Optimal Order Quadrature Error Bounds for Infinite-Dimensional Higher-Order Digital Sequences

Abstract

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.

4. Recent advances in higher order quasi-Monte Carlo methods

“…Now the following result, which can be regarded as a generalization of the result shown in [25,Lemma 3.7], is proven in [38,Lemma 8]. Here we state the result in a slightly more general form.…”

“…Another major step was made in a series of papers [36,37,38], where the authors refined the integration error analysis for smooth functions due to Dick [7,8] and proved that order (2α + 1) digital nets and sequences achieve the best possible order of the worst-case error for a reproducing kernel Hilbert space with dominating mixed smoothness α, which is (log N ) (s−1)/2 /N α . Note that the original work by Dick [8] proves the worst-case error of order (log N ) sα /N α for order α digital nets and sequences, see also [3].…”

“…In order to improve his seminal results, it may be sensible to attempt to exploit some further aspect of the Walsh coefficients. Both the result of [25] and that of [37,38] rely not only on the decay but also on the sparsity of the Walsh coefficients.…”

“…In this article we mainly focus on the papers [25,38] and provide a unified picture on how the Walsh analysis enables recent developments of HoQMC methods both for discrepancy and numerical integration. The rest of this article is organized as follows.…”

“…Thereafter, recent advances in HoQMC methods for discrepancy are described in Section 5, while those for numerical integration are in Section 6. We shall highlight an analogy between the approach by [25] and that of [38], where exploiting both the decay and the sparsity of the Walsh coefficients plays a crucial role. We conclude the article with some future research directions.…”