  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.
