Stochastic Galerkin finite element method (SGFEM) provides an efficient alternative to traditional sampling methods for the numerical solution of linear elliptic partial differential equations with parametric or random inputs. However, computing stochastic Galerkin approximations for a given problem requires the solution of large coupled systems of linear equations. Therefore, an effective and bespoke iterative solver is a key ingredient of any SGFEM implementation. In this paper, we analyze a class of truncation preconditioners for SGFEM. Extending the idea of the mean-based preconditioner, these preconditioners capture additional significant components of the stochastic Galerkin matrix. Focusing on the parametric diffusion equation as a model problem and assuming affine-parametric representation of the diffusion coefficient, we perform spectral analysis of the preconditioned matrices and establish optimality of truncation preconditioners with respect to SGFEM discretization parameters. Furthermore, we report the results of numerical experiments for model diffusion problems with affine and non-affine parametric representations of the coefficient. In particular, we look at the efficiency of the solver (in terms of iteration counts for solving the underlying linear systems) and compare truncation preconditioners with other existing preconditioners for stochastic Galerkin matrices, such as the mean-based and the Kronecker product ones.
翻译:Galerkin Stochastic Galerkin 有限元素法(SGFEM)为具有参数或随机输入的线性椭圆部分偏差方程的数值解决方案提供了一个有效的替代传统抽样方法,然而,为某个特定问题计算随机偏差Galerkin近似值需要解决大量连带线式线性方程系统,因此,一个有效和可口可口的迭代求解器是SGFEM实施的任何关键要素。在本文件中,我们分析SGFEM的脱节前端标准(SGFEM)的一个类别。扩展了基于平均前提的理念,这些前提者捕捉到了Stochetic Galkin 矩阵的更多重要组成部分。我们把偏重于参数的偏差扩散方方方程式作为模型问题,假设扩散系数的偏差分度代表度,我们对先决条件矩阵进行光谱分析,确定与SGFFEM离散性参数有关的脱节性前端标准的最佳性。此外,我们报告了模型扩散问题模型的数值实验结果,与基于正弦和非对系数的非反向性准度准值的模型的描述。我们特别将Slodiceral-suder-requestrequest 的解器的模型作为比较其正前置的模型的数值和定线性定线性定值的模型的比正基底基数。