This paper presents various Iterative Progressive Numerical Methods (IPNMs) for the computation of electromagnetic (EM) wave scattering from many objects. We previously modified the original IPNM from the standpoint of the classical and the IDR-based linear iterative solvers. We demonstrate the performance of the IDR(s)-based IPNMs through some numerical examples of EM wave scattering from regularly placed 27 perfectly electric conducting spheres.

A numerical investigation revealed the relation between the groove randomness of actual-size diffraction gratings and the diffraction efficiencies. The diffraction gratings we treat in this study have around 10000 grooves. When the illumination wavelength is 600 nm, the entire grating size becomes 16.2 mm. The simulation was performed using the difference-field boundary element method (DFBEM). The DFBEM treats the vectorial field with a small amount of memory resources as independent of the grating size. We firstly describe the applicability of DFBEM to a considerably large-sized structure; regularly aligned grooves and a random shallow-groove structure are calculated by DFBEM and compared with the results given by standard BEM and scalar-wave approximation, respectively. Finally we show the relation between the degree of randomness and the diffraction efficiencies for two orthogonal linear polarizations. The relation provides information for determining the tolerance of fabrication errors in the groove structure and measuring the structural randomness by acquiring the irradiance of the diffracted waves.

The recursive transfer method (RTM) is a numerical technique that was developed to analyze scattering phenomena and its formulation is constructed with a difference equation derived from a differential equation by Numerov's discretization method. However, the differential equation to which Numerov's method is applicable is restricted and therefore the application range of RTM is also limited. In this paper, we provide a new discretization scheme to extend RTM formulation using the weak form theory framework. The effectiveness of the proposed formulation is confirmed by microwave scattering induced by a metallic pillar placed asymmetrically in the waveguide. A notable feature of RTM is that it can extract a localized wave from scattering waves even if the tail of the localized wave reaches to the ends of analyzing region. The discrepancy between the experimental and theoretical data is suppressed with in an upper bound determined by the standing wave ratio of the waveguide.

In this paper, we briefly review the scheme of counting statistics, in which a probability of the number of monitored or target transitions in a Markov jump process is evaluated. It is generally easy to construct a master equation for the Markov jump process, and the counting statistics enables us to straightforwardly obtain basic equations of the counting statistics from the master equation; the basic equation is used to calculate the cumulant generating function of the probability of the number of target transitions. For stationary cases, the probability is evaluated from the eigenvalue analysis. As for the nonstationary cases, we review a numerical integration scheme to calculate the statistics of the number of transitions.

This paper presents the iterative progressive numerical methods (IPNMs) based on the induced dimension reduction (IDR) theorem. The IDR theorem is mainly utilized for the development of new nonstationary linear iterative solvers. On the other hand, the use of the IDR theorem enables to revise the classical linear iterative solvers like the Jacobi, the Gauss-Seidel (GS), the relaxed Jacobi, the successive overrelaxation (SOR), and the symmetric SOR (SSOR) methods. The new IPNMs are based on the revised solvers because the original one is similar to the Jacobi method. In the new IPNMs, namely the IDR-based IPNMs, we repeatedly solve linear systems of equations by using a nonstationary linear iterative solver. An initial guess and a stopping criterion are discussed in order to realize a fast computation. We treat electromagnetic wave scattering from 27 perfectly electric conducting spheres and reports comparatively the performance of the IDR-based IPNMs. However, the IDR-based SOR- and the IDR-based SSOR-type IPNMs are not subject to the above numerical test in this paper because of the problem with an optimal relaxation parameter. The performance evaluation reveals that the IDR-based IPNMs are better than the conventional ones in terms of the net computation time and the application range for the distance between objects. The IDR-based GS-type IPNM is the best among the conventional and the IDR-based IPNMs and converges 5 times faster than a standard computation by way of the boundary element method.

This paper presents various types of iterative progressive numerical methods (IPNMs) for the computation of electromagnetic (EM) wave scattering from many objects and reports comparatively the performance of these methods. The original IPNM is similar to the Jacobi method which is one of the classical linear iterative solvers. Then the modified IPNMs are based on other classical solvers like the Gauss-Seidel (GS), the relaxed Jacobi, the successive overrelaxation (SOR), and the symmetric SOR (SSOR) methods. In the original and modified IPNMs, we repeatedly solve linear systems of equations by using a nonstationary iterative solver. An initial guess and a stopping criterion are discussed in order to realize a fast computation. We treat EM wave scattering from 27 perfectly electric conducting (PEC) spheres and evaluate the performance of the IPNMs. However, the SOR- and SSOR-type IPNMs are not subject to the above numerical test in this paper because an optimal relaxation parameter is not possible to determine in advance. The evaluation reveals that the IPNMs converge much faster than a standard BEM computation. The relaxed Jacobi-type IPNM is better than the other types in terms of the net computation time and the application range for the distance between objects.

This paper proposes a new method to numerically obtain Floquet multipliers which characterize stability of periodic orbits of ordinary differential equations. For sufficiently smooth periodic orbits, we can compute Floquet multipliers using some standard numerical methods with enough accuracy. However, it has been reported that these methods may produce incorrect results under some conditions. In this work, we propose a new iterative method to compute Floquet multipliers using eigenvectors of matrix solutions of the variational equations. Numerical examples show effectiveness of the proposed method.

A major step in the numerical solution of electromagnetic scattering problems involves the computation of the convolution based reaction integrals. In this paper a procedure based on the analytical Fourier transform is introduced which allows us to calculate the convolution-based reaction integrals in the spectral domain without evaluating any convolution products directly. A numerical evaluation of the computational cost is presented to show the efficiency of the method when handling electrically large problems.

We propose a progressive transform-based phase unwrapping (PU) technique that employs a recursive structure. Each stage, which is identical with others in the construction, performs PU by FFT method that yields a solution and a residual phase error as well. The residual phase error is then reprocessed by the following stages. This scheme effectively improves the gradient estimate of the noisy wrapped phase image, which is unrecoverable by conventional global PU methods. Additionally, by incorporating computational strength of the transform PU method in a recursive system, we can realize a progressive PU system for prospective near real-time topographic-mapping radar and near real-time medical imaging system (such as MRI thermometry and MRI flow imager). PU performance of the proposed system and the conventional PU methods are evaluated by comparing their residual error quantitatively with a fringe-density-related error metric called FZX (fringe's zero-crossing) number. Experimental results for simulated and real InSAR phase images show significant, progressive improvement over conventional ones of a single-stage system, which demonstrates the high applicability of the proposed method.

The comparative analysis of the well known Double Drift Region (DDR) IMPATT diode structure and the n+pvnp+ structure for the avalanche diode has been realized on the basis of the drift-diffusion nonlinear model. The last type of the diode was named as Double Avalanche Region (DAR) IMPATT diode. This structure includes two avalanche regions inside the diode. The phase delay which was produced by means of the two avalanche zones and the drift zone v is sufficient for the negative resistance obtained for the wide frequency region. The numerical model that is used for the analysis of the various diode structures includes all principal features of the physical phenomena inside the semiconductor structure. The admittance characteristics of both types of the diodes were analyzed in very wide frequency region.

A novel hybrid numerical method, which is based on the extended spectral domain approach combined with the mode-matching method, is applied to evaluate the scattering parameter of waveguide discontinuities. The formulation procedure utilizes the biorthogonal relation in the transformation, and the Green's functions in the spectral domain are obtained easily even in the inhomogeneous lossy regions. The present method does not include the approximate perturbational scheme, and it can evaluate accurately and stably the scattering parameters of either for the thin or thick obstacles made of the wide variety of materials, the lossless dielectrics to highly conductive media, in short computation time. The physical phenomena of transmission through the lossy obstacles are investigated by numerical computations. The results are compared with FEM where FEM computations are feasible, although the FEM computations cannot cover the whole performances of the present method. The good agreement is observed in the corresponding range. The matrix size in this method is smaller than that of other methods. Therefore, the present method is numerically efficient and it would be able to apply for the integrated evaluation of a successive discontinuity. The resonant characteristics of rectangular waveguide cavity are analyzed accurately taking the conductor losses into consideration.

In this paper, we present a numerical method with guaranteed accuracy to solve initial value problems (IVPs) of normal form simultaneous first order ordinary differential equations (ODEs) which have wide domain. Our method is based on the algorithm proposed by Kashiwagi, by which we can obtain inclusions of exact values at several discrete points of the solution curve of ODEs. The method can be regarded as an extension of the Lohner's method. But the algorithm is not efficient for equations which have wide domain, because the error bounds become too wide from a practical point of view. Our purpose is to produce tight bounds even for such equations. We realize it by combining Kashiwagi's algorithm with the mean value form. We also consider the wrapping effects to obtain tighter bounds.

This paper surveys the research topics and results on nonlinear theory and its applications which have been achieved in Japan or by Japanese researchers during the last decade. The paticular emphasis is placed on chaos, neural networks, nonlinear circuit analysis, nonlinear system theory, and numerical methods for solving nonlinear systems.

The method solving Bessel's differential equation for calculating numerical values of the Bessel function Jν(x) is not usually used, but it is made clear here that the differential equation method can give very precise numerical values of Jν(x), and is very effective if we do not mind computing time. Here we improved the differential equation method by using a new transformation of Jν(x). This letter also shows a method of evaluating the errors of Jν(x) calculated by this method. The recurrence method is used for calculating the Bessel function Jν(x) numerically. The convergence of the solutions in this method, however, is not yet examined for all of the values of the complex ν and the real x. By using the differential equation method, this letter will numerically ascertain the convergence of the solutions and the precision of Jν(x) calculated by the recurrence method.

In this paper, an efficient and robust circuit parameter determination method suitable for analog circuit synthesis is presented. The method uses block diagram representation of circuits as implicit design knowledge. Circuit parameter determination is carried out by propagating known values along signal flow in the block diagram. The circuit parameter determination using signal propagation performs successfully when unknown circuit parameters can be solved in one way. However, when the block diagram involves implicit calculation, the propagation stops before all unknown parameters are determined. In order to cope with this problem, we introduced a method that employs a symbolic analysis technique combined with a numerical method. When the propagation of known values stops, one of unknown signals is selected, a unique symbol is assigned to the selected signal, and the signal propagation is restarted. This operation is repeated until there is no unknown signal. When the symbol propagation reaches the signal where the signal value is already set, one nonlinear equation for the signal is obtained by equating both signal values. It can be solved by a numerical method, such as Newton's method. The parameter determination method using procedural description is superior to the optimization based method because it is straightforward to incorporate design knowhow in the description. However, it is burdensome for designers to develop design procedures for each circuit to be synthesized. Because the block diagram based calculation method can be used as subroutine calls during the design procedure development, it simplifies the design procedural description and lowers the burden of designers. The method was applied to the element value determination of bias circuits to demonstrate its effectiveness.

A priori estimation is presented for a computational complexity of the homotopy method applying to a certain class of strongly monotone nonlinear equations. In the present papers, a condition is presented for a certain class of uniquely solvable equations, under which an upper bound of a computational complexity of the Newton type homotopy method can be a priori estimated. In this paper, a condition is considered in a case of linear homotopy equations including the Newton type homotopy equations. In the first place, the homotopy algorithm based on the simplified Newton method is introduced. Then by using Urabe type theorem, which gives a sufficient condition guaranteeing the convergence of the simplified Newton method, a condition is presented under which an upper bound of a computational complexity of the algorithm can be a priori estimated, when it is applied to a certain class of strongly monotone nonlinear equations. The presented condition is demonstrated by numerical experiments.