SIAM/ASA Journal on Uncertainty Quantification, Volume 8, Issue 1, Page 88-113, January 2020.
The paper focuses on numerical solution of parametrized diffusion equations with scalar parameter-dependent coefficient function by the stochastic (spectral) Galerkin method. We study preconditioning of the related discretized problems using preconditioners obtained by modifying the stochastic part of the partial differential equation. We present a simple but general approach for obtaining two-sided bounds to the spectrum of the resulting matrices, based on a particular splitting of the discretized operator. Using this tool and considering the stochastic approximation space formed by classical orthogonal polynomials, we obtain new spectral bounds depending solely on the properties of the coefficient function and the type of the approximation polynomials for several classes of block-diagonal preconditioners. These bounds are guaranteed and applicable to various distributions of parameters. Moreover, the conditions on the parameter-dependent coefficient function are only local, and therefore less restrictive than those usually assumed in the literature.

中文翻译：

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随机Galerkin问题的块预处理：新的双向保证光谱界

SIAM / ASA不确定性量化杂志，第8卷，第1期，第88-113页，2020年1月。
本文重点研究了随机（频谱）Galerkin方法求解带标量参数相关系数函数的参数化扩散方程的数值解。我们使用通过修改偏微分方程的随机部分而获得的预处理器来研究相关离散问题的预处理。我们提出了一种简单但通用的方法，用于基于离散化算子的特定划分来获得所得矩阵谱的两侧边界。使用此工具并考虑由经典正交多项式形成的随机近似空间，我们仅根据系数函数的性质和几类块对角形预处理器的近似多项式的类型来获得新的谱界。这些界限得到保证，并适用于各种参数分布。此外，与参数相关的系数函数上的条件仅是局部的，因此比文献中通常假定的条件限制较少。