25,765 research outputs found
Some Results on the Complexity of Numerical Integration
This is a survey (21 pages, 124 references) written for the MCQMC 2014
conference in Leuven, April 2014. We start with the seminal paper of Bakhvalov
(1959) and end with new results on the curse of dimension and on the complexity
of oscillatory integrals. Some small errors of earlier versions are corrected
Rank-1 lattice rules for multivariate integration in spaces of permutation-invariant functions: Error bounds and tractability
We study multivariate integration of functions that are invariant under
permutations (of subsets) of their arguments. We find an upper bound for the
th minimal worst case error and show that under certain conditions, it can
be bounded independent of the number of dimensions. In particular, we study the
application of unshifted and randomly shifted rank- lattice rules in such a
problem setting. We derive conditions under which multivariate integration is
polynomially or strongly polynomially tractable with the Monte Carlo rate of
convergence . Furthermore, we prove that those tractability
results can be achieved with shifted lattice rules and that the shifts are
indeed necessary. Finally, we show the existence of rank- lattice rules
whose worst case error on the permutation- and shift-invariant spaces converge
with (almost) optimal rate. That is, we derive error bounds of the form
for all , where denotes
the smoothness of the spaces.
Keywords: Numerical integration, Quadrature, Cubature, Quasi-Monte Carlo
methods, Rank-1 lattice rules.Comment: 26 pages; minor changes due to reviewer's comments; the final
publication is available at link.springer.co
Recent advances in higher order quasi-Monte Carlo methods
In this article we review some of recent results on higher order quasi-Monte
Carlo (HoQMC) methods. After a seminal work by Dick (2007, 2008) who originally
introduced the concept of HoQMC, there have been significant theoretical
progresses on HoQMC in terms of discrepancy as well as multivariate numerical
integration. Moreover, several successful and promising applications of HoQMC
to partial differential equations with random coefficients and Bayesian
estimation/inversion problems have been reported recently. In this article we
start with standard quasi-Monte Carlo methods based on digital nets and
sequences in the sense of Niederreiter, and then move onto their higher order
version due to Dick. The Walsh analysis of smooth functions plays a crucial
role in developing the theory of HoQMC, and the aim of this article is to
provide a unified picture on how the Walsh analysis enables recent developments
of HoQMC both for discrepancy and numerical integration
- …