  We demonstrate that the recently developed Optimal Uncertainty Quantification
(OUQ) theory, combined with recent software enabling fast global solutions of
constrained non-convex optimization problems, provides a methodology for
rigorous model certification, validation, and optimal design under uncertainty.
In particular, we show the utility of the OUQ approach to understanding the
behavior of a system that is governed by a partial differential equation --
Burgers' equation. We solve the problem of predicting shock location when we
only know bounds on viscosity and on the initial conditions. Through this
example, we demonstrate the potential to apply OUQ to complex physical systems,
such as systems governed by coupled partial differential equations. We compare
our results to those obtained using a standard Monte Carlo approach, and show
that OUQ provides more accurate bounds at a lower computational cost. We
discuss briefly about how to extend this approach to more complex systems, and
how to integrate our approach into a more ambitious program of optimal
experimental design.
