Stochastic basis adaptation and spatial domain decomposition for partial differential equations with random coefficients
We present a novel approach for high-dimensional stochastic partial differential equations that decomposes the computational domain into smaller disjoint subdomains and reduces the random dimension in each subdomain to save computational costs. The proposed approach combines spatial domain decomposition and stochastic basis adaptation in each subdomain. We adapt the stochastic basis in each subdomain such that the local solution has a low-dimensional random space representation. To compute the stochastic solution in each subdomain, we use the Neumann-Neumann algorithm, where we first estimate the interface solution and 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 $\approx O(N^\alpha)$ to $\approx O(M^\alpha),$ 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 a steady-state diffusion equation with a random space-dependent diffusion coefficient.