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.

翻译：PDDSparse 是一种用于在超级计算机上求解大规模椭圆边值问题的新型混合并行化方案，可视为区域分解的费曼-卡茨公式。其核心在于界面解构成的随机线性稀疏系统，该系统矩阵的条目通过蒙特卡洛模拟生成。在统计误差较小的假设下，我们证明了随机系统矩阵 ${\tilde G}(\omega)$ 接近非奇异 M-矩阵 $G$，即 ${\tilde G}(\omega)+E=G$ 且 $||E||/||G||$ 较小。通过非标准论证方法，我们界定了 $||G^{-1}||$ 和 $G$ 的条件数，表明二者随离散自由度增长而适度增加。此外，易于计算的截断诺伊曼级数 $G^{-1}$ 为 ${\tilde G}(\omega)$ 提供了优良的预条件子基础。这些结论得到了数值实验的支持。