Fast methods for the evaluation of the diffusion kernel with potential extensions to the dissipative kernel

The solution of time-domain integral equations involving the diffusion or dissipative kernel require the evaluation of convolutions in space and time which scale as O(N_s ²N_t ²), where N_s and N_t are the number of spatial and temporal discretizations, respectively. The hybrid time-space acceleration methods proposed here reduce this cost to either O(N_sN_t log(N_t)) or O(N_sN_t) for rank-deficient kernels. These accelerations are achieved by way of a unification of two techniques: (1) accelerated Cartesian expansions (ACE), and (2) fast matrix-vector multiplication methods based upon either fast Fourier transforms (FFT) or matrix decompositions. In particular, the application of this acceleration scheme to the time-domain diffusion equation is presented; however, this set of techniques is particularly well-suited to being adapted to numerous physics problems, and application to dissipative systems are underway.