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 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 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
