Optimal randomized quadrature for weighted Sobolev and Besov classes with the Jacobi weight on the ball
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We consider the numerical integration INT_d(f)=∫_𝔹^df(x)w_μ(x)dx for the weighted Sobolev classes BW^r_p,μ and the weighted Besov classes BB_τ^r(L_p,μ) in the randomized case setting, where w_μ, μ≥0, is the classical Jacobi weight on the ball B^d, 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.
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