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A matrix version of the fast multipole method
 SIAM Review
"... Abstract. We present a matrix interpretation of the threedimensional fast multipole method (FMM). The FMM is for efficient computation of gravitational/electrostatic potentials and fields. It has found various applications and inspired the design of many efficient algorithms. The onedimensional FM ..."
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Abstract. We present a matrix interpretation of the threedimensional fast multipole method (FMM). The FMM is for efficient computation of gravitational/electrostatic potentials and fields. It has found various applications and inspired the design of many efficient algorithms. The onedimensional FMM is well interpreted in terms of matrix computations. The threedimensional matrix version reveals the underlying matrix structures and computational techniques used in FMM. It also provides a unified view of algorithm variants as well as existing and emerging implementations of the FMM.
Adaptive Multiscale Moment Method (AMMM) for Analysis of Scattering From ThreeDimensional Perfectly Conducting Structures
"... Abstract—Adaptive multiscale moment method (AMMM) is presented for the analysis of scattering from threedimensional (3D) perfectly conducting bodies. This algorithm employs the conventional moment method (MM) using the subsectional triangular patch basis functions and a special matrix transformati ..."
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Abstract—Adaptive multiscale moment method (AMMM) is presented for the analysis of scattering from threedimensional (3D) perfectly conducting bodies. This algorithm employs the conventional moment method (MM) using the subsectional triangular patch basis functions and a special matrix transformation, which is derived from solving the Fredholm equation of the first kind by the multiscale technique. This methodology is more suitable for problems where the matrix solution time is much greater than the matrix fill time. The widely used triangular patch vector basis functions developed by Rao et al. are used for expansion and testing functions in the conventional MM. The objective here is to compress the unknowns in existing MM codes, which solves for the currents crossing the edges of the triangular patch basis functions. By use of a matrix transformation, the currents, source terms, and impedance matrix can be arranged in the form of different scales. From one scale to another scale, the initial guess for the solution can be predicted according to the properties of the multiscale technique. AMMM can reduce automatically the size of the linear equations so as to improve the efficiency of the conventional MM. The basic difference between this methodology and the other waveletbased techniques that have been presented so far is that we apply the compression not to the impedance matrix but to the solution itself directly in an iterative fashion even though it is an unknown. Two numerical results are presented, which demonstrate that the AMMM is a useful method for analysis of electromagnetic scattering from arbitrary shaped 3D perfectly conducting bodies. Index Terms—Conducting structures, multiscale moment method, method of moments, multiscale. I.
Sparse Matrix/Canonical Grid Method Applied to 3D Dense Medium Simulations
"... Abstract—The sparse matrix/canonical grid (SMCG) Method, which has been shown to be an efficient method for calculating the scattering from onedimensional and twodimensional random rough surfaces, is extended to threedimensional (3D) dense media scattering. In particular, we study the scattering ..."
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Abstract—The sparse matrix/canonical grid (SMCG) Method, which has been shown to be an efficient method for calculating the scattering from onedimensional and twodimensional random rough surfaces, is extended to threedimensional (3D) dense media scattering. In particular, we study the scattering properties of media containing randomly positioned and oriented dielectric spheroids. Mutual interactions between scatterers are formulated using a Method of Moments solution of the volume integral equation. Iterative solvers for the resulting system matrix normally require P A operations for each matrixvector multiply. The SMCG method reduces this complexity to @ �� � A by defining a neighborhood distance, , by which particle interactions are decomposed into “strong ” and “weak. ” Strong interaction terms are calculated directly requiring @ A operations for each iteration. Weak interaction terms are approximated by a multivariate Taylor series expansion of the 3D background dyadic Green’s function between any given pair of particles. Greater accuracy may be achieved by increasing, using a higher order Taylor expansion, and/or increasing mesh density at the cost of more interaction terms, more fast Fourier transforms (FFTs), and longer FFTs, respectively. Scattering results, computation times, and accuracy for largescale problems with up to 2 gridpoints, 14 14 14 canonical grid size, fifthorder Taylor expansion, and 15 000 discrete scatterers are presented and compared against full solutions. Index Terms—Fast methods, random media, sparse matrix/canonical grid (SMCG), spheroid, threedimensional (3D) scattering. I.
Broadband rational macromodeling based on the adaptive frequency sampling algorithm and the partial element equivalent circuit method
 IEEE Trans. Electromagnetic Compatibility
, 2008
"... Abstract—The increasing operating frequencies in modern designs call for broadband macromodeling techniques. The problem of computing highaccuracy simulation models for highspeed interconnects is of great importance in the modeling arena. Nowadays, many fullwave numerical techniques are availa ..."
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Abstract—The increasing operating frequencies in modern designs call for broadband macromodeling techniques. The problem of computing highaccuracy simulation models for highspeed interconnects is of great importance in the modeling arena. Nowadays, many fullwave numerical techniques are available that provide high accuracy, often at a significant cost in terms of memory storage and computing time. Furthermore, designers are usually only interested in a few electrical quantities such as port voltages and currents. So, model order reduction techniques are commonly used to achieve accurate results in a reasonable time. This paper presents a new technique, based on the partial element equivalent circuit method, which allows to generate reducedorder models by adaptively selecting the complexity (order) of the macromodel and suitable frequency samples. Thus, the proposed algorithm allows to limit the computing time while preserving the accuracy. Validation examples are given. Index Terms—Adaptive frequency sampling (AFS), electromagnetic transient analysis, fitting techniques, frequency response, partial element equivalent circuit (PEEC) method. I.
Xband resistive sensor for high power microwave pulse measurement with flat frequency response
, 2008
"... AbstractA resistive sensor (RS) devoted for high power microwave pulse measurement in cylindrical waveguide is considered. The modeling results of the interaction of the TE 01 (H 01 ) wave with a semiconductor plate with contacts on sidewalls of the plate placed on a wall of the circular waveguide ..."
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AbstractA resistive sensor (RS) devoted for high power microwave pulse measurement in cylindrical waveguide is considered. The modeling results of the interaction of the TE 01 (H 01 ) wave with a semiconductor plate with contacts on sidewalls of the plate placed on a wall of the circular waveguide are presented. A finitedifference timedomain (FDTD) method was employed for the calculation of the electromagnetic field components, reflection coefficient from the semiconductor obstacle, and the average electric field in it. The features of the resonances have been used to engineer the frequency response of the RS. It has been found that such electrophysical parameters of the plate can serve as the prototype of the sensing element (SE) for the circular waveguide RS with flat frequency response.
On the HamiltonWaterloo problem
 Graphs and Combinatorics
, 2002
"... Significant effort has recently been directed towards the development of numerically efficient iterative techniques for the solution of boundary integral equation formulations of time harmonic scattering problems. The primary result of this effort has been the development of several advanced numeric ..."
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Significant effort has recently been directed towards the development of numerically efficient iterative techniques for the solution of boundary integral equation formulations of time harmonic scattering problems. The primary result of this effort has been the development of several advanced numerical techniques which enable the dense matrixvector products associated with the iterative solution of boundary integral equations to be rapidly computed. However, an important aspect of this problem which has yet to be adequately addressed is the development of rapidly convergent iterative techniques to complement the relatively more mature numerical algorithms which expedite the matrixvector product operation. To this end, a class of efficient iterative methods for boundary integral equation formulations of twodimensional scattering problems is presented. This development is based on an attempt to approximately factor (i.e., renormalize) the boundary integral formulation of an arbitrary scattering problem into a product of oneway wave operators and a corresponding coupling operator which accounts for the interactions between oppositely propagating waves on the surface of the scatterer. The original boundary integral formulation of the scattering problem defines the coupling between individual equivalent sources on the surface of the scatterer. The renormalized version of this
Broadband Macromodels for Retarded Partial Element Equivalent Circuit (rPEEC) Method
"... Abstract—The partial element equivalent circuit (PEEC) method is, nowadays, widely used in electromagnetic compatibility and signal integrity problems in both the time and frequency domains. Similar to other integralequationbased techniques, its time domain implementation may suffer from late time ..."
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Abstract—The partial element equivalent circuit (PEEC) method is, nowadays, widely used in electromagnetic compatibility and signal integrity problems in both the time and frequency domains. Similar to other integralequationbased techniques, its time domain implementation may suffer from late time instabilities, especially when considering delays [(Lp,P,R,τ)PEEC] (rPEEC). The cause of the instabilities may be either the numerical technique used for the time integration or problems created by the discrete representation of the electromagnetic continuous problem. In this paper, we concentrate on the latter and show that frequency dispersion plays an important role and must be taken into account in order to preserve accuracy and mitigate instabilities issues. An enhanced formulation of the PEEC method is presented that is based on a more accurate computation of partial elements describing the electric and magnetic field couplings; broadband macromodels are generated incorporating the frequency dependence of such elements, thus, allowing us to obtain better stability properties of the resulting (Lp,P,R,τ)PEEC model. The proposed equivalent circuits resemble those of the standard PEEC formulation but are able to capture the dispersion that, when neglected, might contribute to inaccuracies and late time instabilities. Index Terms—Broadband macromodels for retarded partial element equivalent circuit (rPEEC) method, transient analysis. I.