Extensible lattice sequences or quasi Monte Carlo quadrature

Integration lattices are one of the main types of low discrepancy sets used in quasi-Monte Carlo methods. However, they have the disadvantage of being of fixed size. This article describes the construction of an infinite sequ ence of points, the first b(m) of which forms a lattice for any nonnegative integer m. Thus, if the quadrature error using an initial lattice is too l arge, the lattice can be extended without discarding the original points. G enerating vectors for extensible lattices are found by minimizing a loss fu nction based on some measure of discrepancy or nonuniformity of the lattice . The spectral test used for finding pseudorandom number generators is one important example of such a discrepancy. The performance of the extensible lattices proposed here is compared to that of other methods for some practi cal quadrature problems.