  We present a novel approach for solving steady-state stochastic partial
differential equations (PDEs) with high-dimensional random parameter space. The
proposed approach combines spatial domain decomposition with basis adaptation
for each subdomain. The basis adaptation is used to address the curse of
dimensionality by constructing an accurate low-dimensional representation of
the stochastic PDE solution (probability density function and/or its leading
statistical moments) in each subdomain. Restricting the basis adaptation to a
specific subdomain affords finding a locally accurate solution. Then, the
solutions from all of the subdomains are stitched together to provide a global
solution. We support our construction with numerical experiments for a
steady-state diffusion equation with a random spatially dependent coefficient.
Our results show that highly accurate global solutions can be obtained with
significantly reduced computational costs.
