Adaptive Quasi-Monte Carlo Methods for Cubature

Abstract: High dimensional integrals can be approximated well by quasi-Monte Carlo methods. However, determining the number of function values needed to obtain the desired accuracy is difficult without some upper bound on an appropriate semi-norm of the integrand. This challenge has motivated our recent development of theoretically justified, adaptive cubatures based on digital sequences and lattice nodeset sequences. Our adaptive cubatures are based on error bounds that depend on the discrete Fourier transforms of the integrands. These cubatures are guaranteed for integrands belonging to cones of functions whose true Fourier coefficients decay steadily, a notion that is made mathematically precise. Here we describe these new cubature rules and extend them in two directions. First, we generalize the error criterion to allow both absolute and relative error tolerances. We also demonstrate how to estimate a function of several integrals to a given tolerance. This situation arises in the computation of Sobol' indices. Second, we describe how to use control variates in adaptive quasi-Monte cubature while appropriately estimating the control variate coefficient.