Emergent Mind
Numerical weighted integration of functions having mixed smoothness
(2208.09108)
Published Aug 19, 2022
in
math.NA
and
cs.NA
Abstract
We investigate the approximation of weighted integrals over $\mathbb{R}d$ 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 $d \ge 2$, the upper bound is performed by sparse-grid quadratures with integration nodes on step hyperbolic crosses in the function domain $\mathbb{R}d$.
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