Randomized Polynomial Lattice Rules for Multivariate Integration and Simulation

Lattice rules are among the best methods to estimate integrals in a large numberof dimensions. They are part of the quasi-Monte Carlo set of tools. A new class of lattice rules,defined in a space of polynomials with coefficients in a finite field, is introduced in this paper, anda theoretical framework for these polynomial lattice rules is developed. A randomized version isstudied, implementations and criteria for selecting the parameters are discussed, and examples ofits use as a variance reduction tool in stochastic simulation are provided. Certain types of digitalnet constructions, as well as point sets constructed by taking all vectors of successive output valuesproduced by a Tausworthe random number generator, turn out to be special cases of this method.