University of Waikato
Good lattice rules for numerical multiple integration may be constructed by using a so-called ‘component-by-component’ technique, which is based on successive 1-dimensional searches. The ‘goodness’ of a lattice rule is assessed by the so-called ‘weighted star discrepancy’, which is based on an L∞ maximum error.
In some applications of multiple integrals it is the relative importance of distinct groups of variables that matters. This leads to considering function spaces having ‘general weights’. The first part of the talk will be focused on the main results regarding the construction of rank-1 lattice rules in the ‘general-weighted’ case under the assumption that the number of points of the lattice rule is prime.
In other applications, variables are arranged in the decreasing order of their importance. This leads to considering the ‘product-weighted’ case, for which a recent breakthrough has been obtained under the assumption that the number of lattice points is non-prime. Such a generalisation to the non-prime case is possible by employing asymptotic expansion techniques.
The final part of the talk will focus on the complexity of the component-by-component construction.
Joint work with Stephen Joe (University of Waikato).