136 research outputs found
Component-by-component construction of good intermediate-rank lattice rules
It is known that the generating vector of a rank-1 lattice rule can be constructed component-by-component to achieve strong tractability error bounds in both weighted Korobov spaces and weighted Sobolev spaces. Since the weights for these spaces are nonincreasing, the first few variables are in a sense more important than the rest. We thus propose to copy the points of a rank-1 lattice rule a number of times in the first few dimensions to yield an intermediate-rank lattice rule. We show that the generating vector (and in weighted Sobolev spaces, the shift also) of an intermediate-rank lattice rule can also be constructed component-by-component to achieve strong tractability error bounds. In certain circumstances, these bounds are better than the corresponding bounds for rank-1 lattice rules
Constructing lattice points for numerical integration by a reduced fast successive coordinate search algorithm
In this paper, we study an efficient algorithm for constructing node sets of
high-quality quasi-Monte Carlo integration rules for weighted Korobov, Walsh,
and Sobolev spaces. The algorithm presented is a reduced fast successive
coordinate search (SCS) algorithm, which is adapted to situations where the
weights in the function space show a sufficiently fast decay. The new SCS
algorithm is designed to work for the construction of lattice points, and, in a
modified version, for polynomial lattice points, and the corresponding
integration rules can be used to treat functions in different kinds of function
spaces. We show that the integration rules constructed by our algorithms
satisfy error bounds of optimal convergence order. Furthermore, we give details
on efficient implementation such that we obtain a considerable speed-up of
previously known SCS algorithms. This improvement is illustrated by numerical
results. The speed-up obtained by our results may be of particular interest in
the context of QMC for PDEs with random coefficients, where both the dimension
and the required numberof points are usually very large. Furthermore, our main
theorems yield previously unknown generalizations of earlier results.Comment: 33 pages, 2 figure
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
Successive Coordinate Search and Component-by-Component Construction of Rank-1 Lattice Rules
The (fast) component-by-component (CBC) algorithm is an efficient tool for
the construction of generating vectors for quasi-Monte Carlo rank-1 lattice
rules in weighted reproducing kernel Hilbert spaces. We consider product
weights, which assigns a weight to each dimension. These weights encode the
effect a certain variable (or a group of variables by the product of the
individual weights) has. Smaller weights indicate less importance. Kuo (2003)
proved that the CBC algorithm achieves the optimal rate of convergence in the
respective function spaces, but this does not imply the algorithm will find the
generating vector with the smallest worst-case error. In fact it does not. We
investigate a generalization of the component-by-component construction that
allows for a general successive coordinate search (SCS), based on an initial
generating vector, and with the aim of getting closer to the smallest
worst-case error. The proposed method admits the same type of worst-case error
bounds as the CBC algorithm, independent of the choice of the initial vector.
Under the same summability conditions on the weights as in [Kuo,2003] the error
bound of the algorithm can be made independent of the dimension and we
achieve the same optimal order of convergence for the function spaces from
[Kuo,2003]. Moreover, a fast version of our method, based on the fast CBC
algorithm by Nuyens and Cools, is available, reducing the computational cost of
the algorithm to operations, where denotes the number
of function evaluations. Numerical experiments seeded by a Korobov-type
generating vector show that the new SCS algorithm will find better choices than
the CBC algorithm and the effect is better when the weights decay slower.Comment: 13 pages, 1 figure, MCQMC2016 conference (Stanford
Construction of Good Rank-1 Lattice Rules Based on the Weighted Star Discrepancy
The ‘goodness’ of a set of quadrature points in [0, 1]d may be measured by the weighted star discrepancy. If the weights for the weighted star discrepancy are summable, then we show that for n prime there exist n-point rank-1 lattice rules whose weighted star discrepancy is O(n−1+δ) for any δ>0, where the implied constant depends on δ and the weights, but is independent of d and n. Further, we show that the generating vector z for such lattice rules may be obtained using a component-by-component construction. The results given here for the weighted star discrepancy are used to derive corresponding results for a weighted Lp discrepancy
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