18,978 research outputs found
Good lattice rules with a composite number of points based on the product weighted star discrepancy
Rank-1 lattice rules based on a weighted star discrepancy with weights of a product form have been previously constructed under the assumption that the number of points is prime. Here, we extend these results to the non-prime case. We show that if the weights are summable, there exist lattice rules whose weighted star discrepancy is O(n−1+δ), for any δ > 0, with the implied constant independent of the dimension and the number of lattice points, but dependent on δ and the weights. Then we show that the generating vector of such a rule can be constructed using a component-by-component (CBC) technique. The cost of the CBC construction is analysed in the final part of the paper
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
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
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
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