Stochastic basis adaptation and spatial domain decomposition for PDEs with random coefficients
For researchers solving stochastic PDEs with high-dimensional random inputs, this approach offers a scalable and parallelizable framework to lower computational costs.
This work reduces the computational cost of polynomial chaos methods for high-dimensional stochastic PDEs by decomposing the domain and adapting stochastic bases locally, achieving a reduction from O(N^α) to O(M^α) in operation count. The method is demonstrated on linear and nonlinear equations with random coefficients.
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 $O(N^α)$ to $O(M^α),$ where $N$ is the total number of degrees of freedom, $M$ is the number of degrees of freedom in each subdomain, and the exponent $α>1$ 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.