Digit-by-digit and component-by-component constructions of lattice rules for periodic functions with unknown smoothness

by   Adrian Ebert, et al.

Lattice rules are among the most prominently studied quasi-Monte Carlo methods to approximate multivariate integrals. A rank-1 lattice rule to approximate an s-dimensional integral is fully specified by its generating vector z∈Z^s and its number of points N. While there are many results on the existence of "good" rank-1 lattice rules, there are no explicit constructions for good generating vectors for dimensions s > 3. This is why one usually resorts to computer search algorithms. Motivated by earlier work of Korobov from 1963 and 1982, we present two variants of search algorithms for good lattice rules and show that the resulting rules exhibit a convergence rate in weighted function spaces that can be arbitrarily close to the optimal rate. Moreover, contrary to most other algorithms, we do not need to know the smoothness of our integrands in advance, the generating vector will still recover the convergence rate associated with the smoothness of the particular integrand, and, under appropriate conditions on the weights, the error bounds can be stated without dependence on s. The search algorithms presented in this paper are two variants of the well-known component-by-component (CBC) construction, one of which is combined with a digit-by-digit (DBD) construction. We present numerical results for both algorithms using fast construction algorithms in the case of product weights. They confirm our theoretical findings.


page 1

page 2

page 3

page 4


Stability of lattice rules and polynomial lattice rules constructed by the component-by-component algorithm

We study quasi-Monte Carlo (QMC) methods for numerical integration of mu...

Construction-free median quasi-Monte Carlo rules for function spaces with unspecified smoothness and general weights

We study quasi-Monte Carlo (QMC) integration of smooth functions defined...

A fast probabilistic component-by-component construction of exactly integrating rank-1 lattices and applications

Several more and more efficient component–by–component (CBC) constructio...

Component-by-component construction of randomized rank-1 lattice rules achieving almost the optimal randomized error rate

We study a randomized quadrature algorithm to approximate the integral o...

Fast component-by-component construction of lattice algorithms for multivariate approximation with POD and SPOD weights

In a recent paper by the same authors, we provided a theoretical foundat...

Lattice algorithms for multivariate approximation in periodic spaces with general weight parameters

This paper provides the theoretical foundation for the construction of l...