  Discrete-velocity approximations represent a popular way for computing the
Boltzmann collision operator. The direct numerical evaluation of such methods
involve a prohibitive cost, typically $O(N^{2d+1})$ where $d$ is the dimension
of the velocity space. In this paper, following the ideas introduced in
[27,28], we derive fast summation techniques for the evaluation of
discrete-velocity schemes which permits to reduce the computational cost from
$O(N^{2d+1})$ to $O(\bar{N}^d N^d\log_2 N)$, $\bar{N} << N$, with almost no
loss of accuracy.
