Numerical integration of discontinuous functions in many dimensions

Abstract

We consider the problem of numerically integrating functions with hyperplane discontinuities over the entire Euclidean space in many dimensions. We describe a simple process through which the Euclidean space is partitioned into simplices on which the integrand is smooth, generalising the standard practice of dividing the interval used in one-dimensional problems. Our procedure is combined with existing adaptive cubature algorithms to significantly reduce the necessary number of function valuations and memory requirements of the integrator. The method is embarrassingly parallel and can be trivially scaled across many cores with virtually no overhead. Our method is particularly pertinent to the integration of Green's functions, a problem directly related to the perturbation theory of impurity models. In three spatial dimensions we observe a speed-up of order $100$ which increases with increasing dimensionality.