This study analyzes the nonasymptotic convergence behavior of the quasi-Monte Carlo (QMC) method with applications to linear elliptic partial differential equations (PDEs) with lognormal coefficients. Building upon the error analysis presented in (Owen, 2006), we derive a nonasymptotic convergence estimate depending on the specific integrands, the input dimensionality, and the finite number of samples used in the QMC quadrature. We discuss the effects of the variance and dimensionality of the input random variable. Then, we apply the QMC method with importance sampling (IS) to approximate deterministic, real-valued, bounded linear functionals that depend on the solution of a linear elliptic PDE with a lognormal diffusivity coefficient in bounded domains of $\mathbb{R}^d$, where the random coefficient is modeled as a stationary Gaussian random field parameterized by the trigonometric and wavelet-type basis. We propose two types of IS distributions, analyze their effects on the QMC convergence rate, and observe the improvements.

翻译：本研究分析了准蒙特卡罗（QMC）方法的非渐近收敛行为，并将其应用于具有对数正态系数的线性椭圆型偏微分方程（PDE）。基于(Owen, 2006)提出的误差分析，我们推导出一个依赖于具体被积函数、输入维度以及QMC求积中有限样本数量的非渐近收敛估计，并讨论了输入随机变量的方差和维度的影响。随后，我们将QMC方法与重要性抽样（IS）相结合，用于逼近定义在$\mathbb{R}^d$有界域内、依赖具有对数正态扩散系数线性椭圆型PDE解的确定型实值有界线性泛函，其中随机系数被建模为用三角和小波型基参数化的平稳高斯随机场。我们提出了两种类型的IS分布，分析了它们在QMC收敛速率上的影响，并观察到改进效果。