Optimal uncertainty quantification for legacy data observations of Lipschitz functions
ESAIM: Mathematical Modelling and Numerical Analysis - Modélisation Mathématique et Analyse Numérique, Tome 47 (2013) no. 6, pp. 1657-1689.

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.

DOI : https://doi.org/10.1051/m2an/2013083
Classification : 60E15,  62G99,  65C50,  90C26
Mots clés : uncertainty quantification, probability inequalities, non-convex optimization, Lipschitz functions, legacy data, point observations
     author = {Sullivan, T. J. and McKerns, M. and Meyer, Daniel and Theil, F. and Owhadi, H. and Ortiz, M.},
     title = {Optimal uncertainty quantification for legacy data observations of Lipschitz functions},
     journal = {ESAIM: Mathematical Modelling and Numerical Analysis - Mod\'elisation Math\'ematique et Analyse Num\'erique},
     pages = {1657--1689},
     publisher = {EDP-Sciences},
     volume = {47},
     number = {6},
     year = {2013},
     doi = {10.1051/m2an/2013083},
     mrnumber = {3110491},
     language = {en},
     url = {http://archive.numdam.org/articles/10.1051/m2an/2013083/}
Sullivan, T. J.; McKerns, M.; Meyer, D.; Theil, F.; Owhadi, H.; Ortiz, M. Optimal uncertainty quantification for legacy data observations of Lipschitz functions. ESAIM: Mathematical Modelling and Numerical Analysis - Modélisation Mathématique et Analyse Numérique, Tome 47 (2013) no. 6, pp. 1657-1689. doi : 10.1051/m2an/2013083. http://archive.numdam.org/articles/10.1051/m2an/2013083/

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