This paper aims at developing an efficient preconditioned iterative domain decomposition (DD) method for the sampling of linear stochastic elliptic equations. To this end, we consider a non-overlapping DD method resulting in a Symmetric Positive Definite (SPD) Schur system for almost every sampled problem. To accelerate the iterative solution of the Schur system, we propose a new stochastic preconditioning strategy that produces a preconditioner adapted to each sampled problem and converges toward the ideal preconditioner (i.e., the Schur operator itself) when the numerical parameters increase. The construction of the stochastic preconditioner is trivially parallel and takes place in an off-line stage, while the evaluation of the sample’s preconditioner during the sampling stage has a low and fixed cost. One key feature of the proposed construction is a factorized form combined with Polynomial Chaos expansions of local operators. The factorized form guarantees the SPD character of the sampled preconditioners while the local character of the PC expansions ensures a low computational complexity. The stochastic preconditioner is tested on a model problem in 2 space dimensions. In these tests, the preconditioner is very robust and significantly more efficient than the deterministic median-based preconditioner, requiring, on average, up to 7 times fewer iterations to converge. Complexity analysis suggests the scalability of the preconditioner with the number of subdomains.

本文旨在开发一种有效的预处理迭代域分解（DD）方法，用于线性随机椭圆方程的采样。为此，我们考虑了一种不重叠的DD方法，该方法导致几乎所有采样问题都采用对称正定（SPD）Schur系统。为了加快Schur系统的迭代解决方案，我们提出了一种新的随机预处理策略，该策略可生成适用于每个采样问题的预处理器，并收敛到理想的预处理器（即，则为Schur运算符本身）。随机预处理器的构建过程几乎是并行的，并且是在离线阶段进行的，而在采样阶段对样本预处理器的评估成本较低且固定。拟议结构的一个关键特征是与本地运营商的多项式混沌扩展相结合的因式分解形式。因式分解形式保证了采样的预处理器的SPD特性，而PC扩展的本地特性确保了较低的计算复杂度。随机预处理器在2个空间维度的模型问题上进行了测试。在这些测试中，预处理器非常坚固，并且比确定性的基于中值的预处理器要有效得多，平均而言，预处理器需要 最多可以减少7倍的迭代次数。复杂性分析表明，预处理器具有子域数量的可伸缩性。