A preconditioning strategy is proposed for the iterative solve of large numbers of linear systems with variable matrix and right-hand side which arise during the computation of solution statistics of stochastic elliptic partial differential equations with random variable coefficients sampled by Monte Carlo. Building on the assumption that a truncated Karhunen-Lo\`{e}ve expansion of a known transform of the random variable coefficient is known, we introduce a compact representation of the random coefficient in the form of a Voronoi quantizer. The number of Voronoi cells, each of which is represented by a centroidal variable coefficient, is set to the prescribed number $P$ of preconditioners. Upon sampling the random variable coefficient, the linear system assembled with a given realization of the coefficient is solved with the preconditioner whose centroidal variable coefficient is the closest to the realization. We consider different ways to define and obtain the centroidal variable coefficients, and we investigate the properties of the induced preconditioning strategies in terms of average number of solver iterations for sequential simulations, and of load balancing for parallel simulations. Another approach, which is based on deterministic grids on the system of stochastic coordinates of the truncated representation of the random variable coefficient, is proposed with a stochastic dimension which increases with the number $P$ of preconditioners. This approach allows to bypass the need for preliminary computations in order to determine the optimal stochastic dimension of the truncated approximation of the random variable coefficient for a given number of preconditioners.

翻译：针对蒙特卡洛采样随机变量系数的随机椭圆型偏微分方程解统计量计算过程中产生的大量变矩阵和右端项线性系统的迭代求解，本文提出一种预条件策略。基于已知的随机变量系数变换的截断Karhunen-Loève展开式，我们引入一种紧致的随机系数表示形式——Voronoi量化子。Voronoi单元数量（每个单元由一个质心变量系数表示）被设定为预条件子数目$P$。在采样随机变量系数时，针对特定系数实现组装的线性系统，将使用离该实现最近的质心变量系数对应的预条件子进行求解。我们研究了定义和获取质心变量系数的不同方法，并考察了所提预条件策略在串行模拟中平均求解器迭代次数以及并行模拟中负载均衡方面的特性。此外，基于随机变量系数截断表示的随机坐标系统上的确定性网格，我们提出另一种方法，其随机维度随预条件子数量$P$增加而增长。该方法可绕过为给定预条件子数目确定随机变量系数截断近似的最优随机维度所需的预计算步骤。