M. Vikram

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.

Parallel accelerated cartesian expansions for particle dynamics simulations

Rapid evaluation of potentials in large physical systems plays a crucial role in several fields and has been an intensely studied topic on parallel computers. Computational methods and associated parallel algorithms tend to vary depending on the potential being computed. Real applications often involve multiple potentials, leading to increased complexity and the need to strike a balance between competing data distribution strategies, ultimately resulting in low parallel efficiencies. In this paper, we present a parallel accelerated Cartesian expansion (PACE) method that enables rapid evaluation of multiple forms of potentials using a common Fast Multipole Method (FMM) type framework. In addition, our framework localizes potential dependent computations to one particular operator, allowing reuse of much of the computation across different potentials. We present an implicitly load balanced and communication efficient parallel algorithm and show that it can integrate multiple potentials, multiple time steps and address dynamically evolving physical systems. We demonstrate the applicability of the method by solving particle dynamics simulations using both long-range and Lennard-Jones potentials with parallel efficiencies of 97% on 512 to 1024 processors.

Accelerated cartesian harmonics for fast computation of time and frequency domain low-frequency kernels

Low frequency regime is defined as the domain wherein the spatial size is orders of magnitude smaller than the wavelength. Of course, in time domain, this translates to the smallest wavelength in the simulation. Capturing these geometric features necessitates dense discretization. Methods to overcome the bottlenecks posed by this problem have been an active research topic for a while, both in the time and frequency domain, see Refs. [J. Meng et al., 2006] and [J.S. Zhao and W.C. Chew, 2001] and references therein. The literature in frequency domain is considerably more extensive than time and the paper cited here is only representative of the work. In what follows, we present another scheme, that can be used for simulating low frequency problems in the frequency domain, and with very minor modifications, perform similar analysis in the time domain as well. The algorithms presented here rely on Accelerated Cartesian Expansions (ACE) [B. Shanker and H. He, 2006].

Analysis of scattering from complex, electrically large structures using the generalized method of moments

This paper analyses scattering from PEC structures using a novel methodology based on the Generalized Method of Moments (GMM). The GMM has been introduced as a framework that permits a useful decoupling between the basis functions and the underlying tessellation, a feature that has been the hallmark of the standard Rao-Wilton-Glisson (RWG) basis scheme. This paper extends this method, to include a locally smooth approximation, that accepts any geometry description, from point clouds to standard triangulations. The scheme detailed here has the ability to construct patches of arbitrary sizes and corresponding basis functions, enabling the easy analysis of very complex, realistic geometries. Results will be presented that validate the method and showcase its advantages.

Parallel generalized method of moments for analysis of transient scattering from PEC objects

The state of art of transient simulation using integral equation (TDIE) based methods has grown by leaps and bounds over the past few decades, both in terms of late time stability and computational complexity. This paper aims to address two additional functionalities: (i) development of a framework to enlarge the spatial approximation space using the generalized method of moments (GMM), and (ii) development of a robust parallel framework for accelerating plane wave time domain (PWTD) augmented GMM solvers. Preliminary results using GMM based TDIE solvers demonstrate the first of the desired attributes. Further results on parallelization and its integration with GMM based TDIE solvers will be presented at the conference.

Provably scalable parallel FMM algorithm for multiscale electromagnetic simulations

Few representative results are presented here. First, the parallel algorithm was used in evaluation of scalar potential (1) at N random, uniformly distributed, source/observer pairs within a cubical volume. The algorithm was was executed on 32 to 512 processors using a IBM Bluegene architecture as N and size of the domain was increased from 5 to 80 million and 5 to 80 λ, respectively. In all cases, the smallest box size was chosen such that the average number of points per box is 64. For lack of space, we present results for parallel FMM only. Fig. 1(a) shows the parallel efficiency of the algorithm; it offers efficiency as high as 95% with 512 processors. The time taken for one potential evaluation, averaged across processors, vs. N is shown as log-log plot in figure (1b). The linear slope of the line fits were close to 1, indicating the O(N) scaling of the parallel algorithm. Table I shows the typical time spent on different hierarchical computation steps for N=40 million. The run-time for parallel upward and downward tree traversals are negligible, this is in accordance with the theory following Lemma 1. Fig. 2(a) compares the RCS of a PEC sphere (radius=32λ) computed using the CFIE based parallel solver, run on 256 processor with the Mie series solution. There is excellent agreement between the two solutions. Fig. 2(b) shows the parallel efficiency of the solver for different number of unknowns. With 3.2 million unknowns and 7 level tree, the parallel solver executes with an efficiency as high as 95% on 256 processor and 87% on 512 processors. The solver results were obtained on NCSA Teragrid network.

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.

Fast solvers for electromagnetic simulation of mixed-scale structures

In this paper, we propose a fast solver based on recently developed accelerated Cartesian expansion (ACE) algorithm. ACE is a tree code based algorithm like FMM but based on Cartesian harmonics instead of spherical harmonics. It can be shown that expansion in Cartesian harmonics is stable for low-frequencies unlike conventional FMM. Here, a method to combine ACE and FMM algorithm for efficient analysis of mixed-scale geometries is presented. Results demonstrating the viability of the fast kernel is presented here and their integration with MoM solvers will be presented at the conference.