Optimal Uncertainty Quantification

TL;DR

This paper introduces Optimal Uncertainty Quantification (OUQ), a rigorous framework that derives optimal bounds on uncertainties based on assumptions and information, with applications to concentration inequalities and complex system safety assessments.

Contribution

The paper develops a general OUQ framework that provides finite-dimensional reductions for large optimization problems and applies it to derive new concentration inequalities and analyze uncertainty propagation.

Findings

OUQ yields optimal bounds on uncertainties consistent with assumptions.

New concentration inequalities of Hoeffding and McDiarmid type are developed.

Uncertainty propagation may fail in hierarchical systems with imperfect knowledge.

Abstract

We propose a rigorous framework for Uncertainty Quantification (UQ) in which the UQ objectives and the assumptions/information set are brought to the forefront. This framework, which we call \emph{Optimal Uncertainty Quantification} (OUQ), is based on the observation that, given a set of assumptions and information about the problem, there exist optimal bounds on uncertainties: these are obtained as values of well-defined optimization problems corresponding to extremizing probabilities of failure, or of deviations, subject to the constraints imposed by the scenarios compatible with the assumptions and information. In particular, this framework does not implicitly impose inappropriate assumptions, nor does it repudiate relevant information. Although OUQ optimization problems are extremely large, we show that under general conditions they have finite-dimensional reductions. As an application, we develop \emph{Optimal Concentration Inequalities} (OCI) of Hoeffding and McDiarmid type. Surprisingly, these results show that uncertainties in input parameters, which propagate to output uncertainties in the classical sensitivity analysis paradigm, may fail to do so if the transfer functions (or probability distributions) are imperfectly known. We show how, for hierarchical structures, this phenomenon may lead to the non-propagation of uncertainties or information across scales. In addition, a general algorithmic framework is developed for OUQ and is tested on the Caltech surrogate model for hypervelocity impact and on the seismic safety assessment of truss structures, suggesting the feasibility of the framework for important complex systems. The introduction of this paper provides both an overview of the paper and a self-contained mini-tutorial about basic concepts and issues of UQ.