Fast evaluation of time domain fields in sub-wavelength source/observer distributions using accelerated Cartesian expansions (ACE)

Accelerated Cartesian expansions – A fast method for computing of potentials of the form R−ν for all real ν

“…We have also demonstrated the efficacy of this scheme for other (non-integer) potential functions. Current research is focused on generalizing the proposed methodology to analyze Yukawa and periodic columbic potentials, retarded and Helmholtz potential for sub-wavelength regimes [35], and fast Gauss transform. Finally, application of is scheme to problems in biophysics, electronic structure calculations, and MD codes are currently underway.…”

“…Furthermore, one can exploit the structure of the tensor contractions involved in translation. Some of these improvements have already had a positive impact on the speed of the resulting code [35].…”

Accelerated Cartesian expansion (ACE) based framework for the rapid evaluation of diffusion, lossy wave, and Klein–Gordon potentials

“…1. The computational cost associated with a single matrix vector product Z kÀl S l is OðN 2 s Þ and hence the cost of evaluating P k at all N t time steps scales as OðN 2 s N 2 t Þ. Parenthetically, we note that potential evaluators for the wave equation (in both two and three dimensions) and the lossy wave equation take a form that is similar to (8) [31][32][33]23,29]. With simple algebraic manipulation, it is possible to retrofit these fast schemes into marching on in time (MOT) solvers for solving time domain integral equations.…”

“…However, we must augment our algorithm to include contributions due to K (p) , as they possess no such smooth tail. We instead handle these contributions using methods similar to those given in [29], wherein the ACE algorithm is used to accelerate the computation of the retarded potential in a similar MOT framework. It is significant to note that this is the same problem that arises in the a = 0 limit of Eq.…”

“…It must be noted that, similar methodologies have been introduced earlier [25,26] but they are either not generalizable or offer only some of the advantages of this scheme. ACE has been successfully used to accelerate a number of different kernels, including London [24], low frequency Helmholtz (extensible to Yukawa) [27,28], and retarded wave [29]. The temporal acceleration schemes, both FFT and numerical compression, are also independent of the form of the kernel.…”