Fast multipole method for multivariable integrals

We give a fast numerical algorithm to evaluate a class of multivariable integrals. A direct numerical evaluation of these integrals costs N-m, where m is the number of variables and N is the number of the quadrature points for each variable. For m = 2 and m = 3 and for only one-dimensional variables, we present an algorithm which is able to reduce this cost from N-m to a cost of the order of O((-log epsilon)(mu m) N), where epsilon is the desired accuracy and mu(m) is a constant that depends only on m. Then, we make some comments about possible extensions of such algorithms to number of variables m >= 4 and to higher dimensions. This recursive algorithm can be viewed as an extension of "fast multipole methods" to situations where the interactions between particles are more complex than the standard case of binary interactions. Numerical tests illustrating the efficiency and the limitation of this method are presented.