Lower bounds for the integration error for multivariate functions with mixed smoothness and optimal Fibonacci cubature for functions on the square |
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| Authors: | Dinh D?ng Tino Ullrich |
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| Affiliation: | 1. Vietnam National University, Hanoi, Information Technology Institute, Hanoi, Vietnam;2. Hausdorff‐Center for Mathematics and Institute for Numerical Simulation, 53115 Bonn, Germany |
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| Abstract: | We prove lower bounds for the error of optimal cubature formulae for d‐variate functions from Besov spaces of mixed smoothness  in the case  ,  and  , where  is either the d‐dimensional torus  or the d‐dimensional unit cube  . In addition, we prove upper bounds for QMC integration on the Fibonacci‐lattice for bivariate periodic functions from  in the case  ,  and  . A non‐periodic modification of this classical formula yields upper bounds for  if  . In combination these results yield the correct asymptotic error of optimal cubature formulae for functions from  and indicate that a corresponding result is most likely also true in case  . This is compared to the correct asymptotic of optimal cubature formulae on Smolyak grids which results in the observation that any cubature formula on Smolyak grids can never achieve the optimal worst‐case error. |
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| Keywords: | Quasi‐Monte‐Carlo integration Besov spaces of mixed smoothness Fibonacci lattice B‐spline representations Smolyak grids 41A55 65D32 41A25 41A58 41A63 |
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