For the numerical solution of linear systems that arise from discretized linear partial differential equations, multigrid and domain decomposition methods are well established. Multigrid methods are known to have optimal complexity and domain decomposition methods are in particular useful for parallelization of the implemented algorithm. For linear random operator equations, the classical theory is not directly applicable, since condition numbers of system matrices may be close to degenerate due to non-uniform random input. It is shown that iterative methods converge in the strong, i.e. \(L^p\), sense if the random input satisfies certain integrability conditions. As a main result, standard multigrid and domain decomposition methods are applicable in the case of linear elliptic partial differential equations with lognormal diffusion coefficients and converge strongly with deterministic bounds on the computational work which are essentially optimal. This enables the application of multilevel Monte Carlo methods with rigorous, deterministic bounds on the computational work.

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随机算子方程迭代求解器的强收敛性分析

对于由离散线性偏微分方程产生的线性系统的数值解，已经很好地建立了多重网格和域分解方法。已知多网格方法具有最佳的复杂性，而域分解方法对于实现的算法的并行化特别有用。对于线性随机算子方程，经典理论不能直接应用，因为由于随机输入不均匀，系统矩阵的条件数可能接近退化。结果表明，迭代方法收敛于强值，即\（L ^ p \）表示随机输入是否满足某些可积性条件。作为主要结果，标准的多重网格和域分解方法适用于具有对数正态扩散系数的线性椭圆型偏微分方程，并且在确定性范围内可以收敛，并且在本质上是最优的。这使得在计算工作上具有严格的确定性界限的多级蒙特卡洛方法得以应用。