Stephen Joe

Imbedded lattice rules for multidimensional integration

This paper is concerned with imbedded sequences of lattice rules. The theoretical properties of certain finite imbedded sequences are developed and an efficient search method for finding “good” imbedded sequences of this kind is discussed. An algorithm for computing these sequences is presented, and some numerical results using the search method and this algorithm are given.

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.

Letter section: Randomization of lattice rules for numerical multiple integration

Here we investigate some procedures for the randomization of lattice rules for numerical multiple integration similar to those proposed by Cranley and Patterson (1976) for number-theoretic rules. Currently, there is no easily calculable error estimate available for general lattice rules and the randomization procedures looked at here allow the calculation of confidence intervals for the error.

Component by Component Construction of Rank-1 Lattice Rules HavingO(n-1(In(n))d) Star Discrepancy

The star discrepancy is a quantity for measuring the uniformity of a set of quadrature points and appears in the Koksma-Hlawka inequality. For integrals over [0, 1]d it is known that there exist d-dimensional rank-1 lattice rules having 0(n -1(ln(n))d) star discrepancy, where n is the number of points. Here we show that for n prime such rules may be obtained by constructing their generating vectors component by component. The rules are constructed to satisfy certain bounds on a quantity known as R. Bounds on the star discrepancy in terms of R then yield the desired O(n -1(In(n))d) star discrepancy.

On computing the lattice rule criterion R

Lattice rules are integration rules for approximating integrals of periodic functions over the s-dimensional unit cube. One criterion for measuring the goodness of lattice rules is the quantity R. This quantity is defined as a sum which contains about N[sup s-1] terms, where N is the number of quadrature points. Although various bounds involving R are known, a procedure for calculating R itself does not appear to have been given previously. Here we show how an asymptotic series can be used to obtain an accurate approximation to R. Whereas an efficient direct calculation of R requires O(Nn[sub 1]) operations, where n[sub 1] is the largest invariant of the rule, the use of this asymptotic expansion allows the operation count to be reduced to O(N). A complete error analysis for the asymptotic expansion is given. The results of some calculations of R are also given. 19 refs., 2 tabs.

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.

Implementation of a lattice method for numerical multiple integration

An implementation of a method for numerical multiple integration based on a sequence of imbedded lattice rules is given. Besides yielding an approximation to the integral, this implementation also provides an error estimate which does not require much extra computation. The results of some numerical experiments conclude the paper.

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.

Collocation methods using piecewise polynomials for second kind integral equations

Abstract We consider the numerical solution of second kind Fredholm integral equations in one dimension by using the collocation method and its iterated variant. The collocation solution will be sought in a space of piecewise polynomials of order r. Superconvergence results for the iterated collocation solution are known when discontinuous piecewise polynomials are used and the collocation points are taken to be the r Gaussian points shifted to each subinterval. Here we give a corresponding superconvergence result for the iterated collocation solution when continuous piecewise polynomials with no continuity requirements on the derivatives are used. The collocation points are taken to be the knots plus the r−2 Lobatto points shifted to each subinterval. Some numerical results are also given.

Triangular canonical forms for lattice rules of prime-power order

In this paper the authors develop a theory of t-cycle D-Z representations for s-dimensional lattice rules of prime-power order. Of particular interest are canonical forms which, by definition, have a D-matrix consisting of the nontrivial invariants. Among these is a family of triangular forms, which, besides being canonical, have the defining property that their Z-matrix is a column permuted version of a unit upper triangular matrix. Triangular forms may be obtained constructively using sequences of elementary transformations based on elementary matrix algebra. The authors main result is to define a unique canonical form for prime-power rules. This ultratriangular form is a triangular form, is easy to recognize, and may be derived in a straightforward manner. 12 refs.