Quasi-Monte Carlo finite element methods for elliptic PDEs with lognormal random coefficients
Quasi-Monte Carlo finite element methods for elliptic PDEs with lognormal random coefficients
复制标题
DOI：
10.1007/s00211-014-0689-y
复制
发表时间：
2015-10-01
影响因子：
2.1
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.
在本文中，我们分析了（）中多面体结构域的扩散问题的数值近似值，扩散系数给定为对数正常的随机场，即在其中空间变量是高斯随机场。该分析提出了特定的挑战，因为相应的双线性形式并非远离或超过所有可能的实现。关注计算扩散问题解的线性函数的期望值，我们为根据（1）（1）物理空间中标准连续和分段线性有限元近似构建的方法进行了严格的误差分析； （2）用于计算实现的Karhunen-Losve扩展的截断（导致可能具有高维参数化的确定性扩散问题）； （3）基于晶格的准蒙特卡罗（QMC）正交规则，用于计算积分的参数空间，该规则定义了预期值。本文包含新的错误分析，其中说明了所有三种近似类型的影响。 QMC分析基于对欧几里得空间无界域的高维积分的随机移动晶格规则的最新结果，这表明（在适当条件下）正交点差异差异，在其中，正交点的数量衰减了是任意的，并且在渐近误差中隐含常数的位置独立于整合域的维度。
In this paper we analyze the numerical approximation of diffusion problems over polyhedral domains in (), with diffusion coefficient given as a lognormal random field, i.e., where is the spatial variable and is a Gaussian random field. The analysis presents particular challenges since the corresponding bilinear form is not uniformly bounded away from or over all possible realizations of . Focusing on the problem of computing the expected value of linear functionals of the solution of the diffusion problem, we give a rigorous error analysis for methods constructed from (1) standard continuous and piecewise linear finite element approximation in physical space; (2) truncated Karhunen-LoSve expansion for computing realizations of (leading to a possibly high-dimensional parametrized deterministic diffusion problem); and (3) lattice-based quasi-Monte Carlo (QMC) quadrature rules for computing integrals over parameter space which define the expected values. The paper contains novel error analysis which accounts for the effect of all three types of approximation. The QMC analysis is based on a recent result on randomly shifted lattice rules for high-dimensional integrals over the unbounded domain of Euclidean space, which shows that (under suitable conditions) the quadrature error decays with with respect to the number of quadrature points , where is arbitrarily small and where the implied constant in the asymptotic error bound is independent of the dimension of the domain of integration.