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

01/18/2022
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by   Jiansong Li, et al.
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We consider the numerical integration INT_d(f)=āˆ«_š”¹^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‰¤āˆž, r>(d+2Ī¼)/p, and 0<Ļ„ā‰¤āˆž. 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.

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