7 research outputs found
Lattice rules with random achieve nearly the optimal error independently of the dimension
We analyze a new random algorithm for numerical integration of -variate
functions over from a weighted Sobolev space with dominating mixed
smoothness and product weights
, where the functions are continuous and
periodic when . The algorithm is based on rank- lattice rules
with a random number of points~. For the case , we prove that
the algorithm achieves almost the optimal order of convergence of
, where the implied constant is independent of
the dimension~ if the weights satisfy . The same rate of convergence holds for the more
general case by adding a random shift to the lattice rule with
random . This shows, in particular, that the exponent of strong tractability
in the randomized setting equals , if the weights decay fast
enough. We obtain a lower bound to indicate that our results are essentially
optimal. This paper is a significant advancement over previous related works
with respect to the potential for implementation and the independence of error
bounds on the problem dimension. Other known algorithms which achieve the
optimal error bounds, such as those based on Frolov's method, are very
difficult to implement especially in high dimensions. Here we adapt a
lesser-known randomization technique introduced by Bakhvalov in 1961. This
algorithm is based on rank- lattice rules which are very easy to implement
given the integer generating vectors. A simple probabilistic approach can be
used to obtain suitable generating vectors.Comment: 17 page
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