  Over the last few years there have been dramatic advances in our
understanding of mathematical and computational models of complex systems in
the presence of uncertainty. This has led to a growth in the area of
uncertainty quantification as well as the need to develop efficient, scalable,
stable and convergent computational methods for solving differential equations
with random inputs. Stochastic Galerkin methods based on polynomial chaos
expansions have shown superiority to other non-sampling and many sampling
techniques. However, for complicated governing equations numerical
implementations of stochastic Galerkin methods can become non-trivial. On the
other hand, Monte Carlo and other traditional sampling methods, are
straightforward to implement. However, they do not offer as fast convergence
rates as stochastic Galerkin. Other numerical approaches are the stochastic
collocation (SC) methods, which inherit both, the ease of implementation of
Monte Carlo and the robustness of stochastic Galerkin to a great deal. However,
stochastic collocation and its powerful extensions, e.g. sparse grid stochastic
collocation, can simply fail to handle more levels of complication. The
seemingly innocent Burgers equation driven by Brownian motion is such an
example. In this work we propose a novel enhancement to stochastic collocation
methods using locally improved deterministic model reduction techniques that
can handle this pathological example and hopefully other more complicated
equations like Stochastic Navier-Stokes. Local improvements to reduced-order
models are achieved using sensitivity analysis of the proper orthogonal
decomposition. Our numerical results show that the proposed technique is not
only reliable and robust but also very efficient.
