Laguerre- and Laplace-weighted integration of mixed-smoothness functions

TL;DR

This paper analyzes the approximation of weighted integrals of mixed-smoothness functions using sparse-grid quadratures, providing bounds on convergence rates for functions in Laguerre- or Laplace-weighted Sobolev spaces.

Contribution

It establishes upper and lower bounds for the convergence rates of optimal quadratures for these weighted integrals, utilizing sparse-grid methods with hyperbolic structures.

Findings

Derived bounds for convergence rates of quadratures

Implemented sparse-grid quadratures on hyperbolic corners and crosses

Enhanced understanding of approximation efficiency for weighted Sobolev spaces

Abstract

We investigate the approximation of generalized Laguerre- or Laplace-weighted integrals over $\mathbb{R}^d_+$ or $\mathbb{R}^d$ of functions from generalized Laguerre- or Laplace-weighted Sobolev spaces of mixed smoothness, respectively. We prove upper and lower bounds of the convergence rate of optimal quadratures with respect to $n$ integration nodes for functions from these spaces. The upper bound is performed by sparse-grid quadratures with integration nodes on step hyperbolic corners or hyperbolic crosses in the function domain $\mathbb{R}^d_+$ or $\mathbb{R}^d$, respectively.