Lower bounds for the integration error for multivariate functions with mixed smoothness and optimal Fibonacci cubature for functions on the square

Authors: Dinh Dũng, Tino Ullrich

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....

Title:	Lower bounds for the integration error for multivariate functions with mixed smoothness and optimal Fibonacci cubature for functions on the square
Authors:	Dinh Dũng, Tino Ullrich
Keywords:	Quasi-Monte-Carlo integration; Besov spaces of mixed smoothness; Fibonacci lattice; B-spline representations; Smolyak grids
Issue Date:	2015
Publisher:	Mathematische Nachrichten

URI:	http://repository.vnu.edu.vn/handle/VNU_123/11183
Appears in Collections:	ITI - Papers