1,704 research outputs found
The inverse of the star-discrepancy problem and the generation of pseudo-random numbers
The inverse of the star-discrepancy problem asks for point sets of
size in the -dimensional unit cube whose star-discrepancy
satisfies where
is a constant independent of and . The first existence results in this
direction were shown by Heinrich, Novak, Wasilkowski, and Wo\'{z}niakowski in
2001, and a number of improvements have been shown since then. Until now only
proofs that such point sets exist are known. Since such point sets would be
useful in applications, the big open problem is to find explicit constructions
of suitable point sets .
We review the current state of the art on this problem and point out some
connections to pseudo-random number generators
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
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
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