providing optimal uncertainty quantification (UQ) – and hencerigorous certification – for partially-observed functions. We present a UQ frameworkwithin which the observations may be small or large in number, and need not carryinformation about the probability distribution of the system in operation. The UQobjectives are posed as optimization problems, the solutions of which are optimal boundson the quantities of interest; we consider two typical settings, namely parametersensitivities (McDiarmid diameters) and output deviation (or failure) probabilities. Thesolutions of these optimization problems depend non-trivially (even non-monotonically anddiscontinuously) upon the specified legacy data. Furthermore, the extreme values are oftendetermined by only a few members of the data set; in our principal physically-motivatedexample, the bounds are determined by just 2 out of 32 data points, and the remaindercarry no information and could be neglected without changing the final answer. We proposean analogue of the simplex algorithm from linear programming that uses these observationsto offer efficient and rigorous UQ for high-dimensional systems with high-cardinalitylegacy data. These findings suggest natural methods for selecting optimal (maximallyinformative) next experiments.

Keywords

Uncertainty quantificationprobability inequalitiesnon-convex optimizationLipschitz functionslegacy datapoint observations
Type
Research Article
Information
ESAIM: Mathematical Modelling and Numerical Analysis , Volume 47 , Issue 6 , November 2013 , pp. 1657 - 1689
Copyright
© EDP Sciences, SMAI, 2013

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