The computation of integrals in higher dimensions and on general domains, when no explicit cubature rules are known, can be ”easily” addressed by means of the quasi-Monte Carlo method. The method, simple in its formulation, becomes computationally inefficient when the space dimension is growing and the integration domain is particularly complex. In this paper we present two new approaches to the quasi-Monte Carlo method for cubature based on nonnegative least squares and approximate Fekete points. The main idea is to use less points and especially good points for solving the system of the moments. Good points are here intended as points with good interpolation properties, due to the strict connection between interpolation and cubature. Numerical experiments show that, in average, just a tenth of the points should be used mantaining the same approximation order of the quasi-Monte Carlo method. The method has been satisfactory applied to 2 and 3-dimensional problems on quite complex domains.

It was pointed out by Crisan and Ghazali that the error estimate for the cubature on Wiener space algorithm developed by Lyons and Victoir requires an additional assumption on the drift. In this paper we demonstrate that it is straightforward to adopt the analysis of Kusuoka to obtain a general estimate without an additional assumptions on the drift. In the process we slightly sharpen the bounds derived by Kusuoka.