A LOW-FREQUENCY ELECTROMAGNETIC NEAR-FIELD INVERSE PROBLEM FOR A SPHERICAL SCATTERER

The interior low-frequency electromagnetic dipole excitation of a dielectric sphere is uti- lized as a simplified but realistic model in various biomedical applications. Motivated by these considerations, in this paper, we investigate analytically a near-field inverse scatter- ing problem for the electromagnetic interior dipole excitation of a dielectric sphere. First, we obtain, under the low-frequency assumption, a closed-form approximation of the series of the secondary electric field at the dipole＇s location. Then, we utilize this derived approx- imation in the development of a simple inverse medium scattering algorithm determining the sphere＇s dielectric permittivity. Finally, we present numerical results for a human head model, which demonstrate the accurate determination of the complex permittivity by the developed algorithm.     

Electromagnetic scattering theorems for interior dipole excitation of a layered obstacle

A layered scatterer is excited by a time‐harmonic spherical electromagnetic wave, generated by a dipole located either in the interior or in the exterior of the scatterer. The scatterer's core may be perfect conducting, impedance or dielectric. This paper initiates the investigation of scattering theorems corresponding to the excitation of a layered scatterer by a dipole in its interior. We establish reciprocity and general scattering theorems relating the total electric fields with the corresponding far‐field patterns. The optical theorem, relating the scattering cross‐section with the electric field in the layer containing the dipole, is recovered as a corollary of the general scattering theorem. Furthermore, for a scatterer excited by a spherical and a plane wave, mixed scattering theorems are derived. Besides, all the derived theorems recover known results, concerning the excitation of a homogeneous scatterer by an exterior dipole. We also analyze numerical implementations of the optical theorem in certain concrete scattering applications. Copyright © 2007 John Wiley & Sons, Ltd.     

Low‐frequency electromagnetic scattering by a metal torus in a lossless medium with magnetic dipolar illumination

The present contribution is concerned with an analytical presentation of the low‐frequency electromagnetic fields, which are scattered off a highly conductive ring torus that is embedded within an otherwise lossless ambient and interacting with a time‐harmonic magnetic dipole of arbitrary orientation, located nearby in the three‐dimensional space. Therein, the particular 3‐D scattering boundary value problem is modeled with respect to the solid impenetrable torus‐shaped body, where the toroidal geometry fits perfectly. The incident, the scattered, and the total non‐axisymmetric magnetic and electric fields are expanded in terms of positive integral powers of the real‐valued wave number of the exterior medium at the low‐frequency regime, whereas the static Rayleigh approximation and the first three dynamic terms provide the most significant part of the solution, because all the additional terms are small contributors and, hence, they are neglected. Consequently, the typical Maxwell‐type physical problem is transformed into intertwined either Laplace's or Poisson's potential‐type boundary value problems with the proper conditions, attached to the metallic surface of the torus. The fields of interest assume representations via infinite series expansions in terms of standard toroidal eigenfunctions, obtaining in that way analytical closed‐form solutions in a compact fashion. Although this mathematical procedure leads to infinite linear systems for every single case, these can be readily and approximately solved at a certain level of desired accuracy through standard cut‐off techniques. Copyright © 2016 John Wiley & Sons, Ltd.     

Electromagnetic scattering by a perfectly conducting obstacle in a homogeneous chiral environment: solvability and low‐frequency theory

The scattering of plane time‐harmonic electromagnetic waves propagating in a homogeneous isotropic chiral environment by a bounded perfectly conducting obstacle is studied. The unique solvability of the arising exterior boundary value problem is established by a boundary integral method. Integral representations of the total exterior field, as well as of the left and right electric far‐field patterns are derived. A low‐frequency theory for the approximation of the solution to the above problem, and the derivation of the far‐field patterns is also presented. Copyright © 2002 John Wiley & Sons, Ltd.     

Scattering of electromagnetic plane waves by a spheroid uniformly moving in free space

We investigate the scattering process, generated by a plane electromagnetic field that is incident upon a moving perfectly conducting spheroid. An accurate treatment of the electromagnetic waves interaction with scatterers in uniform motion is based on the special relativity principle. In the object's frame the incident wave is assumed to have a wavelength which is much larger than the characteristic dimension of the scatterer and thus the low‐frequency approximation method is applicable to the scattering problem. For the near electromagnetic field we obtain the zeroth‐order low‐frequency coefficients, while in the far field we calculate the leading terms for the scattering amplitude and scattering cross‐section. Finally, using the inverse Lorentz transform, we obtain the same approximations in the observer's frame. Copyright © 2006 John Wiley & Sons, Ltd.     

The interior transmission problem for anisotropic Maxwell's equations and its applications to the inverse problem

The interior transmission problem appears naturally in the study of the inverse scattering problem of determining the shape of a penetrable medium from a knowledge of the time harmonic incident waves and the far field patterns of the scattered waves. We propose a variational study of this problem in the case of Maxwell's equations in an inhomogeneous anisotropic medium. Then we apply the obtained results to build an ‘extented far field’ operator and give a characterization of the medium from the knowledge of the range of this operator. We then show how the linear sampling method can be viewed as an approximation of this characterization. Copyright © 2004 John Wiley & Sons, Ltd.     

A numerical study of electromagnetic waves in periodic waveguides

Here are considered time‐harmonic electromagnetic waves in a quadratic waveguide consisting of a periodic dielectric core enclosed by conducting walls. The permittivity function may be smooth or have jumps. The electromagnetic field is given by a magnetic vector potential in Lorenz gauge, and defined on a Floquet cell. The Helmholtz operator is approximated by a Chebyshev collocation, Fourier–Galerkin method. Laurent's rule and the inverse rule are employed for the representation of Fourier coefficients of products of functions. The computations yield, for known wavenumbers, values of the first few eigenfrequencies of the field. In general, the dispersion curves exhibit band gaps. Field patterns are identified as transverse electric, TE, transverse magnetic, TM, or hybrid modes. Maxwell's equations are fulfilled. A few trivial solutions appear when the permittivity varies in the guiding direction and across it. The results of the present method are consistent with exact results and with those obtained by a low‐order finite element software. The present method is more efficient than the low‐order finite element method. © 2013 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq 30: 490–513, 2014     

Unique determination of a perfectly conducting ball by a finite number of electric far field data

In this paper,we prove uniqueness in determining a perfectly conducting ball in the inverse electromagnetic scattering problem by a finite number of electric far field patterns with a single incident direction and polarization. It is emphasized that we use only one electric far field pattern datum to uniquely determine the radius of a ball if it is centered at the origin with radius . Furthermore, if its center was not given as a prior information, three more measurement data must be added to uniquely determine its center. The main tool used here is some new results on zeros of spherical Bessel and spherical Neumann functions.     

Tensor Green’s functions for axisymmetrical models of magnetotelluric marine sounding of locally nonhomogeneous layered media

A method for computing the electromagnetic fields of point (dipole) sources is proposed for magnetotelluric sounding (MTS) problems in axisymmetrical conducting layered media. The method expands the tensor Green’s functions of the layered medium in Fourier series in the azimuthal coordinate. For an arbitrary system of point sources we construct algorithms to compute the electromagnetic fields propagating across the plane interface of two conducting half-spaces with different constant conductivities. Translated from Prikladnaya Matematika i Informatika, No. 30, pp. 5–17, 2008.     

Bayesian Inversion of Piecewise Affine Operators in a Gaussian Framework

Piecewise affine inverse problems form a general class of nonlinear inverse problems. In particular inverse problems obeying certain variational structures, such as Fermat's principle in travel time tomography, are of this type. In a piecewise affine inverse problem a parameter is to be reconstructed when its mapping through a piecewise affine operator is observed, possibly with errors. A piecewise affine operator is defined by partitioning the parameter space and assigning a specific affine operator to each part. A Bayesian approach with a Gaussian random field prior on the parameter space is used. Both problems with a discrete finite partition and a continuous partition of the parameter space are considered.

The main result is that the posterior distribution is decomposed into a mixture of truncated Gaussian distributions, and the expression for the mixing distribution is partially analytically tractable. The general framework has, to the authors' knowledge, not previously been published, although the result for the finite partition is generally known.

Inverse problems are currently of large interest in many fields. The Bayesian approach is popular and most often highly computer intensive. The posterior distribution is frequently concentrated close to high-dimensional nonlinear spaces, resulting in slow mixing for generic sampling algorithms. Inverse problems are, however, often highly structured. In order to develop efficient sampling algorithms for a problem at hand, the problem structure must be exploited.

The decomposition of the posterior distribution that is derived in the current work can be used to develop specialized sampling algorithms. The article contains examples of such sampling algorithms. The proposed algorithms are applicable also for problems with exact observations. This is a case for which generic sampling algorithms tend to fail.     

Uniqueness of inverse scattering problem for a penetrable obstacle with rigid core

In this paper, we discuss the inverse scattering problem for a penetrable obstacle with an impenetrable rigid core. Using a generalization of Schiffer's method to nonsmooth domains due to Ramm, we prove that the rigid core is uniquely determined by the far field patterns for a range of interior wavenumbers.     

The problem of recovering the permittivity coefficient from the modulus of the scattered electromagnetic field

Under consideration is the stationary system of equations of electrodynamics relating to a nonmagnetic nonconducting medium. We study the problem of recovering the permittivity coefficient ε from given vectors of electric or magnetic intensities of the electromagnetic field. It is assumed that the field is generated by a point impulsive dipole located at some point y. It is also assumed that the permittivity differs from a given constant ε₀ only inside some compact domain Ω ? R³ with smooth boundary S. To recover ε inside Ω, we use the information on a solution to the corresponding direct problem for the system of equations of electrodynamics on the whole boundary of Ω for all frequencies from some fixed frequency ω ₀ on and for all y ∈ S. The asymptotics of a solution to the direct problem for large frequencies is studied and it is demonstrated that this information allows us to reduce the initial problem to the well-known inverse kinematic problem of recovering the refraction index inside Ω with given travel times of electromagnetic waves between two arbitrary points on the boundary of Ω. This allows us to state uniqueness theorem for solutions to the problem in question and opens up a way of its constructive solution.     

Estimates for the low‐frequency electromagnetic fields scattered by two adjacent metal spheres in a lossless medium

Inductive electromagnetic means, currently employed in real physical applications and dealing with voluminous bodies embedded in lossless media, often call for analytically demanding tools of field calculation at modeling stage and later on at numerical stage. Here, one is considering two closely adjacent perfect conductors, possibly almost touching one another, for which the 3D bispherical geometry provides a good approximation. The particular scattering problem is modeled with respect to the two solid impenetrable metallic spheres, which are excited by a time‐harmonic magnetic dipole, arbitrarily orientated in the 3D space. The incident, the scattered, and the total non‐axisymmetric electromagnetic fields yield rigorous low‐frequency expansions in terms of positive integral powers of the real‐valued wave number in the exterior medium. We keep the most significant terms of the low‐frequency regime, that is, the static Rayleigh approximation and the first three dynamic terms, while the additional terms are small contributors and they are neglected. The typical Maxwell‐type problem is transformed into intertwined either Laplace's or Poisson's potential‐type boundary value problem with impenetrable boundary conditions. In particular, the fields are represented via 3D infinite series expansions in terms of bispherical eigenfunctions, obtaining analytical closed‐form solutions in a compact fashion. This procedure leads to infinite linear systems, which can be solved approximately within any order of accuracy through a cutoff technique.     

The Noise Prediction Model SATIN

This paper presents the noise prediction model SATIN (Statistical Approach to Turbulence Induced Noise) which is based on Lighthill's acoustic analogy. It allows to predict both, the far‐field noise radiation as well as near‐field wall‐pressure fluctuations. Far‐field noise radiation may result from the scattering of wall‐pressure fluctuations at geometrical discontinuities and is therefore important for many practical problems. Within this paper, we focus on the calculation of far‐field noise radiation. The required input values of SATIN are local properties of turbulence, namely the turbulent kinetic energy and the integral length scale which can be obtained by steady solutions of the Reynolds‐averaged Navier‐Stokes equations with a two equation turbulence model. It is assumed that the turbulence is axisymmetric and homogenous, which is taken into account by introducing two anisotropy parameters. The validation of SATIN is done for trailing‐edge noise originating from a thin flat plate using measurements of a phased array. As expected, the anisotropic formulation of SATIN improves the prediction quality considerably compared to isotropic turbulence.     

On entropy stable schemes for degenerate parabolic multispecies kinematic flow models

Entropy stable schemes for the numerical solution of initial value problems of nonlinear, possibly strongly degenerate systems of convection–diffusion equations were recently proposed in Jerez and Parés's study. These schemes extend the theoretical framework of Tadmor's study to convection–diffusion systems. They arise from entropy conservative schemes by adding a small amount of viscosity to avoid spurious oscillations. The main condition for feasibility of entropy conservative or stable schemes for a given model is that the corresponding first‐order system of conservation laws possesses a convex entropy function and corresponding entropy flux, and that the diffusion matrix multiplied by the inverse of the Hessian of the entropy is positive semidefinite. As a new contribution, it is demonstrated in the present work, first, that these schemes can naturally be extended to initial‐boundary value problems with zero‐flux boundary conditions in one space dimension, including an explicit bound on the growth of the total entropy. Second, it is shown that these assumptions are satisfied by certain diffusively corrected multiclass kinematic flow models of arbitrary size that describe traffic flow or the settling of dispersions and emulsions, where the latter application gives rise to zero‐flux boundary conditions. Numerical examples illustrate the behavior and accuracy of entropy stable schemes for these applications.     

Sharp estimate of electric field from a conductive rod and application

We are concerned with the quantitative study of the electric field perturbation due to the presence of an inhomogeneous conductive rod embedded in a homogenous conductivity. We sharply quantify the dependence of the perturbed electric field on the geometry of the conductive rod. In particular, we accurately characterize the localization of the gradient field (i.e., the electric current) near the boundary of the rod where the curvature is sufficiently large. We develop layer‐potential techniques in deriving the quantitative estimates and the major difficulty comes from the anisotropic geometry of the rod. The result complements and sharpens several existing studies in the literature. It also generates an interesting application in EIT (electrical impedance tomography) in determining the conductive rod by a single measurement, which is also known as the Calderón's inverse inclusion problem in the literature.     

An alternative approach to the image reconstruction for parallel data acquisition in MRI

Magnetic resonance imaging with parallel data acquisition requires algorithms for reconstructing the patient's image from a small number of measured k‐space lines. In contrast to well‐known algorithms like SENSE and GRAPPA and its flavours we consider the problem as a non‐linear inverse problem. Fast computation algorithms for the necessary Fréchet derivative and reconstruction algorithms are given. Copyright © 2007 John Wiley & Sons, Ltd.     

Analysis of Direct and Inverse Cavity Scattering Problems

Consider the scattering of a time-harmonic electromagnetic plane wave by an arbitrarily shaped and filled cavity embedded in a perfect electrically conducting infinite ground plane.A method of symmetric coupling of finite element and boundary integral equations is presented for the solutions of electromagnetic scattering in both transverse electric and magnetic polarization cases.Given the incident field,the direct problem is to determine the field distribution from the known shape of the cavity; while the inverse problem is to determine the shape of the cavity from the measurement of the field on an artificial boundary enclosing the cavity.In this paper,both the direct and inverse scattering problems are discussed based on a symmetric coupling method.Variational formulations for the direct scattering problem are presented,existence and uniqueness of weak solutions are studied,and the domain derivatives of the field with respect to the cavity shape are derived.Uniqueness and local stability results are established in terms of the inverse problem.     

High‐performance direct algorithms for computing the sign function of triangular matrices

Algorithms and implementations for computing the sign function of a triangular matrix are fundamental building blocks for computing the sign of arbitrary square real or complex matrices. We present novel recursive and cache‐efficient algorithms that are based on Higham's stabilized specialization of Parlett's substitution algorithm for computing the sign of a triangular matrix. We show that the new recursive algorithms are asymptotically optimal in terms of the number of cache misses that they generate. One algorithm that we present performs more arithmetic than the nonrecursive version, but this allows it to benefit from calling highly optimized matrix multiplication routines; the other performs the same number of operations as the nonrecursive version, suing custom computational kernels instead. We present implementations of both, as well as a cache‐efficient implementation of a block version of Parlett's algorithm. Our experiments demonstrate that the blocked and recursive versions are much faster than the previous algorithms and that the inertia strongly influences their relative performance, as predicted by our analysis.     

Use of specific Green's functions for solving direct problems involving a heterogeneous rigid frame porous medium slab solicited by acoustic waves

A domain integral method employing a specific Green's function (i.e. incorporating some features of the global problem of wave propagation in an inhomogeneous medium) is developed for solving direct and inverse scattering problems relative to slab‐like macroscopically inhomogeneous porous obstacles. It is shown how to numerically solve such problems, involving both spatially‐varying density and compressibility, by means of an iterative scheme initialized with a Born approximation. A numerical solution is obtained for a canonical problem involving a two‐layer slab. Copyright © 2006 John Wiley & Sons, Ltd.