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 this post you can find a Matlab code for constructing digital nets on which was recently proposed in J. Dick, Quasi-Monte Carlo numerical integration on : digital nets and worst-case error. Submitted, 2010. See the previous post where an explanation of the method and a link to the paper can be found. In the numerical example we consider a simple three-dimensional integral. In this example the computation time with the new method is reduced by a factor of ten and additionally the integration error is also reduced. The numerical result in the paper shows that, for the example considered there, that the computation time can be reduced from two and a half minutes to less than two seconds for a certain given error level. 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