  Multi-physics models governed by coupled partial differential equation (PDE)
systems, are naturally suited for partitioned, or modular numerical solution
strategies. Although widely used in tackling deterministic coupled models,
several challenges arise in extending the benefits of modularization to
uncertainty propagation. On one hand, Monte-Carlo (MC) based methods are
prohibitively expensive as the cost of each deterministic PDE solve is usually
quite large, while on the other hand, even if each module contains a moderate
number of uncertain parameters, implementing spectral methods on the combined
high-dimensional parameter space can be prohibitively expensive. In this work,
we present a reduced non-intrusive spectral projection (NISP) based uncertainty
propagation method which separates and modularizes the uncertainty propagation
task in each subproblem using block Gauss-Seidel (BGS) techniques. The overall
computational costs in the proposed method are also mitigated by constructing
reduced approximations of the input data entering each module. These reduced
approximations and the corresponding quadrature rules are constructed via
simple linear algebra transformations. We describe these components of the
proposed algorithm assuming a generalized polynomial chaos (gPC) model of the
stochastic solutions. We demonstrate our proposed method and its computational
gains over the standard NISP method using numerical examples.
