On efficient weighted integration via a change of variables
For researchers in numerical integration, this provides a practical method to handle unbounded-domain integrals with weighted functions, though the improvement is incremental over existing transformations.
This paper develops a change-of-variables technique for weighted integration over unbounded domains that ensures transformed integrands belong to a standard Sobolev space, enabling effective quasi-Monte Carlo and sparse grid methods. Numerical experiments confirm the approach outperforms the standard inverse-CDF transformation.
In this paper, we study the approximation of $d$-dimensional $ρ$-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 $ν$ 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 $ρ$ 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.