Parallel preconditioners and multigrid solvers for stochastic polynomial chaos discretizations of the diffusion equation at the large scale
Parallel preconditioners and multigrid solvers for stochastic polynomial chaos discretizations of the diffusion equation at the large scale
复制标题
用于大规模扩散方程随机多项式混沌离散的并行预处理器和多重网格求解器
DOI：
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发表时间：
2016
影响因子：
4.3
通讯作者：
Barry Lee
中科院分区：
文献类型：
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作者：
Barry Lee
本文提出了并行预处理器和多重网格求解器，用于求解随机系数扩散方程的随机多项式混沌方程组。这些预处理器和求解器是扩展的预处理器在早期的文件中开发的强耦合系统的椭圆型偏微分方程，是正常的系统，可以分解成一个代数耦合组件和对角微分组件。第一个预条件子，它适用于范数等价系统，通过稀疏化的代数耦合分量的逆得到。这种稀疏化导致一个有效的方法来解决这些系统在大规模，即使是大的随机系数的统计变化的问题。这个预条件子的扩展导致了独立的多重网格方法，可以直接应用于实际系统，而不是范数等价系统。这些多重网格方法利用范数等价系统的代数/微分因式分解在粗层次上产生变量解耦系统。此外，这些方法的结构允许通过重用强大的高性能软件（如Hypre库包）轻松实现软件。将建立两个网格矩阵边界，并给出数值结果。版权所有© 2015约翰威利父子有限公司.
This paper presents parallel preconditioners and multigrid solvers for solving linear systems of equations arising from stochastic polynomial chaos formulations of the diffusion equation with random coefficients. These preconditioners and solvers are extensions of the preconditioner developed in an earlier paper for strongly coupled systems of elliptic partial differential equations that are norm equivalent to systems that can be factored into an algebraic coupling component and a diagonal differential component. The first preconditioner, which is applied to the norm equivalent system, is obtained by sparsifying the inverse of the algebraic coupling component. This sparsification leads to an efficient method for solving these systems at the large scale, even for problems with large statistical variations in the random coefficients. An extension of this preconditioner leads to stand‐alone multigrid methods that can be applied directly to the actual system rather than to the norm equivalent system. These multigrid methods exploit the algebraic/differential factorization of the norm equivalent systems to produce variable‐decoupled systems on the coarse levels. Moreover, the structure of these methods allows easy software implementation through re‐use of robust high‐performance software such as the Hypre library package. Two‐grid matrix bounds will be established, and numerical results will be given. Copyright © 2015 John Wiley & Sons, Ltd.