Optimal randomized quadrature for weighted Sobolev and Besov classes with the Jacobi weight on the ball
We consider the numerical integration INT_d(f)=ā«_š¹^df(x)w_Ī¼(x)dx for the weighted Sobolev classes BW^r_p,Ī¼ and the weighted Besov classes BB_Ļ^r(L_p,Ī¼) in the randomized case setting, where w_Ī¼, Ī¼ā„0, is the classical Jacobi weight on the ball B^d, 1ā¤ pā¤ā, r>(d+2Ī¼)/p, and 0<Ļā¤ā. 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.
READ FULL TEXT