84 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
On a projection-corrected component-by-component construction
The component-by-component construction is the standard method of finding
good lattice rules or polynomial lattice rules for numerical integration.
Several authors have reported that in numerical experiments the generating
vector sometimes has repeated components. We study a variation of the classical
component-by-component algorithm for the construction of lattice or polynomial
lattice point sets where the components are forced to differ from each other.
This avoids the problem of having projections where all quadrature points lie
on the main diagonal. Since the previous results on the worst-case error do not
apply to this modified algorithm, we prove such an error bound here. We also
discuss further restrictions on the choice of components in the
component-by-component algorithm
Fast QMC matrix-vector multiplication
Quasi-Monte Carlo (QMC) rules
can be used to approximate integrals of the form , where is a matrix and
is row vector. This type of integral arises for example from
the simulation of a normal distribution with a general covariance matrix, from
the approximation of the expectation value of solutions of PDEs with random
coefficients, or from applications from statistics. In this paper we design QMC
quadrature points
such that for the matrix whose rows are the quadrature points, one can
use the fast Fourier transform to compute the matrix-vector product , , in operations and at most extra additions. The proposed method can be
applied to lattice rules, polynomial lattice rules and a certain type of
Korobov -set.
The approach is illustrated computationally by three numerical experiments.
The first test considers the generation of points with normal distribution and
general covariance matrix, the second test applies QMC to high-dimensional,
affine-parametric, elliptic partial differential equations with uniformly
distributed random coefficients, and the third test addresses Finite-Element
discretizations of elliptic partial differential equations with
high-dimensional, log-normal random input data. All numerical tests show a
significant speed-up of the computation times of the fast QMC matrix method
compared to a conventional implementation as the dimension becomes large
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