A Sparse Composite Collocation Finite Element Method for Elliptic SPDEs
A Sparse Composite Collocation Finite Element Method for Elliptic SPDEs
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DOI：
10.1137/090750743
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发表时间：
2011-11
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通讯作者：
M. Bieri
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文献类型：
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作者：
M. Bieri
本文提出了一种求解具有随机系数和强迫项的椭圆偏微分方程的随机配点法，该偏微分方程假设依赖于有限个随机变量。该方法由空间上的分层小波离散和概率域上的分层搭配算子序列来近似解的统计量。搭配点的选择是基于对每个随机输入变量的概率密度函数的正交多项式的零的Smolyak构造。然后，提出并分析了空间精细水平和随机搭配点的稀疏组合，从而大大降低了总体自由度。与蒙特卡罗方法一样，该算法可以解决许多不耦合的纯确定性椭圆问题，从而可以集成现有的椭圆偏微分方程快速求解器。然后，在二维域上的数值实例将证明这种稀疏复合配置有限元方法相对于“全复合”配置有限元方法和蒙特卡罗方法的优越性。
This work presents a stochastic collocation method for solving elliptic PDEs with random coefficients and forcing term which are assumed to depend on a finite number of random variables. The method consists of a hierarchic wavelet discretization in space and a sequence of hierarchic collocation operators in the probability domain to approximate the solution's statistics. The selection of collocation points is based on a Smolyak construction of zeros of orthogonal polynomials with respect to the probability density function of each random input variable. A sparse composition of levels of spatial refinements and stochastic collocation points is then proposed and analyzed, resulting in a substantial reduction of overall degrees of freedom. Like in the Monte Carlo approach, the algorithm results in solving a number of uncoupled, purely deterministic elliptic problems, which allows the integration of existing fast solvers for elliptic PDEs. Numerical examples on two-dimensional domains will then demonstrate the superiority of this sparse composite collocation finite element method compared to the “full composite” collocation finite element method and the Monte Carlo method.