A Low-Rank Multigrid Method for the Stochastic Steady-State Diffusion Problem
A Low-Rank Multigrid Method for the Stochastic Steady-State Diffusion Problem
复制标题
随机稳态扩散问题的低阶多重网格方法
DOI：
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复制
发表时间：
2016
影响因子：
1.5
通讯作者：
Tengfei Su
中科院分区：
文献类型：
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作者：
H. Elman;Tengfei Su
研究了求解具有张量积结构的大型线性方程组的多重网格方法。这种系统是由随机偏微分方程的随机有限元离散得到的，如随机系数的稳态扩散问题。当问题的方差不太大时，可以用低秩对象很好地近似解。在多重网格算法中，矩阵迭代被截断到低秩，以减少内存需求和计算量。证明了该方法具有解析误差界的收敛性。数值实验表明，与原有的多网格求解器相比，该方法在求解伽辽金系统时具有较好的有效性，特别是在空间离散化相关自由度较大的情况下。
We study a multigrid method for solving large linear systems of equations with tensor product structure. Such systems are obtained from stochastic finite element discretization of stochastic partial differential equations such as the steady-state diffusion problem with random coefficients. When the variance in the problem is not too large, the solution can be well approximated by a low-rank object. In the proposed multigrid algorithm, the matrix iterates are truncated to low rank to reduce memory requirements and computational effort. The method is proved convergent with an analytic error bound. Numerical experiments show its effectiveness in solving the Galerkin systems compared to the original multigrid solver, especially when the number of degrees of freedom associated with the spatial discretization is large.