Accelerated Cartesian Expansions Research Articles

Fast and scalable evaluation of pairwise potentials

Pair potentials or kernels, ψ(|r|), play a critical role in a number of areas; these include biophysics, electrical engineering, fluid dynamics, diffusion physics, solid state physics, and many more. The need to evaluate these potentials rapidly for N particles gives rise to the classical N-body problem. In this paper, we present scalable parallel algorithms for evaluation of these potentials for highly non-uniform distributions. The underlying methodology for evaluating these potentials relies on the accelerated Cartesian expansion (ACE) framework that is quasi-kernel-independent with the requirement that the kernel be differentiable with known derivatives. The results presented demonstrate the accuracy control, low cost, and parallel scalability offered by this method for several example kernels and distributions of up to 5 billion particles on 16384 CPU cores. Potential applications of the algorithm include various disciplines of computational physics, engineering, machine learning, among others.

Differential operator acting on accelerated Cartesian expansion algorithm

The multilevel accelerated Cartesian expansion algorithm (MLACEA) and its incorporation with multilevel fast multipole algorithm (MLFMA) have been studied and used for solving practical problems. The algorithms are successful, but the authors found the truncation error of accelerated Cartesian expansion (ACE) is sensitive with the differential operator. Integral equations in electromagnetics always contain differential operators, so the precision will be influenced when trying to accelerate the method of moments with MLACE algorithm. The reason of why differential operator can influence the precision of the ACE is given, and also proved by numerical experimentation. To meet the requirements of piratical application, electric field integral equation is deeply investigated. The studies also extend to the hybrid algorithm MLFMA–MLACEA and performances are tested on different implementation.

Accelerated Cartesian Expansions for the Rapid Solution of Periodic Multiscale Problems

We present an algorithm for the fast and efficient solution of integral equations that arise in the analysis of scattering from periodic arrays of PEC objects, such as multiband frequency selective surfaces (FSS) or metamaterial structures. Our approach relies upon the method of Accelerated Cartesian Expansions (ACE) to rapidly evaluate the requisite potential integrals. ACE is analogous to FMM in that it can be used to accelerate the matrix vector product used in the solution of systems discretized using MoM. Here, ACE provides linear scaling in both CPU time and memory. Details regarding the implementation of this method within the context of periodic systems are provided, as well as results that establish error convergence and scalability. We also demonstrate the applicability of this algorithm by studying several exemplary electrically dense systems.

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

The development of the multilevel fast multipole algorithm (MLFMA) and its multiscale variants have enabled the use of integral equation (IE) based solvers to compute scattering from complicated structures. Development of scalable parallel algorithms, to extend the reach of these solvers, has been a topic of intense research for about a decade. In this paper, we present a new algorithm for parallel implementation of IE solver that is augmented with a wideband MLFMA and scalable on large number of processors. The wideband MLFMA employed here, to handle multiscale problems, is a hybrid combination of the accelerated Cartesian expansion (ACE) and the classical MLFMA. The salient feature of the presented parallel algorithm is that it is implicitly load balanced and exhibits higher performance. This is achieved by developing a strategy to partition the MLFMA tree, and hence the associated computations, in a self-similar fashion among the parallel processors. As detailed in the paper, the algorithm employs both spatial and direction partitioning approaches in a flexible manner to ensure scalable performance. Plethora of results are presented here to exhibit the scalability of this algorithm on 512 and more processors.

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

Diffusion, 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 as O N s 2 N t 2 , where 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 O ( N s N t log 2 N t ) . Several numerical results are presented for the diffusion equation to validate the accuracy and efficacy of the fast algorithms developed here.

A Novel Wideband FMM for Fast Integral Equation Solution of Multiscale Problems in Electromagnetics

In this paper, we propose a novel scheme to accelerate integral equation solvers when applied to multiscale problems. These class of problems exhibit multiple length/frequency scales and arise when analyzing scattering/radiation from realistic structures where dense discretization is necessary to accurately capture geometric features. Solutions to the discretized integral equations due to these structures is challenging, due to their high computational cost and ill-conditioning of the resulting matrix system. The focus of this paper is on ameliorating the computational cost. Our approach will rely on exploiting the recently developed accelerated Cartesian expansion (ACE) algorithm to arrive at a method that is stable and efficient at low frequencies. These will then be integrated with the well known fast multipole method, thus forming a scheme that is wideband. Rigorous convergence estimates of this method are derived, and convergence and efficiency of the overall fast method is demonstrated. These are then integrated into an existing integral equation solver, whose efficiency is demonstrated for some practical problems.

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

Time domain integral equation solvers for transient scattering from electrically large objects have benefitted significantly from acceleration techniques like the plane wave time domain (PWTD) algorithm; these techniques reduce the asymptotic CPU and memory cost. However, PWTD breaks down when used in the analysis of structures that have subwavelength features or features whose length scales are orders of magnitude smaller than the smallest wavelength in the incident pulse. Instances of these occurring in electromagnetics range from antenna topologies, to feed structures, etc. In this regime, it is the geometric constraints that dictate the computational complexity, as opposed to the wavelength of interest. In this work, we present an approach for efficient analysis of such sub-wavelength source/observer distributions in time domain. The methodology that we seek to exploit is the recently developed algorithm based on Cartesian expansions for accelerating the computation of potentials of the form R^@n. In this paper, we present an efficient methodology for computing these polynomials for two different scenarios; where the size of the domain spans the distance travelled by light in (i) one time step and (ii) multiple time steps. These algorithms are cast within the framework of both uniform and non-uniform distributions. Results that demonstrate the efficiency and convergence of the proposed algorithm are presented.