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
In this post I describe an approach for constructing lattice rules which might be useful for integration in The aim is to use the main ideas from the construction of digital nets over , see here, and apply them to lattice rules. This is the first project mentioned here. 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
In this entry we show how the Walsh degree of a digital net is connected to its t value. This can lead to future research by using ideas developed for finding lattice rules with large trigonometric degree. 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