An efficient algorithm for the evaluation of certain convolution integrals with singular kernels

In this short note, we are concerned with the evaluation of certain convolution integrals with singular kernels. The problem is as follows. Given a kernel function K(t) which is very often singular at the origin and a density function σ(t), evaluate C(t) = ∫ t 0 K(t − τ)σ(τ)dτ for t = ∆t, 2∆t, · · · , T = N∆t. We will oftern write C(tk) = C(k∆t) = Ck in short, and similarly for σ. Direct computation of Ck requires the storage of all previous densities σ1, · · · , σk and O(k) flops at the kth step. Thus it requires on average O(N) storage and flops for each time step and the total amount of flops is O(N2), which forms a bottleneck for long time simulations. We present here an efficient algorithm for such problems. Our algorithm reduces the storage requirement from O(N) to O(log(N)) and the overall computational cost from O(N2) to O(N log(N)). The basic idea is that in the region away from its singular point we may approximate the kernel K accurately by a sum of exponentials with the number of exponentials proportional to log(N), and the computation of the convolution with the exponential functions as the kernel can be sped up using a very simple recurrence relation. The note is organized as follows. In Section 2, we present an outline of the algorithm. In Section 3, we present a numerical example with applications on the Havriliak-Negami model for dielectric medias.