Hiding the weights -- CBC black box algorithms with a guaranteed error bound

TL;DR

This paper introduces two new CBC algorithms that automatically determine weights for lattice rules based on derivative bounds, providing guaranteed error bounds without user-specified weights.

Contribution

The paper presents novel CBC algorithms that eliminate the need for user-chosen weights by using derivative bounds to guarantee error bounds for lattice rules.

Findings

Algorithms produce lattice rules with guaranteed error bounds

Numerical results compare effectiveness under different derivative bounds

Provides rigorous upper bounds on root-mean-square error

Abstract

The component-by-component (CBC) algorithm is a method for constructing good generating vectors for lattice rules for the efficient computation of high-dimensional integrals in the "weighted" function space setting introduced by Sloan and Wo\'zniakowski. The "weights" that define such spaces are needed as inputs into the CBC algorithm, and so a natural question is, for a given problem how does one choose the weights? This paper introduces two new CBC algorithms which, given bounds on the mixed first derivatives of the integrand, produce a randomly shifted lattice rule with a guaranteed bound on the root-mean-square error. This alleviates the need for the user to specify the weights. We deal with "product weights" and "product and order dependent (POD) weights". Numerical tables compare the two algorithms under various assumed bounds on the mixed first derivatives, and provide rigorous upper bounds on the root-mean-square integration error.