A weak form of the conjugate gradient FFT method fortwo-dimensional TE scattering problems

The problem of two-dimensional scattering of a transversal electric polarized wave, by a dielectric object is formulated in terms of a hypersingular integral equation, in which a grad-div operator acts on a vector potential. The vector potential is a spatial convolution of the free-space Green's function and the contrast source over the domain of interest. A weak form of the integral equation for the unknown electric flux density is obtained by testing it with rooftop functions. The vector potential is expanded in a sequence of the rooftop functions and the grad-div operator is integrated analytically over the dielectric object domain only. The method shows excellent numerical performance     

An improved pulse-basis conjugate gradient FFT method for the thinconducting plate problem

A conjugate-gradient fast Fourier transform (CG-FFT) formulation for the scattering by a thin, perfectly conducting plate is presented. Pulse basis functions and a Dirac δ function are used for expansion and testing purposes, respectively. Particular attention is paid to the generation of the terms in the impedance matrix of the resulting matrix equation. Except for the self-coupling terms, all the other terms are computed by explicit integrations. Comparison with two previously reported pulse-basis CG-FFT formulations shows that the present method is more stable when the error tolerance of the solution is reduced. When sufficient cell discretizations are used, smooth distributions can be obtained which resemble those obtained via rooftop-CG-FFT formulation. The numerical results are further validated by comparing the far-field radar cross section with an analytical technique for a circular plate     

The three dimensional weak form of the conjugate gradient FFTmethod for solving scattering problems

The problem of electromagnetic scattering by a three-dimensional dielectric object can be formulated in terms of a hypersingular integral equation, in which a grad-div operator acts on a vector potential. The vector potential is a spatial convolution of the free space Green's function and the contrast source over the domain of interest. A weak form of the integral equation for the relevant unknown quantity is obtained by testing it with appropriate testing functions. The vector potential is then expanded in a sequence of the appropriate expansion functions and the grad-div operator is integrated analytically over the scattering object domain only. A weak form of the singular Green's function has been used by introducing its spherical mean. As a result, the spatial convolution can be carried out numerically using a trapezoidal integration rule. This method shows excellent numerical performance     

A concise conjugate gradient computation of plate problems withmany excitations

One of the major difficulties in the application of the conjugate gradient algorithm for the analysis of electromagnetic scattering problems is the necessity to carry out the calculation separately for each incident wave. In the approach suggested, rather than handling the incident waves directly, a class of possible excitations is represented by a set of strip-type basis functions. For these functions, convergence is predictable and rapid because the majority of the strips are located away from the edges of the scatterer. This choice also facilitates the use of the physical optics approximation as a good initial guess. Once the solutions for all the unit basis functions over the body are known, they can be combined to synthesize the solution for any excitation using the weighting coefficients associated with the expansion of the incident field. Numerical examples are given, and they demonstrate the substantial savings achieved by adopting this approach for the analysis of multiple excitations     

Application of a conjugate gradient FFT method to scattering fromthin planar material plates

The backscatter cross section is calculated for thin material plates with finite electric permittivity, conductivity, and magnetic permeability illuminated by a plane wave. The plates are assumed to be planar with an arbitrary perimeter. The integral equations are formed and solved by a combined conjugate gradient-fast Fourier transform (CG-FFT) method. The CG-FFT method was tested for several geometrics and materials measured and computed backscatter results are compared for a perfectly conducting equilateral triangle plate, a square dielectric and magnetic plate, and a circular dielectric plate. The agreement between measured and computed data is generally good except toward edge-on incidence where several factors cause discrepancies. Accurate approximations to the geometry and far-field integrals become critical near edge-on incidence and, it is postulated that as the incidence angle approaches edge-on, the sampling interval and tolerance should be decreased     

Calculation of electromagnetic scattering from a dielectric cylinder using the conjugate gradient method and FFT

The scattering problem of an axially uniform dielectric cylinder is formulated in terms of the electric field integral equation, where the cylinder is of general cross-sectional shape, inhomogeneity, and anisotropy, and the incident field is arbitrary. Using the pulse-function expansion and the point-matching technique, the integral equation is reduced to a system of simultaneous equations. Then, a published procedure for solving the system using the conjugate gradient method and the fast Fourier transform (FFT) is generalized to the case of oblique-incidence scattering.     

Preconditioned conjugate gradient methods for solving electromagnetic problems

It is shown that appropriately chosen preconditioners can significantly improve the convergence rate of the conjugate gradient (CG) algorithm as applied to electromagnetic problems. Preconditioners are constructed for the problems of scattering from frequency selective surfaces and scattering from infinite cylinders.     

A modified spectral conjugate gradient projection method for signal recovery

In this paper, signal recovery problems are first reformulated as a nonlinear monotone system of equations such that the modified spectral conjugate gradient projection method proposed by Wan et al. can be extended to solve the signal recovery problems. In view of the equations’ analytic properties, an improved projection-based derivative-free algorithm (IPBDF) is developed. Compared with the similar algorithms available in the literature, an advantage of IPBDF is that the search direction is always sufficiently descent as well as being close to the quasi-Newton direction, without requirement of computing the Jacobian matrix. Then, IPBDF is applied into solving a number of test problems for reconstruction of sparse signals and blurred images. Numerical results indicate that the proposed method either can recover signals in less CPU time or can reconstruct the images with higher quality than the other similar ones.     

Improving the convergence rate of the conjugate gradient FFT methodusing subdomain basis functions

A technique to improve the convergence rate of the conjugate gradient-fast Fourier transform (CG-FFT) method is presented. The procedure involves the incorporation of subdomain basis functions associated with the current representation of linear and planar radiating elements. It is shown that significant improvements are achieved in the convergence of the CG-FFT when using sinusoidal basis functions. Numerical results are presented for thin cylindrical dipoles, conducting strips, and material plates of various sizes. In all cases, an increase in the rate of convergence by a factor of two or better was observed     

Application of FFT and the conjugate gradient method for the solution of electromagnetic radiation from electrically large and small conducting bodies

A novel combination of the conjugate gradient (CG) method with the fast Fourier transform technique (FFT) is presented. With this combination, the computational time required to solve large scatterer problems is much less than the time required by the ordinary conjugate gradient method and the method of moments. On the other hand, since the spatial derivatives are replaced with simple multiplications in the transformed domain, some of the computational difficulties present in the ordinary conjugate gradient method and the method of moments do not exist here. Therefore, electrically small structures can also be handled more easily. Finally, since the method is iterative, it is possible to know the accuracy in a problem solution. Two types of scatterers are analyzed: wires (both very long and very short) and square plates (very large and small). The details of the computational procedure are presented along with numerical results for some of the scatterers analyzed.     

预条件共轭梯度法在辐射和散射问题中的应用

用矩量法求解一些辐射和散射问题 ,如线天线辐射和线状体散射等问题时 ,可以产生一个 Toeplitz线性方程组 ,采用预条件共轭梯度法 (PCG)与快速富里叶变换 (FFT)的结合方法 (PCGFFT)来求解该方程组 ,其中预条件器采用 T.Chan的优化循环预条件器。使用 PCGFFT算法 ,可有效地节省内存 ,提高了计算速度。为说明其有效性 ,将 PCGFFT算法与 CGFFT算法以及 Levinson递推算法进行了对比。     

Convergence of the conjugate gradient method when applied to matrix equations representing electromagnetic scattering problems

An iterative procedure based on the conjugate gradient method is used to solve a variety of matrix equations representing electromagnetic scattering problems, in an attempt to characterize the typical rate of convergence of that method. It is found that this rate depends on the cell density per wavelength used in the discretization, the presence of symmetries in the solution, and the degree to which mixed cell sizes are used in the models. Assuming cell densities used in the discretization are in the range of ten per linear wavelength, the iterative algorithm typically requiresN/4toN/2steps to converge to necessary accuracy, whereNis the order of the matrix under consideration.     

A constrained conjugate gradient method for solving the magnetic field boundary integral equation

It is well-known that electromagnetic solutions of boundary integral equations for perfectly electrically conducting scatterers are nonunique for those frequencies which correspond to interior resonances of the scatterer. In this paper a simple and efficient computational method is developed, in which the interior integral representations, required to hold on an interior closed surface, are used as a sufficient constraint to restore uniqueness. We use the interior equations together with the second kind magnetic field integral equation, so that the ill-posedness of the interior equations does not give a problem. We develop a constrained conjugate gradient method that minimizes a cost functional consisting of two terms. The first term is the error norm with respect to the magnetic field boundary integral equation, while the second term is the error norm with respect to the interior equations over a closed interior surface, which is chosen as small as possible. Some numerical examples show the robustness and efficiency of the pertaining computational procedure.     

Solution to some electromagnetic problems using fast Fourier transform with conjugate gradient method

Using the fast Fourier transform (FFT) to compute the convolution integrals that appear in the conjugate-gradient method (CGM), an efficient numerical procedure to solve electromagnetic problems is obtained. In comparison with the method of moments (MM), the proposed FFT-CGM avoids the storage of large matrices and reduces the computer time by orders of magnitude.     

Constrained iterative reconstruction by the conjugate gradient method

The conjugate gradient method incorporating the object-extent constraint is applied to image reconstruction of a three-dimensional object using an incomplete projection-data set. The missing information is recovered by constraining the solution with the knowledge of the outer boundary of the object-extent which may be a priori measured or known. The algorithm is derived from the least-squares criterion as an advanced version of conventional iterative reconstruction algorithms such as SIRT (Simultaneous Iterative Reconstruction Technique) and ILST (Iterative Least Squares Technique). In the case of reconstruction from noisy projection data, a method based on the minimum mean-square error criterion is also proposed. Computer simulated reconstruction images of a phantom using limited angle and number of views are presented. The result shows that the conjugate gradient method incorporating the object-extent constraining provides the fastest convergence and the least error.     

Comments on "Application of FFT and the conjugate gradient method for the solution of electromagnetic radiation from electrically large and small conducting bodies"

In the original paper, (ibid., vol.AP-34, no.5, p.635-40, 1986) T.K. Sarkar et al. describe how the fast Fourier transform (FFT) and the conjugate gradient method (CGM) can be used to efficiently make use of the convolutional form of the electric field integral equation for straight wire antennas. The commentators agree that the CGM converges more rapidly than the previously used method of steepest descent and the spectral iteration technique. They also point out that experience with the dyadic Green's function encountered in the two-dimensional transverse electric (TE) and three-dimensional equations for dielectric bodies is not so encouraging. It is found that for these cases the CGM convergence can be quite slow, degrading the efficiency of the algorithm. In addition, the solution obtained using the pulse-basis point-matching method to discretize the integral equation are very inaccurate for the two-dimensional TE and three-dimensional cases. A reply to these comments from the authors of the original paper is included     

Application of conjugate gradient method for static problemsinvolving conductors of arbitrary shape

In this paper, two implementations of the Conjugate Gradient Method (CGM) for the solution of problems in electrostatics involving conductors of arbitrary shapes have been discussed. The first method uses a least squares approximation for the computation of the pertinent integral operator and is referred to as LSD. A second implementation, referred to as Point Matching Discretisation (PMD) effects considerable saving in the computer time since it uses the midpoint rule for the integral arising in LSD. Both these techniques require O(N) storage, where N is the number of patches used to model the conductor. Further, a matrix interpretation of the present formulation has been derived. This has facilitated the comparison of the techniques described in this paper with the well known Method of Moments (MoM) formulation and has led to better understanding of the convergence of the results. Using illustrative examples of canonical (square and circular discs) and arbitrary shape (a pyramid mounted on a cube), the convergence of and the computer time for both the implementations have been investigated. It has been shown that both the techniques yielded monotonically convergent results for all the examples considered and that the LSD offers better estimate of the capacitance than PMD with lower number of patches     

A conjugate gradient algorithm for the treatment of multipleincident electromagnetic fields

An iterative method based on the conjugate gradient (CG) algorithm is developed for the efficient treatment of equations involving multiple excitations. Examples show that significant time savings can be obtained as compared to treating each excitation individually with the conjugate gradient algorithm. However, these savings are not obtained without the drawback of increased memory requirements to store the additional excitations, residuals, and solutions. The efficiency of this algorithm tends to increase as additional excitations are added     

NOMA系统中利用共轭梯度法的功率分配方案

通过功率分配,5G通信的关键技术——非正交多址（NOMA）实现发射功率域的多用户复用,有效提高了频谱效率。不同的功率分配方案直接影响系统的吞吐量,针对NOMA下行链路现有功率分配算法存在的局部最优问题,提出了一种利用共轭梯度法的最优功率分配方案,采用共轭梯度法求解用户的加权和速率最大化的优化问题。现有理论证明,该方法可以收敛到全局最优解。仿真结果表明,该方法性能优于已有的固定功率分配（FPA）算法和分数阶发射功率分配（FTPA）算法,且此非正交多址（NOMA）系统性能明显优于正交多址（OMA）系统。      

Effect of sampling rate on the conjugate gradient method applied tosignal extrapolation

A different response to changes in sampling rate has been observed for conjugate gradients as opposed to SVD (singular-value decomposition) applied to signal extrapolation. The conjugate gradient solution is described in terms of the SVD, which leads to an explanation of the behavior of the conjugate gradient method in this context. Formulating the conjugate gradient method as a spectral filtering method and analyzing it in terms of the spectral filter functions generated as the iteration progresses have made it possible to explain the difference in the effect of sampling rate changes on the conjugate gradients approach to signal extrapolation as compared to the truncated SVD approach discussed by Sullivan and Liu (1984)