Optimal uncertainty quantification for legacy data observations of Lipschitz functions

¹ Mathematics Institute, University of Warwick, Coventry, CV4 7AL, UK.
Tim.Sullivan@warwick.ac.uk
² Center for Advanced Computing Research, California Institute of Technology, 1200 East California Boulevard, Mail Code 158-79, Pasadena, CA 91125, USA.
mmckerns@caltech.edu
³ Lehrstuhl für Numerische Mechanik, Technische Universität München, Boltzmannstrasse 15, 85747, Garching bei München, Germany.
meyer@lnm.mw.tum.de
⁴ Mathematics Institute, University of Warwick, Coventry, CV4 7AL, UK.
f.theil@warwick.ac.uk
⁵ 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
⁶ 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

Received: 9 February 2012

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.