Tag Archives: higher order digital sequence

Digital Nets and Sequences Preprint

A complete preprint of the book

is now available here. The final published version can be obtained directly from Cambridge University Press here.

The preprint version differs of course from the final version, for instance, the page numbers are different. However, the numbering of Chapters, Sections, Theorems, Lemmas, Corollaries, Definitions and Examples is the same in both versions. The list of corrections is for the published version. We do not have a separate list for the preprint version (though the corrections for the published version also apply to the preprint version).

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Some possible simplifications of the notation of higher order nets and sequences

In this entry I discuss the definition of (digital) higher order nets and sequences and some possible simplifications of the notation.

Digital higher order nets and sequences have been introduced in

whereas higher order nets have been introduced in

  • [2] J. Dick and J. Baldeaux, Equidistribution properties of generalized nets and sequences. In: Proceedings of the MCQMC’08 conference, Montreal, Canada, P. L’Ecuyer and A. Owen (eds.), pp. 305–323, 2009. doi: 10.1007/978-3-642-04107-5_19 An earlier version can be found here.

There are several parameters occurring in the definition of higher order nets, namely t, \alpha,\beta, n, m, s, b and for higher order sequences we have the parameters t, \alpha, \beta, \sigma, s, b.

(Digital) higher order nets and sequences are point sets \{\boldsymbol{x}_0,\ldots, \boldsymbol{x}_{b^m-1}\} \subset [0,1)^s and sequences \{\boldsymbol{x}_0,\boldsymbol{x}_1, \ldots, \} such that

\displaystyle \left|\int_{[0,1]^s} f(\boldsymbol{x}) \,\mathrm{d} \boldsymbol{x} - \frac{1}{N} \sum_{n=0}^{N-1} f(\boldsymbol{x}_n) \right| = \mathcal{O} (N^{-\alpha} (\log N)^{\alpha s}),

where {}N =b^m is the number of quadrature points and {} \alpha is the smoothness of the integrand {}f. Continue reading

Higher order scrambling

Recently I uploaded the paper

This paper deals with a generalization of Owen’s scrambling algorithm which improves on the convergence rate of the root mean square error for smooth integrands. The bound on the root mean square error is best possible (apart from the power of the \log N factor) and this can also be observed from some simple numerical examples shown in the paper (note that the figures in the paper show the standard deviation (or root mean square error) and not the variance of the estimator). In this post you can also find Matlab programs which generate the quadrature points introduced in this paper and a program to generate the numerical results shown in the paper. Continue reading

How to generate higher order Sobol points in Matlab and some numerical examples

Higher order digital nets and sequences are quasi-Monte Carlo point sets \{\boldsymbol{x}_{n-1}\}_{n=1}^N, where N=b^m for digital nets and N=\infty for digital sequences (for more background information on this topic see Chapters 1-4 and Chapter 8 in the book; several sample chapters of this book can be downloaded here), which satisfy the following property:

\displaystyle \left|\int_{[0,1]^s} f(\boldsymbol{x}) \, \mathrm{d} \boldsymbol{x}  -  \frac{1}{b^m} \sum_{n=0}^{b^m-1} f(\boldsymbol{x}_n) \right|  \le C_{s,\alpha} \frac{(\log N)^{s \alpha}}{N^\alpha}, \hspace{2cm} (1)

where \alpha \ge 1 is the smoothness of the integrand {}f and C_{s,\alpha} \textgreater 0 is a constant which only depends on {}s and {}\alpha (but not on {}N). For instance, if {}f has square integrable partial mixed derivatives up to order {}2 in each variable, then we get a convergence rate of \mathcal{O}(N^{-2+\varepsilon}), where {}N is the number of quadrature points used in the approximation and \varepsilon >0 can be chosen arbitrarily small. If {}f has {}3 derivatives then we get a convergence of \mathcal{O}(N^{-3+\varepsilon}), and so on.

This method has been introduced in the papers:

The first paper only deals with periodic functions, whereas the second paper also includes nonperiodic functions.

For some heuristic explanation how higher order digital nets and sequences work see the paper:

  • J. Dick, On quasi-Monte Carlo rules achieving higher order convergence. In: Proceedings of the MCQMC’08 conference, Montreal, Canada, P. L’Ecuyer and A. Owen (eds.), pp. 73–96, 2009. doi: 10.1007/978-3-642-04107-5_5 An earlier version can be found here.

In this entry we provide a Matlab program which generates point sets which satisfy the property (1). Continue reading