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Lattice rules are among the most prominently studied quasi-Monte Carlomethods to approximate multivariate integrals. A rank-1 lattice rule toapproximate an $s$-dimensional integral is fully specified by its generatingvector $\mathbf{z} \in \mathbb{Z}^s$ and its number of points $N$. While thereare many results on the existence of "good" rank-1 lattice rules, there are noexplicit constructions for good generating vectors for dimensions $s \ge 3$.This is why one usually resorts to computer search algorithms. Motivated byearlier work of Korobov from 1963 and 1982, we present two variants of searchalgorithms for good lattice rules and show that the resulting rules exhibit aconvergence rate in weighted function spaces that can be arbitrarily close tothe optimal rate. Moreover, contrary to most other algorithms, we do not needto know the smoothness of our integrands in advance, the generating vector willstill recover the convergence rate associated with the smoothness of theparticular integrand, and, under appropriate conditions on the weights, theerror bounds can be stated without dependence on $s$. The search algorithmspresented in this paper are two variants of the well-knowncomponent-by-component (CBC) construction, one of which is combined with adigit-by-digit (DBD) construction. We present numerical results for bothalgorithms using fast construction algorithms in the case of product weights.They confirm our theoretical findings.

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