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：本研究分析了拟蒙特卡洛（QMC）方法在具有对数正态系数的线性椭圆偏微分方程（PDEs）应用中的非渐近收敛行为。基于（Owen, 2006）中提出的误差分析，我们推导了一个非渐近收敛估计，该估计依赖于具体的被积函数、输入维度以及QMC求积中使用的有限样本数量。我们讨论了输入随机变量的方差和维度的影响。随后，我们将结合重要性采样（IS）的QMC方法应用于逼近确定性、实值、有界的线性泛函，这些泛函依赖于定义在$\mathbb{R}^d$有界域上、具有对数正态扩散系数的线性椭圆PDE的解，其中随机系数被建模为由三角函数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，分析了它们对QMC收敛率的影响，并观察到了改进效果。