Sparse Matrix Equation Research Articles

A surface integral equation formulation for electromagnetic scattering from a conducting cylinder coated with multilayers of homogeneous materials

A surface integral equation formulation is presented for electromagnetic scattering by a conducting cylinder coated with multilayers of homogeneous materials. Each layer may have a nonunity relative permittivity and permeability. Both the TM and TE polarizations are considered. The surface equivalence principle is utilized to model the problem where each layer is replaced by equivalent surface currents residing on the enclosing boundaries. A systematic procedure is developed to generate a set of coupled integral equations for an arbitrary number of layers. The method of moments is invoked to convert these equations into a sparse matrix equation which can be solved using sparse matrix routines. Numerical results are presented to demonstrate the accuracy and efficiency of the proposed method. The performance of the method is compared with that of the volume-surface integral equation formulation where a great saving in memory storage and computation time is achieved.

An Efficient Domain Decomposition Laguerre-FDTD Method for Two-Dimensional Scattering Problems

In this paper, an efficient domain decomposition technique is introduced into the unconditionally stable finite-difference time-domain (FDTD) method based on weighted Laguerre polynomials to solve two-dimensional (2-D) electromagnetic scattering problems. The whole computational space is decomposed into multiple subdomains where there is no direct field coupling between any two different subdomains. For the large sparse matrix equation generated by the implicit scheme, the domain decomposition technique transforms this large scale equation into some independent smaller equations. With the total-field/scattered-field boundary and Mur's second-order absorbing boundary condition, the radar cross sections of two 2-D structures are calculated. The numerical examples verify the accuracy and efficiency of the proposed method.

Bethe ansatz and ordinary differential equation correspondence for degenerate Gaudin models

In this work, we generalize the numerical approach to Gaudin models developed earlier by us to degenerate systems showing that their treatment is surprisingly convenient from a numerical point of view. In fact, high degeneracies not only reduce the number of relevant states in the Hilbert space by a non negligible fraction, they also allow to write the relevant equations in the form of sparse matrix equations. Moreover, we introduce a new inversion method based on a basis of barycentric polynomials which leads to a more stable and efficient root extraction which most importantly avoids the necessity of working with arbitrary precision. As an example we show the results of our procedure applied to the Richardson model on a square lattice.

Wavelet Based MBPE Technique for Fast RCS Computation of DNG Metamaterials

Described by the drude model, the DNG metamaterials is essentially a dispersive media, the scattering analysis of it over broad band by asymptotic waveform evaluation (AWE) technique is difficult since the computation of high order differential coefficients for the impedance matrix elements, to reduce the differential order, the model-based parameter estimation (MBPE) technique based on two expanded points is applied to analysis the scattering characters of double-negative (DNG) metamaterials. A wavelet matrix transform (WMT) is used to get a sparse matrix equation in wavelet-domain, which is solved over a broad frequency band by the MBPE. As compared with AWE, the MBPE technique obtained more accurate results over a broader frequency band.

Implementation of the moving particle semi-implicit method on GPU

The Moving Particle Semi-implicit (MPS) method performs well in simulating violent free surface flow and hence becomes popular in the area of fluid flow simulation. However, the implementations of searching neighbouring particles and solving the large sparse matrix equations (Poisson-type equation) are very time-consuming. In order to utilize the tremendous power of parallel computation of Graphics Processing Units (GPU), this study has developed a GPU-based MPS model employing the Compute Unified Device Architecture (CUDA) on NVIDIA GTX 280. The efficient neighbourhood particle searching is done through an indirect method and the Poisson-type pressure equation is solved by the Bi-Conjugate Gradient (BiCG) method. Four different optimization levels for the present general parallel GPU-based MPS model are demonstrated. In addition, the elaborate optimization of GPU code is also discussed. A benchmark problem of dam-breaking flow is simulated using both codes of the present GPU-based MPS and the original CPU-based MPS. The comparisons between them show that the GPU-based MPS model outperforms 26 times the traditional CPU model.

Efficient Implementation for 3-D Laguerre-Based Finite-Difference Time-Domain Method

When the Laguerre-based finite-difference time-domain (FDTD) method is used for electromagnetic problems, a huge sparse matrix equation results, which is very expensive to solve. We previously introduced an efficient algorithm for implementing an unconditionally stable 2-D Laguerre-based FDTD method. We numerically verified that the efficient algorithm can save CPU time and memory storage greatly while maintaining comparable computational accuracy. This paper presents new efficient algorithm for implementing unconditionally stable 3-D Laguerre-based FDTD method. To do so, a factorization-splitting scheme using two sub-steps is adopted to solve the produced huge sparse matrix equation. For a full update cycle, the presented scheme solves six tri-diagonal matrices for the electric field components and computes three explicit equations for the magnetic field components. A perfectly matched layer absorbing boundary condition is also extended to this approach. In order to demonstrate the accuracy and efficiency of the proposed method, numerical examples are given.

PML Absorbing Boundary Condition for Efficient 2-D WLP-FDTD Method

In this letter, Berenger's perfectly matched layer (PML) absorbing boundary condition is presented for the efficient two-dimensional (2-D) finite-difference time-domain (FDTD) method with weighted Laguerre polynomials. Through adding a perturbation term, the huge sparse matrix equation is solved with a factorization-splitting scheme. To verify the validity of the proposed formulations, a numerical example of scattering from a 2-D rectangular conductor is given.

Finite-element contrast source inversion method for microwave imaging

With respect to the microwave imaging of the dielectric properties in an imaging region, the full derivation of a new inversion algorithm based on the contrast source inversion (CSI) algorithm and a finite-element method (FEM) discretization of the Helmholtz differential operator formulation for the scattered electromagnetic field is presented. The unknown dielectric properties are represented as nodal values on a two-dimensional (2D) arbitrary triangular mesh using linear basis functions. The use of FEM to represent the Helmholtz operator allows for the flexibility of having an inhomogeneous background medium, as well as the ability to accurately model any boundary shape or type: both conducting and absorbing. The resulting sparse and symmetric FEM matrix equation can be solved efficiently, and it is shown how its solution can be used to calculate the gradient operators required in the conjugate-gradient CSI update without storing the inverse of the FEM matrix. The inversion algorithm is applied to conductive-enclosures of various shapes and unbounded-region microwave tomography configurations where the 2D transverse magnetic (TM) approximation can be applied.

Efficient Implementation for the Unconditionally Stable 2-D WLP-FDTD Method

This letter presents an efficient algorithm for the unconditionally stable two-dimensional finite-difference time-domain method with weighted Laguerre polynomials (2-D WLP-FDTD). The huge sparse matrix equation is solved with a factorization-splitting scheme. This leads to much less CPU time and memory storage than those in the conventional implementation. To verify the accuracy and efficiency of the proposed method, two numerical examples are given.

An Efficient Algorithm for Implementing the Crank–Nicolson Scheme in the Mixed Finite-Element Time-Domain Method

In this paper, an efficient algorithm for implementing Crank-Nicolson scheme in the finite-element time-domain (FETD) method is presented. Based on a direct discretization of the first-order coupled Maxwell curl equations, this algorithm employs edge elements (Whitney 1-form) to expand the electric field and face elements (Whitney 2-form) for the magnetic field. Since the curl of an edge-element is the linear combination of those face elements whose faces contain the given edge, only the Maxwell-Ampere equation composes a sparse linear matrix equation for the electric field update; the Maxwell-Faraday equation is explicit. The Crank-Nicolson scheme is implemented leading to an unconditionally stable vector FETD method and the matrix inverse is not required to be computed explicitly. Therefore, only one matrix equation is required to be solved at each time step. Numerical results demonstrate that the proposed method is efficient when compared with the conventional leap-frog mixed FETD method and the Crank-Nicolson FDTD method.

Sparsification of the Impedance Matrix in the Solution of the Integral Equation by Using the Maximally Orthogonalized Basis Functions

The higher order vector basis functions defined in large patches have been utilized in the numerical solution of integral equations in this paper to sparsify the impedance matrix and relieve the memory pressure. The physical explanation for the sparsification of the impedance matrix is also elucidated. Furthermore, the maximally orthogonalized bases have been applied to improve the condition number of the impedance matrix. The scaling factor was reformed to speed up the iteration convergence in the numerical solution. Finally, the iterative method for sparse matrix equations is applied to improve the solution efficiency. Some numerical results are provided to illustrate the excellent performance both in the sparsification of the impedance matrix and solution efficiency for numerical analysis of the scattering problem.

Wind velocity simulation of spatial three-dimensional fields based on autoregressive model

This paper adopts autoregressive (AR) model to simulate the wind velocity of spatial three-dimensional fields in accordance with the time and space dependent characteristics of the 3-D fields. Based on the built MATLAB programming, this paper discusses in detail the issues of the AR model deduced by matrix form in the simulation and proposes the corresponding solving methods: the over-relaxation iteration to solve the large sparse matrix equations produced by large number of degrees of freedom of structures; the improved Gauss formula to calculate the numerical integral equations which integral functions contain oscillating functions; the mixed congruence and central limit theorem of Lindberg-Levy to generate random numbers. This paper also develops a method of ascertaining the rank of the AR model. The numerical examples show that all those methods are stable and reliable, which can be used to simulate the wind velocity of all large span structures in civil engineering.

Unconditionally stable Crank–Nicolson finite-different time-domain method for simulation of three-dimensional microwave circuits

An unconditionally stable Crank–Nicolson finite-difference time-domain (CN-FDTD) algorithm is presented for three-dimensional microwave circuit analysis. First, Mur's first–order absorbing boundary condition is applied this CN-FDTD algorithm. A symmetric successive over relaxation–preconditioned biconjugate-gradient algorithm is also proposed to solve the large sparse matrix equation obtained in the CN-FDTD method. Resonant cavity and several planar microstrip circuits are presented to illustrate the versatility of this technique. Numerical results indicate that with a time-step size excessively larger than the Courant–Friedrich–Levy limit, the accuracy of CN-FDTD is still much higher than that of the alternating-direction implicit FDTD.

On the use of piecewise linear wavelets for fast construction of sparsified moment matrices in solving the thin-wire EFIE

Multiresolution wavelet expansion technique has been successfully used in the method of moments (MoM), and sparse matrix equations have been attained. Solving boundary integral equations arising in electromagnetic (EM) problems by the wavelet-based moment method (WMM) involves a time-consuming double numerical integration for each entry of the resultant matrix which in turn can outweigh the advantages of achieving a sparse matrix. The paper presents an alternative computational model to speed up the WMM by excluding double numerical integrations in the evaluation of matrix elements. In this regard, pieces of linear wavelet bases are replaced by proper sinusoidal functions for which closed-form analytical expressions are available. In addition, by introducing approximate closed-form expressions for radiating EM fields of wavelet current elements, the thresholding procedure is modified so that one can compute only the matrix elements of interest. To demonstrate the effectiveness of the proposed method, the thin-wire electric field integral equation (EFIE) is numerically solved by non-orthogonal linear spline wavelet bases.

An Adaptive Algebraic Multigrid Algorithm for Micromagnetism

We present an adaptive algebraic multigrid algorithm. The method is intended for large sparse matrix equations that arise from finite-element discretizations of the stray field in three-dimensional micromagnetism on nonuniform or unstructured grids. It uses a varying threshold value to control the grid ratio, trying to optimize the overall efficiency of the algebraic multigrid solver. We present numerical results and compare them with the preconditioned conjugate gradient method.

Parameters extraction for on-chip transmission lines via novel fast sparsification method: FDHT

In order to fast extract the frequency-dependent parameters of on-chip interconnects with lossy substrate, the fast discrete Hadamard transform (FDHT) is adopted to sparsify the dense matrix formed by the complex dielectric image method which is accelerated by the ϵ-algorithm of the Padé approximation and GMRES iteration is used to solve the resulted sparse matrix equations.

Low rank approximate solutions to large Sylvester matrix equations

In the present paper, we propose an Arnoldi-based method for solving large and sparse Sylvester matrix equations with low rank right hand sides. We will show how to extract low-rank approximations via a matrix Krylov subspace method. We give some theoretical results such an expression of the exact solution and upper bounds for the norm of the error and for the residual. Numerical experiments will also be given to show the effectiveness of the proposed method.

Optimization of the sources in local hyperthermia using a combined finite element-genetic algorithm method

This article describes an optimization process specially designed for local and regional hyperthermia in order to achieve the desired specific absorption rate in the patient. It is based on a genetic algorithm coupled to a finite element formulation. The optimization method is applied to real human organs meshes assembled from computerized tomography scans. A 3D finite element formulation is used to calculate the electromagnetic field in the patient, achieved by radiofrequency or microwave sources. Space discretization is performed using incomplete first order edge elements. The sparse complex symmetric matrix equation is solved using a conjugate gradient solver with potential projection pre-conditionning. The formulation is validated by comparison of calculated specific absorption rate distributions in a phantom to temperature measurements. A genetic algorithm is used to optimize the specific absorption rate distribution to predict the phases and amplitudes of the sources leading to the best focalization. The objective function is defined as the specific absorption rate ratio in the tumour and healthy tissues. Several constraints, regarding the specific absorption rate in tumour and the total power in the patient, may be prescribed. Results obtained with two types of applicators (waveguides and annular phased array) are presented and show the faculties of the developed optimization process.

Application of wavelets to singular integral scattering equations

The use of orthonormal wavelet basis functions for solving singular integral scattering equations is investigated. It is shown that these basis functions lead to sparse matrix equations which can be solved by iterative techniques. The scaling properties of wavelets are used to derive an efficient method for evaluating the singular integrals. The accuracy and efficiency of the wavelet transforms are demonstrated by solving the two-body $T$-matrix equation without partial wave projection. The resulting matrix equation which is characteristic of multiparticle integral scattering equations is found to provide an efficient method for obtaining accurate approximate solutions to the integral equation. These results indicate that wavelet transforms may provide a useful tool for studying few-body systems.

Wavelet-like bases for thin-wire integral equations in electromagnetics

In this paper, wavelets are used in solving, by the method of moments, a modified version of the thin-wire electric field integral equation, in frequency domain. The time domain electromagnetic quantities, are obtained by using the inverse discrete fast Fourier transform. The retarded scalar electric and vector magnetic potentials are employed in order to obtain the integral formulation. The discretized model generated by applying the direct method of moments via point-matching procedure, results in a linear system with a dense matrix which have to be solved for each frequency of the Fourier spectrum of the time domain impressed source. Therefore, orthogonal wavelet-like basis transform is used to sparsify the moment matrix. In particular, dyadic and M-band wavelet transforms have been adopted, so generating different sparse matrix structures. This leads to an efficient solution in solving the resulting sparse matrix equation. Moreover, a wavelet preconditioner is used to accelerate the convergence rate of the iterative solver employed. These numerical features are used in analyzing the transient behavior of a lightning protection system. In particular, the transient performance of the earth termination system of a lightning protection system or of the earth electrode of an electric power substation, during its operation is focused. The numerical results, obtained by running a complex structure, are discussed and the features of the used method are underlined.