  Coupled partial differential equation (PDE) systems, which often represent
multi-physics models, are naturally suited for modular numerical solution
methods. However, several challenges yet remain in extending the benefits of
modularization practices to the task of uncertainty propagation. Since the cost
of each deterministic PDE solve can be usually expected to be quite
significant, statistical sampling based methods like Monte-Carlo (MC) are
inefficient because they do not take advantage of the mathematical structure of
the problem, and suffer for poor convergence properties. 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 due to the curse of dimensionality. In this
work, we present a module-based and efficient intrusive spectral projection
(ISP) method for uncertainty propagation. In our proposed method, each
subproblem is separated and modularized via block Gauss-Seidel (BGS)
techniques, such that each module only needs to tackle the local stochastic
parameter space. Moreover, the computational costs are significantly mitigated
by constructing reduced chaos approximations of the input data that enter each
module. We demonstrate implementations of our proposed method and its
computational gains over the standard ISP method using numerical examples.
