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The structure and average discrepancies of lattice rules for numerical integration

Lattice rules are equal-weight quadrature rules which are used in the approximation of multidimensional integrands over the s-dimensional unit cube [0,1]ˢ. One of the problems encountered in the study of such rules is the unavailability of a unique representation. It is known that any lattice rule may be expressed in a canonical D - Z form in which D is a diagonal matrix whose diagonal entries are known as the invariants and Z is an integer matrix. Although D is unique in this canonical form, Z may be chosen in many different ways. Except for the case of so-called projection-regular and prime-power rules, no such unique Z is available. In the latter case of prime-power rules, the unique D - Z form developed is known as an ultratriangular form. Associated with each ultratriangular form is a set of column indices. Any lattice rule may be decomposed into prime-power components. In this thesis, a unique D - Z form is defined for a special class of lattice rules for which the component prime-power rules have a consistent set of column indices. This new unique form includes the known unique forms for projection-regular and prime-power rules as special cases. We also use the ultratriangular form for prime-power lattice rules to derive a formula to calculate the number of prime-power rules having a given set of invariants and column indices. The existing theory of lattice rules that is based on the generator matrix of the dual lattice has made the assumption that its representation in the so-called Hermite normal form is upper triangular. However, since projection-regular rules have a unique Z-matrix which is unit upper triangular, the corresponding generator matrix for the dual lattice is lower triangular. This suggests that a lower triangular Hermite normal form might be appropriate for study. We consider this situation and give the conditions on the lower triangular Hermite normal form which allow a projection-regular rule to be easily recognized. Number-theoretic rules are a class of lattice rules which are known to be particularly suitable for the approximation of multidimensional integrals in which the integrands are periodic. In the case of non-periodic integrands there is numerical evidence that the average L₂ discrepancy for these rules is smaller than the expected value for Monte-Carlo rules when the dimension s is less than 18. For non-periodic integrands, a vertex-modified version of the number-theoretic rule has been previously proposed. In s-dimensions these vertex-modified rules contain 2ˢ weights which may be chosen optimally so that the discrepancy is minimized. We shall compare the average discrepancy for these optimal vertex-modified number-theoretic rules with that for normal number-theoretic and Monte-Carlo rules. A similar comparison is also carried out between the averages for number-theoretic rules and for 2ˢ copy rules with approximately the same number of points. In the case of periodic integrands it has been shown that the average of Pα and the values of R for 2ˢ copy rules are smaller than those for number-theoretic rules. For this periodic case, we use an analogue of the L₂ discrepancy to carry out a similar comparison.
Type of thesis
The University of Waikato
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