# Tag Archives: numerical integration over R^s

## QMC rules over R^s: Matlab code and numerical example

In this post you can find a Matlab code for constructing digital nets on $\mathbb{R}^s$ which was recently proposed in J. Dick, Quasi-Monte Carlo numerical integration on $\mathbb{R}^s$: 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

## Quasi-Monte Carlo for R^s: digital nets and worst-case error

Recently I submitted the paper titled Quasi-Monte Carlo numerical integration over $\mathbb{R}^s$: digital nets and worst-case error. I will give some heuristic explanation of the results in this paper. Interesting in this context is also the sparse grid approach from the PhD thesis of Markus Holtz (supervised by Prof. Michael Griebel).

Therein I aim to develop a theory on quasi-Monte Carlo integration over $\mathbb{R}^s$. The idea is to transform a digital net from $[0,1]^s$ to $\mathbb{R}^s$ such that in elementary intervals of the form $\displaystyle \prod_{i=1}^s [A b^j, (A+1) b^j)$ one has a digitally shifted digital net, and at the same time, globally one has a given (discretized) distribution. Continue reading