Optimal order quasi-Monte Carlo integration in weighted Sobolev spaces of arbitrary smoothness

Abstract

We investigate quasi-Monte Carlo integration using higher order digital nets in weighted Sobolev spaces of arbitrary fixed smoothness $\alpha \in \mathbb{N}$, $\alpha \ge 2$, defined over the $s$-dimensional unit cube. We prove that randomly digitally shifted order $\beta$ digital nets can achieve the convergence of the root mean square worst-case error of order $N^{-\alpha }(\log N{)}^{(s-1)/2}$ when $\beta \ge 2\alpha$. The exponent of the logarithmic term, i.e., $(s-1)/2$, is improved compared to the known result by Baldeaux and Dick, in which the exponent is $s\alpha /2$. Our result implies the existence of a digitally shifted order $\beta$ digital net achieving the convergence of the worst-case error of order $N^{-\alpha }(\log N{)}^{(s-1)/2}$, which matches a lower bound on the convergence rate of the worst-case error for any cubature rule using $N$ function evaluations and thus is best possible.

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• Change of variable in spaces of mixed smoothness and numerical integration of multivariate functions on the unit cube
• 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
• Quasi-Monte Carlo integration using digital nets with antithetics
• Recent advances in higher order quasi-Monte Carlo methods
• Richardson extrapolation allows truncation of higher order digital nets and sequences
• Möbius-Transformed Trapezoidal Rule