15 research outputs found
Efficient calculation of the worst-case error and (fast) component-by-component construction of higher order polynomial lattice rules
We show how to obtain a fast component-by-component construction algorithm
for higher order polynomial lattice rules. Such rules are useful for
multivariate quadrature of high-dimensional smooth functions over the unit cube
as they achieve the near optimal order of convergence. The main problem
addressed in this paper is to find an efficient way of computing the worst-case
error. A general algorithm is presented and explicit expressions for base~2 are
given. To obtain an efficient component-by-component construction algorithm we
exploit the structure of the underlying cyclic group.
We compare our new higher order multivariate quadrature rules to existing
quadrature rules based on higher order digital nets by computing their
worst-case error. These numerical results show that the higher order polynomial
lattice rules improve upon the known constructions of quasi-Monte Carlo rules
based on higher order digital nets
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