Multi-level Monte Carlo Finite Element method for elliptic PDEs with stochastic coefficients
Multi-level Monte Carlo Finite Element method for elliptic PDEs with stochastic coefficients
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DOI：
10.1007/s00211-011-0377-0
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发表时间：
2011-09-01
影响因子：
2.1
通讯作者：
Zollinger, Nathaniel
中科院分区：
文献类型：
--
作者：
Barth, Andrea;Schwab, Christoph;Zollinger, Nathaniel
在蒙特卡洛方法中，样本量的四倍，将误差减半。在随机部分微分方程（SPDE）的模拟中，总工作是样本量次偏差方程实例的解决方案成本。引入了多层蒙特卡洛方法，在某些情况下，可以将整体工作减少到确定性PDE实例的离散化。模型问题是带有随机系数的椭圆方程。多级蒙特卡洛错误和工作估计是针对解决方案的平均值和更高时刻的。事实证明，计算均值场的总体复杂性以及随机解的K点相关性在单个多级别的求解确定性椭圆问题的未知数中具有对数线性的复杂性。数值示例完成了理论分析。
In Monte Carlo methods quadrupling the sample size halves the error. In simulations of stochastic partial differential equations (SPDEs), the total work is the sample size times the solution cost of an instance of the partial differential equation. A Multi-level Monte Carlo method is introduced which allows, in certain cases, to reduce the overall work to that of the discretization of one instance of the deterministic PDE. The model problem is an elliptic equation with stochastic coefficients. Multi-level Monte Carlo errors and work estimates are given both for the mean of the solutions and for higher moments. The overall complexity of computing mean fields as well as k-point correlations of the random solution is proved to be of log-linear complexity in the number of unknowns of a single Multi-level solve of the deterministic elliptic problem. Numerical examples complete the theoretical analysis.