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:
- J. Dick, Explicit constructions of quasi-Monte Carlo rules for the numerical integration of high dimensional periodic functions. SIAM J. Numer. Anal., 45, 2141–2176, 2007.
- J. Dick, Walsh spaces containing smooth functions and quasi-Monte Carlo rules of arbitrary high order. SIAM J. Numer. Anal., 46, 1519–1553, 2008.
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