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 extreme 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, contrary to the classical sensitivity analysis
paradigm, these results show that uncertainties in input parameters do not
necessarily propagate to output uncertainties.

In addition, a general algorithmic framework is developed for OUQ and is
tested on the Caltech surrogate model for hypervelocity impact, suggesting the
feasibility of the framework for important complex systems.