In this paper we give a new Koksma-Hlawka type inequality for Quasi-Monte Carlo (QMC) integration. QMC integration of a function f : [0, 1) s → R by a finite point set P ⊂ [0, 1) s is the approximation of the integralWe treat a certain class of point sets P called digital nets. A Koksma-Hlawka type inequality is an inequality bounding the integration error Err(f ; P) := I(f ) − IP(f ) by a bound of the form |Err(f ; P)| ≤ C · f · D(P). We can obtain a Koksma-Hlawka type inequality by estimating bounds on |f (k)|, wheref (k) is a generalized Fourier coefficient with respect to the Walsh system. In this paper we prove bounds on Walsh coefficientsf (k) by introducing an operator called 'dyadic difference' ∂i,n. By converting dyadic differences ∂i,n to derivatives ∂ ∂x i , we get a new bound on |f (k)| for a function f whose mixed partial derivatives up to order α in each variable are continuous. This new bound is smaller than the known bound on |f (k)| under some condition. The new Koksma-Hlawka inequality is derived using this new bound on the Walsh coefficients.
Quasi-Monte Carlo (QMC) quadrature rules using higher order digital nets and sequences have been shown to achieve the almost optimal rate of convergence of the worst-case error in Sobolev spaces of arbitrary fixed smoothness α ∈ N, α ≥ 2. In a recent paper by the authors, it was proved that randomly-digitally-shifted order 2α digital nets in prime base b achieve the best possible rate of convergence of the root mean square worst-case error of order N −α (log N ) (s−1)/2 for N = b m , where N and s denote the number of points and the dimension, respectively, which implies the existence of an optimal order QMC rule. More recently, the authors provided an explicit construction of such an optimal order QMC rule by using Chen-Skriganov's digital nets in conjunction with Dick's digit interlacing composition. These results were for fixed number of points. In this paper we give a more general result on an explicit construction of optimal order QMC rules for arbitrary fixed smoothness α ∈ N including the endpoint case α = 1. That is, we prove that the projection of any infinite-dimensional order 2α + 1 digital sequence in prime base b onto the first s coordinates achieves the best possible rate of convergence of the worst-case error of order N −α (log N ) (s−1)/2 for N = b m . The explicit construction presented in this paper is not only easy to implement but also extensible in both N and s.
We investigate quasi-Monte Carlo integration using higher order digital nets in weighted Sobolev spaces of arbitrary fixed smoothness α ∈ N, α ≥ 2, defined over the s-dimensional unit cube. We prove that randomly digitally shifted order β digital nets can achieve the convergence of the root mean square worst-case error of order N −α (log N ) (s−1)/2 when β ≥ 2α. 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 sα/2. Our result implies the existence of a digitally shifted order β digital net achieving the convergence of the worst-case error of order N −α (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.
We study multivariate numerical integration of smooth functions in weighted Sobolev spaces with dominating mixed smoothness α ≥ 2 defined over the s-dimensional unit cube. We propose a new quasi-Monte Carlo (QMC)-based quadrature rule, named extrapolated polynomial lattice rule, which achieves the almost optimal rate of convergence. Extrapolated polynomial lattice rules are constructed in two steps: i) construction of classical polynomial lattice rules over F b with α consecutive sizes of nodes, b m−α+1 , . . . , b m , and ii) recursive application of Richardson extrapolation to a chain of α approximate values of the integral obtained by consecutive polynomial lattice rules.We prove the existence of good extrapolated polynomial lattice rules achieving the almost optimal order of convergence of the worst-case error in Sobolev spaces with general weights. Then, by restricting to product weights, we show that such good extrapolated polynomial lattice rules can be constructed by the fast component-by-component algorithm under a computable quality criterion. The required total construction cost is of order (s + α)N log N , which improves the currently known result for interlaced polynomial lattice rule, that is of order sαN log N . We also study the dependence of the worst-case error bound on the dimension.A big advantage of our method compared to interlaced polynomial lattice rules is that the fast QMC matrix vector method can be used in this setting, while still achieving the same rate of convergence. Such a method was previously not known.Numerical experiments for test integrands support our theoretical result.
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