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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.
@article{M2AN_2013__47_6_1657_0,
author = {Sullivan, T. J. and McKerns, M. and Meyer, D. 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 },
pages = {1657--1689},
publisher = {EDP-Sciences},
volume = {47},
number = {6},
year = {2013},
doi = {10.1051/m2an/2013083},
mrnumber = {3110491},
language = {en},
url = {http://geodesic.mathdoc.fr/articles/10.1051/m2an/2013083/}
}
TY - JOUR AU - Sullivan, T. J. AU - McKerns, M. AU - Meyer, D. AU - Theil, F. AU - Owhadi, H. AU - Ortiz, M. TI - Optimal uncertainty quantification for legacy data observations of Lipschitz functions JO - ESAIM: Mathematical Modelling and Numerical Analysis PY - 2013 SP - 1657 EP - 1689 VL - 47 IS - 6 PB - EDP-Sciences UR - http://geodesic.mathdoc.fr/articles/10.1051/m2an/2013083/ DO - 10.1051/m2an/2013083 LA - en ID - M2AN_2013__47_6_1657_0 ER -
%0 Journal Article %A Sullivan, T. J. %A McKerns, M. %A Meyer, D. %A Theil, F. %A Owhadi, H. %A Ortiz, M. %T Optimal uncertainty quantification for legacy data observations of Lipschitz functions %J ESAIM: Mathematical Modelling and Numerical Analysis %D 2013 %P 1657-1689 %V 47 %N 6 %I EDP-Sciences %U http://geodesic.mathdoc.fr/articles/10.1051/m2an/2013083/ %R 10.1051/m2an/2013083 %G en %F M2AN_2013__47_6_1657_0
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 , Tome 47 (2013) no. 6, pp. 1657-1689. doi: 10.1051/m2an/2013083
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