On a reduced digit-by-digit component-by-component construction of lattice point sets

TL;DR

This paper introduces an efficient reduced digit-by-digit component-by-component algorithm for constructing lattice point sets for quasi-Monte Carlo integration, achieving near-optimal error bounds with reduced computational effort, especially in high dimensions.

Contribution

The paper presents a novel reduced CBC-DBD algorithm that significantly speeds up lattice point set construction while maintaining near-optimal error bounds, especially for functions with rapidly decaying weights.

Findings

The new algorithm reduces computational effort independent of dimension under certain conditions.

Constructed lattice rules achieve near-optimal convergence rates.

Numerical experiments demonstrate substantial speed-up over previous methods.

Abstract

In this paper, we study an efficient algorithm for constructing point sets underlying quasi-Monte Carlo integration rules for weighted Korobov classes. The algorithm presented is a reduced fast component-by-component digit-by-digit (CBC-DBD) algorithm, which useful for to situations where the weights in the function space show a sufficiently fast decay. The advantage of the algorithm presented here is that the computational effort can be independent of the dimension of the integration problem to be treated if suitable assumptions on the integrand are met. The new reduced CBC-DBD algorithm is designed to work for the construction of lattice point sets, and the corresponding integration rules (so-called lattice rules) can be used to treat functions in different kinds of function spaces. We show that the integration rules constructed by our algorithm satisfy error bounds of almost optimal convergence order. Furthermore, we give details on an efficient implementation such that we obtain a considerable speed-up of a previously known CBC-DBD algorithm that has been studied before. This improvement is illustrated by numerical results.