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Optimal uncertainty quantification for legacy data observations of 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.. Tim.Sullivan@warwick.ac.uk
M. McKerns
Affiliation:
Center for Advanced Computing Research, California Institute of Technology, 1200 East California Boulevard, Mail Code 158-79, Pasadena, CA 91125, USA.; mmckerns@caltech.edu
D. Meyer
Affiliation:
Mathematics Institute, University of Warwick, Coventry, CV4 7AL, UK.; f.theil@warwick.ac.uk
F. Theil
Affiliation:
Mathematics Institute, University of Warwick, Coventry, CV4 7AL, UK.; f.theil@warwick.ac.uk
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.; owhadi@caltech.edu
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.; ortiz@caltech.edu
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Abstract

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.

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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