Preconditioning Stochastic Galerkin Saddle Point Systems
Preconditioning Stochastic Galerkin Saddle Point Systems
复制标题
预处理随机伽辽金鞍点系统
DOI：
10.1137/090777797
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发表时间：
2010
期刊：
影响因子：
--
通讯作者：
E. Ullmann
中科院分区：
文献类型：
--
作者：
C. Powell;E. Ullmann
确定性二阶椭圆偏微分方程的混合有限元离散导致鞍点系统的迭代求解器和预处理器的研究是成熟的。随机二阶椭圆型偏微分方程的解的Galerkin逼近，将物理空间中的标准混合有限元离散与概率空间上的全局多项式逼近相耦合，也会产生具有熟悉鞍点结构的线性系统。对于随机非线性问题，这类系统的解决方案提出了一个严重的计算挑战。块是与两个不同的离散化相关的矩阵对的克罗内克积的总和，并且系统很大，反映了大多数随机近似方案中固有的维数灾难。此外，对于本文考虑的问题，鞍点矩阵的前导块是块稠密的，并且矩阵向量积的成本是非平凡的。我们实现了一个随机Galerkin离散的稳态扩散问题写为一个混合的一阶系统。扩散系数被假定为对数正态随机场，通过有限数量的高斯随机变量的非线性函数近似。我们研究由此产生的鞍点系统和调查的Schur补和增广型的块对角预处理器的效率与最小残差方法（MINRES）使用。通过引入所谓的克罗内克积预条件，我们提高了廉价的，基于平均值的预条件的随机非线性扩散系数的统计特性的鲁棒性。
Mixed finite element discretizations of deterministic second-order elliptic PDEs lead to saddle point systems for which the study of iterative solvers and preconditioners is mature. Galerkin approximation of solutions of stochastic second-order elliptic PDEs, which couple standard mixed finite element discretizations in physical space with global polynomial approximation on a probability space, also give rise to linear systems with familiar saddle point structure. For stochastically nonlinear problems, the solution of such systems presents a serious computational challenge. The blocks are sums of Kronecker products of pairs of matrices associated with two distinct discretizations, and the systems are large, reflecting the curse of dimensionality inherent in most stochastic approximation schemes. Moreover, for the problems considered herein, the leading blocks of the saddle point matrices are block-dense, and the cost of a matrix vector product is nontrivial. We implement a stochastic Galerkin discretization for the steady-state diffusion problem written as a mixed first-order system. The diffusion coefficient is assumed to be a lognormal random field, approximated via a nonlinear function of a finite number of Gaussian random variables. We study the resulting saddle point systems and investigate the efficiency of block-diagonal preconditioners of Schur complement and augmented type for use with the minimal residual method (MINRES). By introducing so-called Kronecker product preconditioners, we improve the robustness of cheap, mean-based preconditioners with respect to the statistical properties of the stochastically nonlinear diffusion coefficients.