Quasi-Monte Carlo integration over $\mathbb{R}^s$ with boundary-damping importance sampling

Abstract

This paper proposes a new importance sampling (IS) that is tailored to quasi-Monte Carlo (QMC) integration over $\mathbb{R}^s$. IS introduces a multiplicative adjustment to the integrand by compensating the sampling from the proposal instead of the target distribution. Improper proposals result in severe adjustment factor for QMC. Our strategy is to first design a adjustment factor to meet desired regularities and then determine a tractable transport map from the standard uniforms to the proposal for using QMC quadrature points as inputs. The transport map has the effect of damping the boundary growth of the resulting integrand so that the effectiveness of QMC can be reclaimed. Under certain conditions on the original integrand, our proposed IS enjoys a fast convergence rate independently of the dimension $s$, making it amenable to high-dimensional problems.