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Discrepancy Bounds for Mixed Sequences
A mixed sequence is a sequence in the -dimensional unit cube
which one obtains by concatenating a -dimensional low-discrepancy
sequence with an -dimensional random sequence.
We discuss some probabilistic bounds on the star discrepancy of
mixed sequences
L_p- and S_{p,q}^rB-discrepancy of (order 2) digital nets
Dick proved that all order digital nets satisfy optimal upper bounds of
the -discrepancy. We give an alternative proof for this fact using Haar
bases. Furthermore, we prove that all digital nets satisfy optimal upper bounds
of the -discrepancy for a certain parameter range and enlarge that
range for order digitals nets. -, - and -discrepancy is considered as well
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
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