Optimal quasi-Monte Carlo rules on order 2 digital nets for the numerical integration of multivariate periodic functions

Abstract

We investigate quasi-Monte Carlo rules for the numerical integration of multivariate periodic functions from Besov spaces $S_{p,q}^{r}B({\mathbb{T}}^d)$ with dominating mixed smoothness $1/p<r<2$. We show that order 2 digital nets achieve the optimal rate of convergence $N^{-r}(\log N{)}^{(d-1)(1-1/q)}$. The logarithmic term does not depend on $r$ and hence improves the known bound provided by J. Dick for the special case of Sobolev spaces $H_{\text{mix}}^r({\mathbb{T}}^d)$. Secondly, the rate of convergence is independent of the integrability $p$ of the Besov space, which allows for sacrificing integrability in order to gain Besov regularity. Our method combines characterizations of periodic Besov spaces with dominating mixed smoothness via Faber bases with sharp estimates of Haar coefficients for the discrepancy function of higher order digital nets. Moreover, we provide numerical computations which indicate that this bound also holds for the case $r=2$.

arXiv comment7 figures in Num. Mathem., online 2015

References in corpus (6)

• Walsh spaces containing smooth functions and quasi-Monte Carlo rules of arbitrary high order
• Higher order QMC Galerkin discretization for parametric operator equations
• Explicit constructions of quasi-Monte Carlo rules for the numerical integration of high dimensional periodic functions
• Quasi-Monte Carlo methods for integration of functions with dominating mixed smoothness in arbitrary dimension
• Discrepancy and integration in function spaces with dominating mixed smoothness
• L_p- and S_{p,q}^rB-discrepancy of (order 2) digital nets

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• Optimal order quadrature error bounds for infinite-dimensional higher order digital sequences
• An explicit construction of optimal order quasi-Monte Carlo rules for smooth integrands
• On "Upper error bounds for quadrature formulas on function classes" by K. K. Frolov
• A higher order Faber spline basis for sampling discretization of functions
• Richardson extrapolation allows truncation of higher order digital nets and sequences