Comments on “An improved pulse-basis conjugate gradient FFTmethod for the thin conducting plate problem” [with reply]

Peter and Zwamborn comment on the paper by Tran and McCowen (IEEE Trans. Antenn. Propag., vol.41, p.185-90, 1993) which presents pulse-basis expansion functions and Dirac δ testing functions within a conjugate gradient FFT (CGFFT) formulation. In the conclusions, Tran and McCowen state: “Although these spurious effects do not appear in the CGFFT formulation using the rooftop basis function proposed by Zwamborn and van den Berg (1991), there are more computational costs associated with their method than with ours.” Peter and Zwamborn disagree with this statement. A reply is given by Tran and McCowen     

A weak form of the conjugate gradient FFT method for plate problems

A number of electromagnetic field problems for planar structures 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 convolution of the free-space Green's function and some surface current density over the domain of interest. A weak form of this integral equation is obtained by testing it with subdomain basis functions defined over the plate domain only. As a next step, the vector potential is expanded in a sequence of subdomain basis functions and the grad-div operator is integrated analytically over the plate domain only. For the problem of electromagnetic scattering by a plate, the method shows excellent numerical performance. The numerical difficulties encountered in some previous conjugate gradient fast Fourier transform (CGFFT) methods have been eliminated     

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     

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.     

An improved method for a redundancy optimization problem

The present note suggests an improvement over an earlier method for a redundancy optimization problem. There is a drastic reduction in the number of steps required to arrive at the optimal allocation. A numerical example illustrates the technique.     

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.     

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.     

An adaptive nulling system for a narrow-band signal with alook-direction constraint utilizing the conjugate gradient method

The problem of iterative adjustments of the weights of a phased array with a look-direction constraint in the presence of jammers is presented. The technique described uses the adaptive conjugate method instead of the popularly used method of steepest descent to eliminate the jammer components, thus minimizing the error between the received signal and the actual one. This iterative method minimized the L₂ norm (the mean square error) and is guaranteed to converge in a finite number of steps which is not available for the present techniques utilized to solve this problem. Improvement in rate of convergence is thus achieved at the expense of algorithmic complexity     

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     

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.     

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     

An improved quasi-static finite-difference scheme for induced field evaluation based on the biconjugate gradient method

This paper presents a biconjugate gradient (BiCG) method that can significantly improve the performance of the quasi-static finite-difference scheme, which has been widely used to model field induction phenomena in voxel phantoms. The proposed BiCG method offers remarkable computational advantages in terms of convergence performance and memory consumption over the conventional iterative, successive overrelaxation algorithm. The scheme has been validated against other known solutions on a lossy, multilayered ellipsoid phantom excited by an ideal coil loop. The wide application capability and computational performance of the BiCG method is demonstrated by modeling the exposure of MRI healthcare workers to fields produced by pulsed field gradients. This is an important topic of research in light of the Physical Agents Directive 2004/40/EC because a variety of realistic operator postures near the bore entrance of an MRI system are modeled.     

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.     

A simplified texture gradient method for improved image segmentation

Inspired by the probability of boundary (Pb) algorithm, a simplified texture gradient method has been developed to locate texture boundaries within grayscale images. Despite considerable simplification, the proposed algorithm’s ability to locate texture boundaries is comparable with Pb’s texture boundary method. The proposed texture gradient method is also integrated with a biologically inspired model, to enable boundaries defined by discontinuities in both intensity and texture to be located. The combined algorithm outperforms the current state-of-art image segmentation method (Pb) when this method is also restricted to using only local cues of intensity and texture at a single scale.     

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     

基于预处理共轭梯度法的低复杂度信号检测算法

大规模多输入多输出系统中,最小均方误差信号检测算法是近似最优的,但由于其涉及矩阵求逆,计算复杂度随着天线数量增加呈指数增长.提出了低复杂度的预处理共轭梯度信号检测算法,该算法通过预处理技术降低矩阵条件数,从而加快共轭梯度信号检测算法的收敛速度.仿真结果显示,该算法在小数量的迭代中能够达到和最小均方误差检测算法相似的误码率,算法复杂度下降了一个数量级.相比直接用共轭梯度法,能够更快收敛到最佳值.     

求解DEC-POMDP问题的改进遗传算法

分布式部分可观测马尔科夫模型(Decentralized partially observable Markov decision progress,DEC-POMDP)是研究不确定性情况下多主体协同决策的重要模型。由于其求解难度是NEXP-complete,所以迄今为止尚没有有效的算法能求出其最优解。但是存在一部分近似求解的算法可以解决规模较小的问题。针对此问题,在遗传算法的基础上,通过引入最佳起始状态和最佳收益状态提出改进的遗传算法(Improved Genetic Algorithms,IGA),算法将问题的求解分为两个步骤,首先求解从给定起始状态到最佳起始状态的近似最优策略,然后求解在最佳收益状态之间转换的策略。通过实验可以看出IGA压缩了要搜索的策略空间,减小了编码长度,是求解DEC-POMDP的有效算法。