  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.
