Efficient Iterative Solvers for Stochastic Galerkin Discretizations of Log-Transformed Random Diffusion Problems
Efficient Iterative Solvers for Stochastic Galerkin Discretizations of Log-Transformed Random Diffusion Problems
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DOI：
10.1137/110836675
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发表时间：
2012-04
期刊：
影响因子：
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通讯作者：
E. Ullmann;H. Elman;O. Ernst
中科院分区：
文献类型：
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作者：
E. Ullmann;H. Elman;O. Ernst
本文考虑了扩散系数为随机场指数的稳态扩散问题的数值解。标准的随机Galerkin制定这个问题是计算上的要求，因为它的不确定组件的非线性结构，我们认为这个问题的一个重新制定的版本作为一个随机对流扩散问题的随机对流速度线性依赖于一个固定数量的独立截断高斯随机变量。相关的伽辽金矩阵是非对称的，但稀疏，并允许快速矩阵向量乘法与最佳的复杂性。我们构造和分析两个块对角预条件，这个Galerkin矩阵使用Krylov子空间方法，如广义最小残差法。我们测试所提出的预处理方法的效率，并比较迭代求解器的性能在扩散和对流扩散配方的模型问题。
We consider the numerical solution of a steady-state diffusion problem where the diffusion coefficient is the exponent of a random field. The standard stochastic Galerkin formulation of this problem is computationally demanding because of the nonlinear structure of the uncertain component of it. We consider a reformulated version of this problem as a stochastic convection-diffusion problem with random convective velocity that depends linearly on a fixed number of independent truncated Gaussian random variables. The associated Galerkin matrix is nonsymmetric but sparse and allows for fast matrix-vector multiplications with optimal complexity. We construct and analyze two block-diagonal preconditioners for this Galerkin matrix for use with Krylov subspace methods such as the generalized minimal residual method. We test the efficiency of the proposed preconditioning approaches and compare the iterative solver performance for a model problem posed in both diffusion and convection-diffusion formulations.