On Solving Stochastic Collocation Systems with Algebraic Multigrid
On Solving Stochastic Collocation Systems with Algebraic Multigrid
复制标题
用代数多重网格求解随机配置系统
DOI：
10.1093/imanum/drr034
复制
发表时间：
2012
影响因子：
2.1
通讯作者：
C. Powell
中科院分区：
文献类型：
--
作者：
A. Gordon;C. Powell
随机配置方法有助于使用随机数据对偏微分方程 (PDE) 进行数值求解，并产生类似线性系统的长序列。当具有随机扩散系数的椭圆偏微分方程在物理域中使用混合有限元方法离散化时，我们获得鞍点系统。单独考虑时，这些问题解决起来都是微不足道的。挑战在于利用它们的相似性来回收信息并最小化解决整个序列的成本。我们将随机搭配应用于模型随机椭圆问题，并使用 Raviart?Thomas 元素在物理空间中进行离散化。我们为由此产生的线性系统提出了一种有效的解决策略，该策略比文献中的任何其他系统都更加稳健。特别是，我们表明，如果重用关键设置信息，则使用微调代数多重网格预处理是可行的。所提出的求解器对于随机线性和非线性数据的离散化和统计参数的变化具有鲁棒性。
Stochastic collocation methods facilitate the numerical solution of partial differential equations (PDEs) with random data and give rise to long sequences of similar linear systems. When elliptic PDEs with random diffusion coefficients are discretized with mixed finite element methods in the physical domain we obtain saddle point systems. These are trivial to solve when considered individually; the challenge lies in exploiting their similarities to recycle information and minimize the cost of solving the entire sequence. We apply stochastic collocation to a model stochastic elliptic problem and discretize in physical space using Raviart?Thomas elements. We propose an efficient solution strategy for the resulting linear systems that is more robust than any other in the literature. In particular, we show that it is feasible to use finely-tuned algebraic multigrid preconditioning if key set-up information is reused. The proposed solver is robust with respect to variations in the discretization and statistical parameters for stochastically linear and nonlinear data.