Numerical weighted integration of functions having mixed smoothness

TL;DR

This paper analyzes the convergence rates of optimal quadrature methods for weighted integrals of functions with mixed smoothness in multiple dimensions, providing bounds and specific results for one and higher dimensions.

Contribution

It establishes upper and lower bounds for the convergence rates of optimal quadratures for weighted Sobolev spaces with mixed smoothness, including the use of sparse-grid quadratures in higher dimensions.

Findings

Optimal convergence rate in 1D for weighted Sobolev spaces.

Upper bounds achieved by sparse-grid quadratures in higher dimensions.

Convergence bounds depend on the dimension and smoothness of the integrand.

Abstract

We investigate the approximation of weighted integrals over $\mathbb{R}^d$ for integrands from weighted Sobolev spaces of mixed smoothness. We prove upper and lower bounds of the convergence rate of optimal quadratures with respect to $n$ integration nodes for functions from these spaces. In the one-dimensional case $(d=1)$, we obtain the right convergence rate of optimal quadratures. For $d \ge 2$, the upper bound is performed by sparse-grid quadratures with integration nodes on step hyperbolic crosses in the function domain $\mathbb{R}^d$.