  We present a novel way of deciding when and where to refine a mesh in
probability space in order to facilitate the uncertainty quantification in the
presence of discontinuities in random space. A discontinuity in random space
makes the application of generalized polynomial chaos expansion techniques
prohibitively expensive. The reason is that for discontinuous problems, the
expansion converges very slowly. An alternative to using higher terms in the
expansion is to divide the random space in smaller elements where a lower
degree polynomial is adequate to describe the randomness. In general, the
partition of the random space is a dynamic process since some areas of the
random space, particularly around the discontinuity, need more refinement than
others as time evolves. In the current work we propose a way to decide when and
where to refine the random space mesh based on the use of a reduced model. The
idea is that a good reduced model can monitor accurately, within a random space
element, the cascade of activity to higher degree terms in the chaos expansion.
In terms, this facilitates the efficient allocation of computational sources to
the areas of random space where they are more needed. For the Kraichnan-Orszag
system, the prototypical system to study discontinuities in random space, we
present theoretical results which show why the proposed method is sound and
numerical results which corroborate the theory.
