In this paper, we study the approximation of $d$-dimensional $\rho$-weighted integrals over unbounded domains $\mathbb{R}_+^d$ or $\mathbb{R}^d$ using a special change of variables, so that quasi-Monte Carlo (QMC) or sparse grid rules can be applied to the transformed integrands over the unit cube. We consider a class of integrands with bounded $L_p$ norm of mixed partial derivatives of first order, where $p\in[1,+\infty].$ The main results give sufficient conditions on the change of variables $\nu$ which guarantee that the transformed integrand belongs to the standard Sobolev space of functions over the unit cube with mixed smoothness of order one. These conditions depend on $\rho$ and $p$. The proposed change of variables is in general different than the standard change based on the inverse of the cumulative distribution function. We stress that the standard change of variables leads to integrands over a cube; however, those integrands have singularities which make the application of QMC and sparse grids ineffective. Our conclusions are supported by numerical experiments.

中文翻译：

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通过变量的变化进行有效的加权积分

在本文中，我们研究了无界域 $\mathbb{R}_+^d$ 或 $\mathbb{R}^d$ 上的 $d$ 维 $\rho$ 加权积分的近似值变量，以便准蒙特卡罗 (QMC) 或稀疏网格规则可以应用于单位立方体上的转换被积函数。我们考虑一类具有一阶混合偏导数的有界$L_p$范数的被积函数，其中$p\in[1,+\infty].$主要结果给出了变量$\nu$变化的充分条件，其中保证变换后的被积函数属于具有一阶混合平滑度的单位立方体上函数的标准 Sobolev 空间。这些条件取决于 $\rho$ 和 $p$。建议的变量变化一般不同于基于累积分布函数倒数的标准变化。我们强调变量的标准变化导致立方体上的被积函数；然而，这些被积函数具有奇异性，使得 QMC 和稀疏网格的应用无效。我们的结论得到了数值实验的支持。