We study numerical integration over bounded regions in $\mathbb{R}^s, s\ge1$ with respect to some probability measure. We replace random sampling with quasi-Monte Carlo methods, where the underlying point set is derived from deterministic constructions that aim to fill the space more evenly than random points. Such quasi-Monte Carlo point sets are ordinarily designed for the uniform measure, and the theory only works for product measures when a coordinate-wise transformation is applied. Going beyond this setting, we first consider the case where the target density is a mixture distribution where each term in the mixture comes from a product distribution. Next we consider target densities which can be approximated with such mixture distributions. We require the approximation to be a sum of coordinate-wise products and the approximation to be positive everywhere (so that they can be re-scaled to probability density functions). We use tensor product hat function approximations for this purpose here, since a hat function approximation of a positive function is itself positive. We also study more complex algorithms, where we first approximate the target density with a general Gaussian mixture distribution and approximate the mixtures with an adaptive hat function approximation on rotated intervals. The Gaussian mixture approximation allows us to locate the essential parts of the target density, whereas the adaptive hat function approximation allows us to approximate the finer structure of the target density. We prove convergence rates for each of the integration techniques based on quasi-Monte Carlo sampling for integrands with bounded partial mixed derivatives. The employed algorithms are based on digital $(t,s)$-sequences over the finite field $\mathbb{F}_2$ and an inversion method. Numerical examples illustrate the performance of the algorithms for some target densities and integrands.

翻译：我们研究在 $\mathbb{R}^s, s\ge1$ 有界区域上关于某概率测度的数值积分问题。我们采用拟蒙特卡罗方法替代随机采样，其中底层点集由确定性构造生成，旨在比随机点更均匀地填充空间。此类拟蒙特卡罗点集通常针对均匀测度设计，且理论仅适用于经坐标变换后的乘积测度。超越这一设定，我们首先考虑目标密度为混合分布的情形，其中混合项均来自乘积分布。其次，我们研究可通过此类混合分布近似的目标密度。我们要求近似解为坐标乘积之和，且处处为正（从而可缩放为概率密度函数）。为此采用张量积帽函数近似，因为正函数的帽函数近似结果本身为正。我们还研究了更复杂的算法：先用一般高斯混合分布近似目标密度，再通过旋转区间上的自适应帽函数近似逼近混合分布。高斯混合近似使我们能定位目标密度的关键区域，而自适应帽函数近似则能捕捉目标密度的精细结构。针对具有有界混合偏导数的被积函数，我们证得各基于拟蒙特卡罗采样的积分技术的收敛速率。采用算法基于有限域 $\mathbb{F}_2$ 上的数字 $(t,s)$-序列及逆变换法。数值算例展示了算法对特定目标密度与积分函数的性能表现。