Quasi-Monte Carlo methods for elliptic PDEs with random coefficients and applications
Quasi-Monte Carlo methods for elliptic PDEs with random coefficients and applications
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DOI：
10.1016/j.jcp.2011.01.023
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发表时间：
2011-05-10
影响因子：
4.1
通讯作者：
Sloan, I. H.
中科院分区：
文献类型：
--
作者：
Graham, I. G.;Kuo, F. Y.;Sloan, I. H.
我们设计和实施了准蒙特卡洛方法，用于计算具有随机系数的一类椭圆形偏微分方程解决方案的非线性功能的期望。我们的动机来自随机多孔培养基中的流体流动，其中相关功能包括空间任何点的流体压力/速度或速度场传输的污染羽流的突破性时间。我们的重点是需要大量随机变量来建模系数场。作为经典蒙特卡洛的替代方案，我们在这里采用了准蒙特卡罗方法，这些方法在适当（通常是高度）参数空间中使用确定性选择的样品点。 PDE解决方案的每个实现都需要空间中的有限元（Fe）近似，这是通过将系数场的实现限制在适当的常规空间网格（不一定与Fe网格相同的）中完成的。在统计均匀的情况下，可以对相应的协方差矩阵进行对角度化，并且可以使用FFT有效地计算所需的系数实现。通过这种方式，我们避免使用截短的karhunen-loeve扩展，但在参数空间中引入了高标称维度。报道了具有二维粗糙随机场，高方差和较小长度量表的数值实验，表明准蒙特卡罗方法始终优于蒙特卡洛法，误差较小，明显好于O（N-1/2） ）收敛率，其中n是样品数量。此外，随着标称维数的增加，准蒙特卡洛法的收敛速率似乎不会降解。报道了尺寸高至106的示例。 （c）2011 Elsevier Inc.保留所有权利。
We devise and implement quasi-Monte Carlo methods for computing the expectations of nonlinear functionals of solutions of a class of elliptic partial differential equations with random coefficients. Our motivation comes from fluid flow in random porous media, where relevant functionals include the fluid pressure/velocity at any point in space or the breakthrough time of a pollution plume being transported by the velocity field. Our emphasis is on situations where a very large number of random variables is needed to model the coefficient field. As an alternative to classical Monte Carlo, we here employ quasi-Monte Carlo methods, which use deterministically chosen sample points in an appropriate (usually high-dimensional) parameter space. Each realization of the PDE solution requires a finite element (FE) approximation in space, and this is done using a realization of the coefficient field restricted to a suitable regular spatial grid (not necessarily the same as the FE grid). In the statistically homogeneous case the corresponding covariance matrix can be diagonalized and the required coefficient realizations can be computed efficiently using FFT. In this way we avoid the use of a truncated Karhunen-Loeve expansion, but introduce high nominal dimension in parameter space. Numerical experiments with 2-dimensional rough random fields, high variance and small length scale are reported, showing that the quasi-Monte Carlo method consistently outperforms the Monte Carlo method, with a smaller error and a noticeably better than O(N-1/2) convergence rate, where N is the number of samples. Moreover, the rate of convergence of the quasi-Monte Carlo method does not appear to degrade as the nominal dimension increases. Examples with dimension as high as 106 are reported. (C) 2011 Elsevier Inc. All rights reserved.