Model-based parameter estimation in electromagnetics. III.Applications to EM integral equations

For pt.II see ibid., vol.40, no.2, p.51-66 (1998). This article has considered the rationale and illustrated the application of model-based parameter estimation (MBPE) to achieve reduced-order representations of electromagnetic observables via fitting models, the model-based part of MBPE, that derive from the physics of EM fields. The parameter-estimation part of MBPE is the process of obtaining numerical values for the coefficients of the fitting model by matching or fitting it to sampled values of the EM observable of interest. Although a wider range of fitting models are feasible, attention here is focused on what are termed waveform-domain models, comprised of exponential series, and spectral-domain models, comprised of pole series. These kinds of fitting models are shown to provide natural basis functions for many kinds of EM observables, whether these observables are based on experimental measurement or numerical computation     

Mathematical foundations of neurocomputing

An attempt is made to establish a mathematical theory that shows the intrinsic mechanisms, capabilities, and limitations of information processing by various architectures of neural networks. A method of statistically analyzing one-layer neural networks is given, covering the stability of associative mapping and mapping by totally random networks. A fundamental problem of statistical neurodynamics is considered in a way that is different from the spin-glass approach. A dynamic analysis of associative memory models and a general theory of neural learning, in which the learning potential function plays a role, are given. An advanced theory of learning and self-organization is proposed, covering backpropagation and its generalizations as well as the formation of topological maps and neural representations of information     

Mathematical foundations of minimal cutsets

Since their introduction in the reliability field, binary decision diagrams have proved to be the most efficient tool to assess Boolean models such as fault trees. Their success increases the need of sound mathematical foundations for the notions that are involved in reliability and dependability studies. This paper clarifies the mathematical status of the notion of minimal cutsets which have a central role in fault-tree assessment. Algorithmic issues are discussed. Minimal cutsets are distinct from prime implicants and they have a great interest from both a computation complexity and practical viewpoint. Implementation of BDD algorithms is explained. All of these algorithms are implemented in the Aralia software, which is widely used. These algorithms and their mathematical foundations were designed to assess efficiently a very large noncoherent fault tree that models the emergency shutdown system of a nuclear reactor     

New boundary integral equations for CAD of waveguide circuits:guided-mode extracted integral equations

Novel boundary integral equations which are applicable to the analysis of many kinds of waveguide circuits are presented. The new integral equations can treat the waveguide discontinuity problems like the scattering by the isolated finite-sized metallic objects or cavity problems and do not use normal-mode expansion techniques. They are suitable for the basic theory of CAD software for various waveguide circuits. The two-port and H-plane waveguide discontinuity problems which satisfy the single-mode and two-mode conditions are treated. The case of waveguide corner bend is considered as an example. Numerical examples are shown in order to confirm the validity of the new integral equations     

Mathematical problems in error calculations for interference

Several treatments of interference problems are examined with a view to determine whether doubtful mathematical assumptions about the existence of probability density functions affect the conclusions. Most derivations can be recast so that the existence of a density function is not necessary for the conclusions to be valid     

Time domain adaptive integral method for surface integral equations

An efficient marching-on-in-time (MOT) scheme is presented for solving electric, magnetic, and combined field integral equations pertinent to the analysis of transient electromagnetic scattering from perfectly conducting surfaces residing in an unbounded homogenous medium. The proposed scheme is the extension of the frequency-domain adaptive integral/pre-corrected fast-Fourier transform (FFT) method to the time domain. Fields on the scatterer that are produced by space-time sources residing on its surface are computed: 1) by locally projecting, for each time step, all sources onto a uniform auxiliary grid that encases the scatterer; 2) by computing everywhere on this grid the transient fields produced by the resulting auxiliary sources via global, multilevel/blocked, space-time FFTs; 3) by locally interpolating these fields back onto the scatterer surface. As this procedure is inaccurate when source and observer points reside close to each other; and 4) near fields are computed classically, albeit (pre-)corrected, for errors introduced through the use of global FFTs. The proposed scheme has a computational complexity and memory requirement of O(N/sub t/N/sub s/log/sup 2/N/sub s/) and O(N/sub s//sup 3/2/) when applied to quasiplanar structures, and of O(N/sub t/N/sub s//sup 3/2/log/sup 2/N/sub s/) and O(N/sub s//sup 2/) when used to analyze scattering from general surfaces. Here, N/sub s/ and N/sub t/ denote the number of spatial and temporal degrees of freedom of the surface current density. These computational cost and memory requirements are contrasted to those of classical MOT solvers, which scale as O(N/sub t/N/sub s//sup 2/) and O(N/sub s//sup 2/), respectively. A parallel implementation of the scheme on a distributed-memory computer cluster that uses the message-passing interface is described. Simulation results demonstrate the accuracy, efficiency, and the parallel performance of the implementation.     

Investigation of electromagnetics problems for strip structures by means of the integral equation method

A method is proposed for the analysis of radiating and guiding strip structures. The method involves reducing the original electromagnetics problems to special integral equations that allow the development of efficient and universal simulation algorithms. Examples of the numerical analysis of the aforementioned practically important strip structures are presented.     

Sparse inverse preconditioning of multilevel fast multipole algorithm for hybrid Integral equations in electromagnetics

In computational electromagnetics, the multilevel fast multipole algorithm (MLFMA) is used to reduce the computational complexity of the matrix vector product operations. In iteratively solving the dense linear systems arising from discretized hybrid integral equations, the sparse approximate inverse (SAI) preconditioning technique is employed to accelerate the convergence rate of the Krylov iterations. We show that a good quality SAI preconditioner can be constructed by using the near part matrix numerically generated in the MLFMA. The main purpose of this study is to show that this class of the SAI preconditioners are effective with the MLFMA and can reduce the number of Krylov iterations substantially. Our experimental results indicate that the SAI preconditioned MLFMA maintains the computational complexity of the MLFMA, but converges a lot faster, thus effectively reduces the overall simulation time.     

Model-based parameter estimation in electromagnetics. I. Backgroundand theoretical development

This article first provides a background and motivation for using model based parameter estimation (MBPE) in electromagnetics, focusing on the use of fitting models that are described by exponential and pole series. How data obtained from various kinds of sampling procedures can be used to quantify such models, i.e., to determine numerical values for their coefficients is also presented. The paper continues by illustrating applications of MBPE to various kinds of EM observables. It concludes by discussing how MBPE might be used to improve the efficiency of first-principles models based on frequency-domain integral equations     

A survey of various numerical techniques to solve operator equations in electromagnetics

The objective of this paper is to survey many of the popular methods utilized in solving numerical problems arising in electromagnetics. Historically, the matrix methods have been quite popular. One of the primary objectives of this paper is to introduce a new class of iterative methods, which have advantages over the classical matrix methods in the sense that a given problem may be solved to a prespecified degree of accuracy. Also, these iterative methods (particularly conjugate gradient methods) converge to the solution in a finite number of steps irrespective of the initial starting guess. Numerical examples have been presented to illustrate the principles.     

The foundations of set theoretic estimation

Explains set theoretic estimation, which is governed by the notion of feasibility and produces solutions whose sole property is to be consistent with all information arising from the observed data and a priori knowledge. Each piece of information is associated with a set in the solution space, and the intersection of these sets, the feasibility set, represents the acceptable solutions. The practical use of the set theoretic framework stems from the existence of efficient techniques for finding these solutions. Many scattered problems in systems science and signal processing have been approached in set theoretic terms over the past three decades. The author synthesizes a single, general framework from these various approaches, examines its fundamental philosophy, goals, and analytical techniques, and relates it to conventional methods     

Volume integral equations for analysis of dielectric branchingwaveguides

Novel forms of volume integral equations are developed for the exact treatment of wave propagation in two-dimensional dielectric branching waveguides. The integral equations can be obtained by considering the condition at a point far away from the junction section. An approximate solution by the Born approximation and a numerical solution by the moment method establish the validity of the new volume integral equations. The numerical results are discussed from the viewpoint of energy conservation and reciprocity. The solution is exact if sufficiently large computer memory and computational time are used. The method can be extended to problems of a more general nature (i.e. the incident TM mode), and complex configurations of branching waveguides. The basic idea is also applicable to techniques using boundary (surface) integral equations which are applicable to three-dimensional problems     

Model-based parameter estimation in electromagnetics. II.Applications to EM observables

Discusses spectral domain model based parameter estimation (MBPE), waveform-dominated MBPE, and adapting and optimizing the sampling of the generating model. These are discussed with reference to antenna theory including scattering     

Theoretical error in the integral equation MEI

The theoretical, or non-numerical error in the integral equation MEI (IE-MEI) is analysed and compared to the numerical error in the method of moments (MoM). It is found that, although this error does not decrease with the discretisation step, it is smaller for the usual discretisation than the error in the MoM     

Dual-surface integral equations in electromagnetic scattering

A brief review is given of the derivation and application of dual-surface integral equations, which eliminate the spurious resonances from the solution to the original electric-field and magnetic-field integral equations applied to perfectly electrically conducting scatterers. Emphasis is placed on numerical solutions of the dual-surface electric-field integral equation for three-dimensional perfectly electrically conducting scatterers.     

Timing error estimation for baseband multicarrier modulation

In this paper, the topic of timing error estimation for baseband discrete multitone modulation is addressed in the context of high‐speed digital subscriber line applications. To the authors' best knowledge, this problem is sparsely and not in depth treated in the literature. In this study the modified Cramer–Rao lower bound is derived, two conventional estimators are considered and a new one is proposed. Behavior of these estimators is evaluated in terms of bias, variance and computational complexity in a wide variety of scenarios, concluding that the proposed estimator appears to be superior in all instances. Copyright © 2008 John Wiley & Sons, Ltd.     

Well-conditioned boundary integral equations for three-dimensional electromagnetic scattering

We introduce a new version of the combined field integral equation (CFIE) for the solution of electromagnetic scattering problems in three dimensions. Unlike the conventional CFIE, the new CFIE is well-conditioned, meaning that it is a second kind integral equation that does not suffer from spurious resonances and does not become ill conditioned for fine discretizations (the so-called "low-frequency problem"). The new CFIE combines the standard magnetic field integral operator with an analytically preconditioned electric field integral operator. We also report numerical results showing that the new formulation stabilizes the number of iterations needed to solve the CFIE on closed surfaces. This is in contrast to the conventional CFIE, where the number of iterations grows as the discretization is refined.     

Error estimation for numerical differential equations

Given a capable human being and a computer, it is possible to make an approximation to the solution of a nonlinear differential equation. However, under the (usually correct) assumption that the equation is analytically intractable, the result of the computation is not the exact solution; indeed it may be so far from the exact solution as to be completely useless. We are interested in the relationship between the effort expended by the human and the computer, and the duality of the computed approximation to a partial or ordinary differential equation. To be specific, we would like to think in terms of a cost-benefit analysis. The cost of the computation is a combination of the human effort and computer resources used to obtain the approximation. The benefit includes, of course, the computed approximation, but it also includes an estimate of the quality of the approximation, that is, an error estimate. It is our opinion that in computational science, as with the experimental sciences, results should always be presented with some estimate of their accuracy. In addition, however, there is another facet to error estimation: one cannot even attempt a cost-benefit analysis or efficiency comparison of methods without an error estimate to evaluate the results     

The optimal error exponent for Markov order estimation

We consider the problem of estimating the order of a stationary ergodic Markov chain. Our focus is on estimators which satisfy a generalized Neyman-Pearson criterion of optimality. Specifically, the optimal estimator minimizes the probability of underestimation among all estimators with probability of overestimation not exceeding a given value. Our main result identifies the best exponent of asymptotically exponential decay of the probability of underestimation. We further construct a consistent estimator, based on Kullback-Leibler divergences, which achieves the best exponent. We also present a consistent estimator involving a recursively computable statistic based on appropriate mixture distributions; this estimator also achieves the best exponent for underestimation probability