The authors present a way to reduce the errors of the quasi-Monte Carlo method discrepancy by integrand smoothing and by further “dimension reduction”. The first approach is to replace the integration of a discontinuous integrand with a continuous decision function. An alternative is to use weighted uniform sampling: to each sample point a weight equal to its acceptance probability is assigned. Computational examples are presented that show root mean square error reduction for the proposed methods. The results can also be improved by an alternative discretization (called dimension reduction) as shown in an example for the estimation of the solution of a linear parabolic differential equation (Feynman-Kac formula). This is based on rearrangement of variables so that the principal variations of the integrand occur over the lower dimensions.