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

• Main Author: Dinh Dũng, Tino Ullrich
• Format: Book Book chapter Dataset
• Published: Mathematische Nachrichten 2016
• Subjects: Quasi-Monte-Carlo integration; Besov spaces of mixed smoothness; Fibonacci lattice; B-spline representations; Smolyak grids
• Online Access: http://repository.vnu.edu.vn/handle/VNU_123/11183
• Institution: Vietnam National University, Hanoi

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.