Reliable Adaptive Cubature Using Digital Sequences

Abstract

Quasi-Monte Carlo cubature methods often sample the integrand using Sobol’ (or other digital) sequences to obtain higher accuracy than IID sampling. An important question is how to conservatively estimate the error of a digital sequence cubature so that the sampling can be terminated when the desired tolerance is reached. We propose an error bound based on the discrete Walsh coefficients of the integrand and use this error bound to construct an adaptive digital sequence cubature algorithm. The error bound and the corresponding algorithm are guaranteed to work for integrands lying in a cone defined in terms of their true Walsh coefficients. Intuitively, the inequalities defining the cone imply that the ordered Walsh coefficients do not dip down for a long stretch and then jump back up. An upper bound on the cost of our new algorithm is given in terms of the unknown decay rate of the Walsh coefficients.