Quasi-Monte Carlo Finite Element Methods for a Class of Elliptic Partial Differential Equations with Random Coefficients

作者：Frances Y. Kuo, Christoph Schwab, Ian H. Sloan · 发表于：SIAM Journal on Numerical Analysis · 年份：2012 · DOI：10.1137/110845537 · 被引用次数：247 · 研究领域：Probabilistic and Robust Engineering Design、Mathematical Approximation and Integration、Advanced Numerical Methods in Computational Mathematics

In this paper quasi-Monte Carlo (QMC) methods are applied to a class of elliptic partial differential equations (PDEs) with random coefficients, where the random coefficient is parametrized by a countably infinite number of terms in a Karhunen--Loève expansion. Models of this kind appear frequently in numerical models of physical systems, and in uncertainty quantification. The method uses a QMC method to estimate expected values of linear functionals of the exact or approximate solution of the PDE, with the expected value considered as an infinite dimensional integral in the parameter space corresponding to the randomness induced by the random coefficient. The error analysis, arguably the first rigorous application of the modern theory of QMC in weighted spaces, exploits the regularity with respect to both the physical variables (the variables in the physical domain) and the parametric variables (the parameters corresponding to randomness). In the weighted-space theory of QMC methods, “weights,” describing the varying difficulty of different subsets of the variables, are introduced in order to make sure that the high-dimensional integration problem is tractable. It turns out that the weights arising from the present analysis are of a nonstandard kind, being of neither product nor order dependent form, but instead a hybrid of the two---we refer to these as “product and order dependent weights,” or “POD weights” for short. These POD weights are of a simple enough form to permit...