An adaptive approach to cube-based quasi-Monte Carlo integration on Rd

The standard domain for quasi-Monte Carlo approximations is the unit cube. Recently, much research has been done to make quasi-Monte Carlo methods applicable to the real space. Mathé and Wei proposed an algorithm that splits Rd into cubes. One of the difficulties with their approach is that the user needs to know the decay factor of the problem beforehand. We propose an adaptive approach where the algorithm itself determines how to distribute the points. We also prove an optimal distribution of N points over several quasi-Monte Carlo integrations.