Analysis and preconditioning of a probabilistic domain decomposition algorithm for elliptic boundary value problems

Abstract

PDDSparse is a new hybrid parallelisation scheme for solving large-scale elliptic boundary value problems on supercomputers, which can be described as a Feynman-Kac formula for domain decomposition. At its core lies a stochastic linear, sparse system for the solutions on the interfaces, whose entries are generated via Monte Carlo simulations. Assuming small statistical errors, we show that the random system matrix ${\tilde G}(\omega)$ is near a nonsingular M-matrix $G$, i.e. ${\tilde G}(\omega)+E=G$ where $||E||/||G||$ is small. Using nonstandard arguments, we bound $||G^{-1}||$ and the condition number of $G$, showing that both of them grow moderately with the degrees of freedom of the discretisation. Moreover, the truncated Neumann series of $G^{-1}$ -- which is straightforward to calculate -- is the basis for an excellent preconditioner for ${\tilde G}(\omega)$. These findings are supported by numerical evidence.