Optimal quadrature for weighted function spaces on multivariate domains

TL;DR

This paper investigates optimal numerical integration methods for weighted Sobolev and Besov function spaces on spheres, balls, and simplices, demonstrating when randomized algorithms outperform deterministic ones in convergence speed.

Contribution

It derives optimal quadrature errors for weighted Sobolev and Besov spaces on multivariate domains, highlighting the advantages of randomized algorithms under certain weight conditions.

Findings

Optimal quadrature errors for weighted Sobolev spaces on spheres.

Upper bounds for quadrature errors using least $ ext{l}_p$ approximation and Monte Carlo.

Randomized algorithms can achieve faster convergence than deterministic methods when $p>1$.

Abstract

Consider the numerical integration $${\rm Int}_{\mathbb S^d,w}(f)=\int_{\mathbb S^d}f({\bf x})w({\bf x}){\rm d}\sigma({\bf x}) $$ for weighted Sobolev classes $BW_{p,w}^r(\mathbb S^d)$ with a Dunkl weight $w$ and weighted Besov classes $BB_\gamma^\Theta(L_{p,w}(\mathbb S^d))$ with the generalized smoothness index $\Theta $ and a doubling weight $w$ on the unit sphere $\mathbb S^d$ of the Euclidean space $\mathbb R^{d+1}$ in the deterministic and randomized case settings. For $BW_{p,w}^r(\mathbb S^d)$ we obtain the optimal quadrature errors in both settings. For $BB_\gamma^\Theta(L_{p,w}(\mathbb S^d))$ we use the weighted least $\ell_p$ approximation and the standard Monte Carlo algorithm to obtain upper estimates of the quadrature errors which are optimal if $w$ is an $A_\infty$ weight in the deterministic case setting or if $w$ is a product weight in the randomized case setting. Our results show that randomized algorithms can provide a faster convergence rate than that of the deterministic ones when $p>1$. Similar results are also established on the unit ball and the standard simplex of $\mathbb R^d$.