  We present a novel uncertainty quantification approach for high-dimensional
stochastic partial differential equations that reduces the computational cost
of polynomial chaos methods by decomposing the computational domain into
non-overlapping subdomains and adapting the stochastic basis in each subdomain
so the local solution has a lower dimensional random space representation. The
local solutions are coupled using the Neumann-Neumann algorithm, where we first
estimate the interface solution then evaluate the interior solution in each
subdomain using the interface solution as a boundary condition. The interior
solutions in each subdomain are computed independently of each other, which
reduces the operation count from $O(N^\alpha)$ to $O(M^\alpha),$ where $N$ is
the total number of degrees of freedom, $M$ is the number of degrees of freedom
in each subdomain, and the exponent $\alpha>1$ depends on the uncertainty
quantification method used. In addition, the localized nature of solutions
makes the proposed approach highly parallelizable. We illustrate the accuracy
and efficiency of the approach for linear and nonlinear differential equations
with random coefficients.
