On a projection-corrected component-by-component construction

TL;DR

This paper introduces a modified component-by-component algorithm for constructing lattice rules that ensures all components are distinct, preventing projection issues, and provides theoretical error bounds for this new approach.

Contribution

It proposes a novel variation of the classical algorithm that enforces component uniqueness and proves associated worst-case error bounds.

Findings

The modified algorithm avoids projections with all points on the diagonal.

Error bounds are established for the new construction method.

The approach improves the quality of lattice rules by ensuring component diversity.

Abstract

The component-by-component construction is the standard method of finding good lattice rules or polynomial lattice rules for numerical integration. Several authors have reported that in numerical experiments the generating vector sometimes has repeated components. We study a variation of the classical component-by-component algorithm for the construction of lattice or polynomial lattice point sets where the components are forced to differ from each other. This avoids the problem of having projections where all quadrature points lie on the main diagonal. Since the previous results on the worst-case error do not apply to this modified algorithm, we prove such an error bound here. We also discuss further restrictions on the choice of components in the component-by-component algorithm.