Numerical weighted integration of functions having mixed smoothness

08/19/2022

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by   Dinh Dũng, et al.

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We investigate the approximation of weighted integrals over ℝ^d for integrands from weighted Sobolev spaces of mixed smoothness. We prove upper and lower bounds of the convergence rate of optimal quadratures with respect to n integration nodes, for functions from these spaces. In the one-dimensional case (d=1), we obtain the right convergence rate of optimal quadratures . For d ≥ 2, the upper bound is performed by sparse-grid quadratures with integration nodes on step hyperbolic crosses in the function domain ℝ^d.