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

Abstract

We consider the numerical integrationINTd(f)=∫Bdf(x)wμ(x)dx for the weighted Sobolev classes BWp,μr and the weighted Besov classes BBτr(Lp,μ) in the randomized case setting, where wμ,μ≥0, is the classical Jacobi weight on the ball Bd, 1≤p≤∞, r>(d+2μ)/p, and 0<τ≤∞. 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.