Quasi-Monte Carlo methods for integration of functions with dominating mixed smoothness in arbitrary dimension

Abstract

In a celebrated construction, Chen and Skriganov gave explicit examples of point sets achieving the best possible $L_2$-norm of the discrepancy function. We consider the discrepancy function of the Chen-Skriganov point sets in Besov spaces with dominating mixed smoothness and show that they also achieve the best possible rate in this setting. The proof uses a $b$-adic generalization of the Haar system and corresponding characterizations of the Besov space norm. Results for further function spaces and integration errors are concluded.

arXiv admin note: text overlap with arXiv:1109.4548

References in corpus (3)

• Discrepancy of generalized Hammersley type point sets in Besov spaces of dominating mixed smoothness
• On lower bounds for the L_2-discrepancy
• Discrepancy and integration in function spaces with dominating mixed smoothness

Cited by in corpus (8)

• Calculation of Discrepancy Measures and Applications
• The role of Frolov's cubature formula for functions with bounded mixed derivative
• Optimal quasi-Monte Carlo rules on order 2 digital nets for the numerical integration of multivariate periodic functions
• Optimal order quasi-Monte Carlo integration in weighted Sobolev spaces of arbitrary smoothness
• Optimal $L_p$-discrepancy bounds for second order digital sequences
• $L_p$-discrepancy of the symmetrized van der Corput sequence
• L_p- and S_{p,q}^rB-discrepancy of the symmetrized van der Corput sequence and modified Hammersley point sets in arbitrary bases
• Lower bounds for the integration error for multivariate functions with mixed smoothness and optimal Fibonacci cubature for functions on the square