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

Abstract

a b s t r a c tDiffusion, lossy wave, and Klein-Gordon equations find numerous applications in practical problems across a range of diverse disciplines. The temporal dependence of all three Green's functions are characterized by an infinite tail. This implies that the cost complexity of the spatio-temporal convolutions, associated with evaluating the potentials, scales aswhere N s and N t are the number of spatial and temporal degrees of freedom, respectively. In this paper, we discuss two new methods to rapidly evaluate these spatio-temporal convolutions by exploiting their block-Toeplitz nature within the framework of accelerated Cartesian expansions (ACE). The first scheme identifies a convolution relation in time amongst ACE harmonics and the fast Fourier transform (FFT) is used for efficient evaluation of these convolutions. The second method exploits the rank deficiency of the ACE translation operators with respect to time and develops a recursive numerical compression scheme for the efficient representation and evaluation of temporal convolutions. It is shown that the cost of both methods scales as OðN s N t log 2 N t Þ. Several numerical results are presented for the diffusion equation to validate the accuracy and efficacy of the fast algorithms developed here.

An O(N) method for the rapid analysis of periodic problems using Accelerated Cartesian Expansions (ACE)

“…As ACE is not wedded to addition theorems for special functions, it is possible to apply these to the rapid evaluation of many different potentials. To date, this has been done for potentials of the form r −ν (ν ∈ R) [17], Lienard-Wiechert potentials [18], and diffusion, Klein-Gordon, and lossy wave potentials [19]. Likewise, ACE has also been implemented together with FMM for the wideband analysis of electromagnetic phenomena [20], with analytically derived error bounds that have been demonstrated via numerical experimentation.…”

A Thermodynamically Consistent Fractional Visco-Elasto-Plastic Model with Memory-Dependent Damage for Anomalous Materials

“…Nevertheless, non-smooth loading/unloading conditions pose additional challenges to develop high-order schemes for the model. In terms of efficiency, the computational bottleneck lies in the free-energy discretization, which needs further improvements before employing fast schemes for the fractional derivatives, e.g., fast convolution [35,61] and fast multi-pole approaches [57]. Variants of the developed model can be incorporated in a straightforward fashion.…”

A Scalable Parallel Wideband MLFMA for Efficient Electromagnetic Simulations on Large Scale Clusters

“…ACE is an almost kernel independent method as (a) all quantities of the ACE algorithm, except the translation operator, are independent of the form of the kernel and (b) the ACE expansions are rapidly converging for any non-oscillatory function [5], [38]- [40]. Readers interested in the details and other salient features of the ACE algorithm are referred to [38].…”