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).
Posted in Quasi-Monte Carlo, Research
Tagged cyclic digital nets, digital net, discrepancy, discrepancy theory, duality theory, Duality Theory for Digital Nets, fast component by component, geometric discrepancy, Higher order digital net, higher order digital sequence, higher order polynomial lattice rule, higher order Sobol sequence, hyperplane nets, Niederreiter sequence, numerical integration, polynomial lattice rule, Propagation Rule, quasi-Monte Carlo, randomised quasi-Monte Carlo, Sobol sequence, uniform distribution, Walsh function
In the following we first describe a connection of -nets and Sudoku and then generalize the Sudoku in analogy to higher order -nets in base 3. A new type of Sudoku puzzle is presented at the end. Continue reading
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
-  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 and for higher order sequences we have the parameters
(Digital) higher order nets and sequences are point sets and sequences such that
where is the number of quadrature points and is the smoothness of the integrand Continue reading
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 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
Posted in Open problems, Quasi-Monte Carlo, Research
Tagged digital net, digital sequence, Higher order digital net, higher order digital sequence, higher order scrambled Sobol sequence, higher order scrambling, higher order Sobol sequence, Matlab higher order scrambling, randomized quasi-Monte Carlo, scrambled Sobol sequence, scrambling, Sobol sequence
Higher order digital nets and sequences are quasi-Monte Carlo point sets where for digital nets and 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:
where is the smoothness of the integrand and is a constant which only depends on and (but not on For instance, if has square integrable partial mixed derivatives up to order in each variable, then we get a convergence rate of where is the number of quadrature points used in the approximation and can be chosen arbitrarily small. If has derivatives then we get a convergence of 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
We recently submitted the manuscript
This paper fits into the work on higher order quasi-Monte Carlo rules which started with  and . It can be viewed as the higher order extension of , where classical polynomial lattice rules were considered.
1. Construction Algorithms
As stated in the title of the manuscript, we present construction algorithms for higher order polynomial lattice rules. The construction is based on the worst-case error rather than the quality parameter . This allows us to find good higher order polynomial lattice rules for weighted function spaces. It also presents a feasible alternative to the direct construction introduced in  (and  for the periodic case).
We use a variety of approaches to achieve our results. Continue reading
We (J. Baldeaux, J.D., F. Pillichshammer) just submitted a manuscript on Duality Theory and Propagation Rules for Higher Order Nets (you can download the manuscript by clicking on the title). In a nutshell, propagation rules are methods to construct, from given nets with some given parameters, new nets with at least one parameter different from the given nets. Duality theory on the other hand is the vehicle which allows us to analyse those propagation rules. It is an analytic description (using Walsh functions in our case) of the geometric properties of nets.
The manuscript is a continuation of the paper with P. Kritzer on Duality Theory and Propagation Rules for Generalized Digital Nets . (Note that in some papers the notion `generalized (digital) net’ was used instead of the term `(digital) higher order net’.) The difference is that we do not assume that the point sets are constructed by the digital construction scheme.
Most propagation rules for classical -nets have a higher order analogue. This applies to digital as well as geometric nets. Additionally there are also some further propagation rules which do not exist for classical nets (for example the higher order to higher order construction). The higher order to higher order propagation rule is, as of now, still the only method to obtain higher order nets from classical nets. Fortunately it works in the digital as well as geometric case.
1. A subnet propagation rule for digital higher order nets
We give an example of a propagation rule in the following. In the classical case there is a propagation rule which states that: Continue reading