Optimal randomized quadrature for weighted Sobolev and Besov classes with the Jacobi weight on the ball

TL;DR

This paper analyzes the optimal randomized numerical integration methods for weighted Sobolev and Besov classes on the ball with Jacobi weights, showing that randomness can improve convergence rates over deterministic methods.

Contribution

It derives the order of the optimal quadrature errors in the randomized setting for weighted Sobolev and Besov classes, demonstrating improved convergence compared to deterministic methods.

Findings

Randomized quadrature errors decay as n^{-r/d-1/2+(1/p-1/2)_+}.

Randomness improves convergence rates when p>1.

Optimal error bounds are established for weighted classes on the ball.

Abstract

We consider the numerical integration $${\rm INT}_d(f)=\int_{\mathbb{B}^{d}}f(x)w_\mu(x)dx $$ for the weighted Sobolev classes $BW^{r}_{p,\mu}$ and the weighted Besov classes $BB_\tau^r(L_{p,\mu})$ in the randomized case setting, where $w_\mu, \,\mu\ge0,$ is the classical Jacobi weight on the ball $\Bbb B^d$, $1\le p\le \infty$, $r>(d+2\mu)/p$, and $0<\tau\le\infty$. For the above two classes, we obtain the orders of the optimal quadrature errors in the randomized case setting are $n^{-r/d-1/2+(1/p-1/2)_+}$. Compared to the orders $n^{-r/d}$ of the optimal quadrature errors in the deterministic case setting, randomness can effectively improve the order of convergence when $p>1$.