Hiding the weights -- CBC black box algorithms with a guaranteed error bound
For researchers in high-dimensional numerical integration, this work removes the burden of weight selection in CBC algorithms while providing guaranteed error bounds.
The paper introduces two new CBC algorithms that, given bounds on mixed first derivatives, produce randomly shifted lattice rules with guaranteed root-mean-square error bounds, eliminating the need for users to specify weights. Numerical tables provide rigorous error bounds under various derivative bounds.
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 Woźniakowski. 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.