129 research outputs found
The construction of good lattice rules and polynomial lattice rules
A comprehensive overview of lattice rules and polynomial lattice rules is
given for function spaces based on semi-norms. Good lattice rules and
polynomial lattice rules are defined as those obtaining worst-case errors
bounded by the optimal rate of convergence for the function space. The focus is
on algebraic rates of convergence for
and any , where is the decay of a series representation
of the integrand function. The dependence of the implied constant on the
dimension can be controlled by weights which determine the influence of the
different dimensions. Different types of weights are discussed. The
construction of good lattice rules, and polynomial lattice rules, can be done
using the same method for all ; but the case is special
from the construction point of view. For the
component-by-component construction and its fast algorithm for different
weighted function spaces is then discussed
Construction of quasi-Monte Carlo rules for multivariate integration in spaces of permutation-invariant functions
We study multivariate integration of functions that are invariant under the
permutation (of a subset) of their arguments. Recently, in Nuyens,
Suryanarayana, and Weimar (Adv. Comput. Math. (2016), 42(1):55--84), the
authors derived an upper estimate for the th minimal worst case error for
such problems, and showed that under certain conditions this upper bound only
weakly depends on the dimension. We extend these results by proposing two
(semi-) explicit construction schemes. We develop a component-by-component
algorithm to find the generating vector for a shifted rank- lattice rule
that obtains a rate of convergence arbitrarily close to
, where denotes the smoothness of our
function space and is the number of cubature nodes. Further, we develop a
semi-constructive algorithm that builds on point sets which can be used to
approximate the integrands of interest with a small error; the cubature error
is then bounded by the error of approximation. Here the same rate of
convergence is achieved while the dependence of the error bounds on the
dimension is significantly improved
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
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