Linear inverse problems in wave motion: nonsymmetric first-kindintegral equations

We present a general framework to study the solution of first-kind integral equations. The integral operator is assumed to be compact and nonself-adjoint and the integral equation can possess a nonempty null space. An approach is presented for adding contributions from the null space to the minimum-energy solution of the integral equation through the introduction of weighted Hilbert spaces. Stability, accuracy, and nonuniqueness of the solution are discussed through the use of model resolution, data fit, and model covariance operators. The application of this study is to inverse problems that exhibit nonuniqueness     

Iterative computational techniques in scattering based upon the integrated square error criterion

An iterative technique is developed to rigorously compute the electromagnetic time- and frequency-domain scattering problems. The method is based upon a wave-function expansion technique (this also includes the integral-representation techniques), in which the electromagnetic field equations and causality conditions are satisfied analytically, while the boundary conditions or the constitutive relations have to be satisfied in a computational manner. The latter is accomplished by an iterative minimization of the integrated square error. For the solution of an integral equation, it is shown how to obtain optimum convergence. Some numerical results pertaining to a number of representative problems illustrate the numerical advantages and disadvantages of the iterative method.     

Variational Solution of Integral Equations

A variational solution of the Fredholm integral equation of the first kind resulting from Laplace's equation with Dirichlet boundary conditions is discussed. Positive-definiteness of the integral operator is used to guarantee convergence. The square parallel plate capacitor is given as an example with several different types of trial functions. Special singular functions to handle known field behavior are shown to result in improved accuracy with reduced computing cost. The air-dielectric interface condition is related to a general Neumann-mixed boundary condition for which a variational method with a positive-definite integral operator is presented. Multiple boundary conditions are handled by mutually constraining separate variational expressions for each boundary condition. A T-shaped conductor on a dielectric slab, representative of quasi-static solutions of microstrip discontinuities, is presented as a three-dimensional example with multiple boundary conditions. Generally, it is shown how the finite-element method for the solution of partial differential equations may be extended to handle integral equation formulations.     

Application of the integral equation method of smoothing to random surface scattering

The integral equation method of smoothing (IEMS) is applied to the magnetic field integral equation (MFIE) weighted by the exponentialexp (jk_{1}zeta)wherezetais the stochastic surface height. An integral equation in coordinate space for the average of the product of the surface current and the exponential factor is developed. The exact closed-form solution of this integral equation is obtained based on the specularity of the average scattered field. The complex amplitude of the average scattered field is thus determined by an algebraic equation which clearly shows the effects of multiple scattering on the surface. In addition, it is shown how the incoherent scattered power can be obtained using this method. Comparisons with the Kirchhoff approximation and the dishonest approach are presented, and the first-order smoothing result is shown to be superior to both.     

Electromagnetic scattering by indented screens

The problem of three dimensional electromagnetic scattering from a perfectly conducting screen with a bounded indentation is formulated as a system of boundary integral equations for the electric current density on the cavity wall and the interface between the cavity and free space. It is shown how the fictitious current density on the interface may be eliminated resulting in an integral equation of the second kind for the current density on the cavity wall only, with no integration over the infinite screen. In addition, integral representations are derived that represent the field everywhere in space in terms of the current density on the cavity wall only. Furthermore, asymptotic expressions for the far field are also presented. The equations and representations simplify considerably in the two-dimensional scalar case and results are presented for both TE and TM polarization     

Considerations on Matrix Methods and Estimation of Their Errors

Two-dimensional field equations are reduced to Fredholm integral equations of the second kind. The integral equations are solved by matrix methods. The convergence of the matrix solutions is discussed. The matrix methods are applied to calculating the cutoff wavenumbers of waveguides. A method of estimating the errors is proposed. A method of correcting the matrix solutions is described and applied to a field problem in which the boundary is large compared with the wavelength. It is pointed out that for the commonest method of solving integral equations numerically (the method of subsections), the accuracy depends strongly on the position in each subsection of the point to which the field is referred. The dependence of the error on position is examined quantitatively.     

A mapping approach to rate-distortion computation and analysis

In rate-distortion theory, results are often derived and stated in terms of the optimizing density over the reproduction space. In this paper, the problem is reformulated in terms of the optimal mapping from the unit interval with Lebesgue measure that would induce the desired reproduction probability density. This results in optimality conditions that are “random relatives” of the known Lloyd (1982) optimality conditions for deterministic quantizers. The validity of the mapping approach is assured by fundamental isomorphism theorems for measure spaces. We show that for the squared error distortion, the optimal reproduction random variable is purely discrete at supercritical distortion (where the Shannon (1948) lower bound is not tight). The Gaussian source is thus the only source that produces continuous reproduction variables for the entire range of positive rate. To analyze the evolution of the optimal reproduction distribution, we use the mapping formulation and establish an analogy to statistical mechanics. The solutions are given by the distribution at isothermal statistical equilibrium, and are parameterized by the temperature in direct correspondence to the parametric solution of the variational equations in rate-distortion theory. The analysis of an annealing process shows how the number of “symbols” grows as the system undergoes phase transitions. Thus, an algorithm based on the mapping approach often needs but a few variables to find the exact solution, while the Blahut (1972) algorithm would only approach it at the limit of infinite resolution. Finally, a quick “deterministic annealing” algorithm to generate the rate-distortion curve is suggested. The resulting curve is exact as long as continuous phase transitions in the process are accurately followed     

A path integral time-domain method for electromagnetic scattering

A new full wave time-domain formulation for the electromagnetic field is obtained by means of a path integral. The path integral propagator is derived via a state variable approach starting with Maxwell's differential equations in tensor form. A numerical method for evaluating the path integral is presented and numerical dispersion and stability conditions are derived and numerical error is discussed. An absorbing boundary condition is demonstrated for the one-dimensional (1-D) case. It is shown that this time domain method is characterized by the unconditional stability of the path integral equations and by its ability to propagate an electromagnetic wave at the Nyquist limit, two numerical points per wavelength. As a consequence the calculated fields are not subject to numerical dispersion. Other advantages in comparison to presently popular time-domain techniques are that it avoids time interval interleaving and it does not require the methods of linear algebra such as basis function selection or matrix methods     

基于近区后向散射场幅度的二维导体目标的逆散射

本文给出了一种改进的基于多方向多入射频率的平均波照射下的近区散射场度测量值反演导体目标轮廓的逆散射场的幅度值反演导体目标轮廓的逆散射方法。     

Diffraction of plane waves by a resistive strip residing between two impedance half-planes

A uniform asymptotic solution is presented for the diffraction of E_z polarized plane waves by a resistive strip residing between two impedance half-planes. The analysis proceeds from triple integral equations approach which leads to a system of uncoupled modified Wiener-Hopf equations (MWHE). This system is then reduced to two pairs of Fredholm integral equations of the second kind which are solved by successive approximations. Diffracted field expressions are derived up to the third order terms which include the surface wave field effects in a uniform manner.     

Application of wavelet transforms for solving integral equationsthat arise in rough-surface scattering

We consider the problem of scattering a plane wave from a periodic rough surface. The scattered field is evaluated once the field on the boundary is calculated. The latter is the solution of an integral equation. In fact, different integral equation formulations are available in both coordinate and spectral space. We solve these equations using standard numerical techniques, and compare the results to corresponding solutions of the equations using wavelet transform methods for "sparsification" of the impedance matrix. Using an energy check, the methods are shown to be highly accurate. We limit the discussion to the Dirichlet problem (scalar), or the TE-polarized case for a one-dimensional surface. The boundary unknown is thus the normal derivative of the total (scalar) field or, equivalently, the surface current. We illustrate two conclusions. First, sparsification (using "thresholded" wavelet transforms) can significantly reduce accuracy. Second, the wavelet transform did not speed up the overall solution. For our examples, the solution time was considerably increased when thresholded wavelet transforms were used     

A comparison of integral equations with unique solution in theresonance region for scattering by conducting bodies

A comparison of integral equations, for problems involving scattering by arbitrary-shape conducting bodies, having a unique solution in the resonance region is presented. The augmented electric and magnetic field integral equations and the combined field integral equation, in their exact and approximate versions, are considered. The integral equations and the basis and test functions used in the method of moments to solve them are reviewed. Their implementation in a computer code is analyzed, mainly the relation between the matrix properties and the CPU time and memory. Numerical results (condition number and backscattering cross section) are presented for the cube. It is shown that the combined field integral equation, and the approximate (symmetric) combined field integral equation, are the most efficient equations to use in the neighborhood of resonant frequencies, because the overdetermined augmented integral equations require an extra matrix multiplication     

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.     

Regularized integral equations and curvilinear boundary elementsfor electromagnetic wave scattering in three dimensions

The boundary integral equations (BIEs), in their original forms, which govern the electromagnetic (EM) wave scattering in three-dimensional space contain at least a hypersingularity (1/R³ ) or a Cauchy-singularity (1/R²), usually both. Thus, obtaining reliable numerical solutions using such equations requires considerable care, especially when developing systematic numerical integration procedures for realistic problems. Regularized BIEs for the numerical computation of time-harmonic EM scattering fields due to arbitrarily-shaped scatterers are introduced. Two regularization approaches utilizing an isolation method plus a mapping are presented to remove all singularities prior to numerical integration. Both approaches differ from all existing approaches to EM scattering problems. Both work for integral equations initially containing either hypersingularities or Cauchy-singularities, without the need to introduce surface divergences or other derivatives of the EM fields on the boundary. Also, neither approach is limited to flat surfaces nor flat-element models of curved surfaces. The Muller linear combination of the electric- and magnetic-field integral equations (EFIE) and (MFIE) is used to avoid the resonance difficulty that is usually associated with integral equation-based formulations. Some preliminary numerical results for EM scattering due to single and multiple dielectric spheres are presented and compared with analytical solutions     

Simplifications in the stochastic Fourier transform approach to random surface scattering

Further developments in the application of the stochastic Fourier transform approach (SFTA) to random surface scattering are presented. It is first shown that the infinite dimensional integral equation for the stochastic Fourier transform of the surface current can be reduced to the three dimensions associated with the random surface height and slopes. A three-dimensional integral equation of the second kind is developed for the average scattered field in stochastic Fourier transform space using conditional probability density functions. Various techniques for determining the transformed current (and, subsequently, the incoherent scattered power) from the average scattered field in stochastic Fourier transform space are developed and studied from the point of view of computational suitability. The case of vanishingly small surface correlation length is reexamined and the SFTA is found to provide erroneous results for the average scattered field due to the basic failure of the magnetic field integral equation (MFIE) in this limit.     

The well—posedness of the formulation of diffraction problems reduced to Fredholm integral equations of the first kind with a smooth kernel

The well-posedness of diffraction problems that are reduced to Fredholm integral equations of the first kind with a smooth kernel is analyzed. The auxiliary source method and the method of extended boundary conditions, both of which involve solution of Fredholm integral equations of the first kind with a smooth kernel, are applied to show for specific examples that algorithms of calculation of all physically significant quantities—the scattering pattern, the field at an arbitrary spatial point except current-carrier points, etc.—are quite stable and allow for computation of the aforementioned quantities with a preassigned accuracy.     

A higher order FDTD method in Integral formulation

We present a fourth-order (4, 4) finite-difference time-domain (FDTD)-like algorithm based on the integral form of Maxwell's equations. The algorithm, which is called the integro-difference time-domain (IDTD) method, achieves its fourth-order accuracy in space and time by taking into account the spatial and temporal variations of electromagnetic fields within each computational cell. In the algorithm, the electromagnetic fields within each cell are represented by space and time integrals (or integral averages) of the fields, i.e., the electric and magnetic fluxes (D,B) are represented by the surface-integral average, and the electric and magnetic fields (E,H) by the line and time integral average. In order to relate the integral average fields in the staggered update equations, we have obtained constitutive relations for these fields. It is shown that the IDTD update equations combined with the constitutive relations are fourth-order accurate both in space and time. The fourth-order correction terms are represented by the modified coefficients in the update equations; the numerical structure remains the same as the conventional second-order update equations and more importantly does not require the storage of field variables at the previous time steps to obtain the fourth-order accuracy in time. Furthermore, the Courant-Friedrichs-Lewy (CFL) stability criteria of this fourth-order algorithm turns out to be identical to the stability limits of conventional second-order FDTD scheme based on differential formulation.     

Printed antennas analysis by a Gabor frame-based method of moments

Gabor frame-based discretization is proposed for the first time as a fully rigorous and flexible tool in the context of antenna analysis. A rigorous discretization procedure based on frame theory is presented and applied to integral equations solution through a method of moments (MoM). In this approach, the unknown field or current is expanded in a set of spatially and spectrally translated elementary functions. The use of a Gaussian window function as basis element allows for the representation of radiated fields as a superposition of shifted and rotated Gaussian beams. By exploiting the well understood propagation and transformation features of Gaussian beams, the fields can be evaluated by summations of analytic terms, at any observation point. This method seems well suited to model antennas embedded in complex systems including arbitrary interfaces. Numerical results are presented for slot antennas at the interface between two dielectric half spaces and compared to a standard MoM to validate the approach and illustrate its attractive characteristics.     

On the scattering of electromagnetic waves by bodies buried in a half-space with locally rough interface

A method for the scattering of electromagnetic waves from cylindrical bodies of arbitrary materials and cross sections buried beneath a rough interface is presented. The problem is first reduced to the solution of a Fredholm integral equation of the second kind through the Green's function of the background medium. The integral equation is treated here by an application of the method of moments (MoM). The Green's function of the two-part space with rough interface is obtained by a novel approach which is based on the assumption that the perturbations of the rough surface from a planar one are objects located at both sides of the planar boundary. Such an approach allows one to formulate the problem as a scattering of cylindrical waves from buried cylindrical bodies which is solved by means of MoM. The method is effective for surfaces having a localized and arbitrary roughness. Numerical simulations are carried out to validate the results and to show the effects of some parameters on the total field. The present formulation permits one to get the near and far field expression of the scattered wave.     

Green's function for arbitrarily shaped dielectric bodies of revolution

In this paper, a solution is developed to calculate the electric field at one point in space due to an electric dipole exciting an arbitrarily shaped dielectric body of revolution (BOR). Specifically, the electric field is determined from the solution of coupled surface integral equations (SIE) for the induced surface electric and magnetic currents on the dielectric body excited by an elementary electric current dipole source. Both the interior and exterior fields to the dielectric BOR may be accurately evaluated via this approach. For a highly lossy dielectric body, the numerical Green's function is also obtainable from an approximate integral equation (AIE) based on a surface boundary condition. If this equation is solved by the method of moments, significant numerical efficiency over SIE is realized. Numerical results obtained by both SIE and AIE approaches agree with the exact solution for the special case of a dielectric sphere. With this numerical Green's function, the complicated radiation and scattering problems in the presence of an arbitrarily shaped dielectric BOR are readily solvable by the method of moments.