This paper is concerned with the electromagnetic inverse scattering problem that aims to determine the location and shape of anisotropic scatterers from far field data (at a fixed frequency). We study the orthogonality sampling method which is a simple, fast and robust imaging method for solving the electromagnetic inverse shape problem. We first provide a theoretical foundation for the sampling method and a resolution analysis of its imaging functional. We then establish an equivalent relation between the orthogonality sampling method and direct sampling method as well as resolution analysis for the latter. The analysis used to justify the Factorization method for the far field operator plays an important role in the justifications. Finally, we present some numerical examples to validate the performance of the sampling methods for anisotropic scatterers in three dimensions.
We propose a method to reconstruct the electrical current density from acoustically-modulated boundary measurements of time-harmonic electromagnetic fields. We show that the current can be uniquely reconstructed with Lipschitz stability. We also report numerical simulations to illustrate the analytical results.
We study an inverse uniqueness with a knowledge of spectral data in the interior transmission problem defined by an index of refraction in a simple domain. We expand the solution in such a domain into a series of one dimensional problems. For each one dimensional problem, we apply a value distribution theory in complex analysis to describe the eigenvalues of the system. By the orthogonality of the one dimensional system, we consider the uniqueness on the perturbation along each given incident angle.
The inverse electromagnetic scattering problem for anisotropic media in general does not have a unique solution. A possible approach to this problem is through the use of appropriate "target signatures," i.e. eigenvalues associated with the direct scattering problem that are accessible to measurement from a knowledge of the scattering data. In this paper we shall consider three different sets of eigenvalues that can be used as target signatures: 1) eigenvalues of the electric far field operator, 2) transmission eigenvalues and 3) Stekloff eigenvalues.
In this paper, we study time-harmonic electromagnetic scattering in two scenarios, where the anomalous scatterer is either a pair of electromagnetic sources or an inhomogeneous medium, both with compact supports. We are mainly concerned with the geometrical inverse scattering problem of recovering the support of the scatterer, independent of its physical contents, by a single far-field measurement. It is assumed that the support of the scatterer (locally) possesses a conical singularity. We establish a local characterisation of the scatterer when invisibility/transparency occurs, showing that …
The electroseismic model describes the coupling phenomenon of the electromagnetic waves and seismic waves in fluid immersed porous rock. Electric parameters have better contrast than elastic parameters while seismic waves provide better resolution because of the short wavelength. The combination of theses two different waves is prominent in oil exploration. Under some assumptions on the physical parameters, we derived a Hölder stability estimate to the inverse problem of recovery of the electric parameters and the coupling coefficient from the knowledge of the fields in a small open domain nea…
Consider the problem of scattering of electromagnetic waves by a doubly periodic structure. The medium above the structure is assumed to be inhomogeneous characterized completely by an index of refraction. Below the structure is a perfect conductor or an imperfect conductor partially coated with a dielectric. Having established the well-posedness of the direct problem by the variational approach, we prove the uniqueness of the inverse problem, that is, the unique determination of the doubly periodic grating with its physical property and the index of refraction from a knowledge of the scattere…
In this paper, we consider inverse time-harmonic acoustic and electromagnetic scattering from locally perturbed rough surfaces in three dimensions. The scattering interface is supposed to be the graph of a Lipschitz continuous function with compact support. It is proved that an acoustically sound-soft or sound-hard surface can be uniquely determined by the far-field pattern of infinite number of incident plane waves with distinct directions. Moreover, a single point source or plane wave can be used to uniquely determine a scattering surface of polyhedral type. These uniqueness results apply to…
This paper is concerned with uniqueness for reconstructing a periodic inhomogeneous medium covered on a perfectly conducting plate. We deal with the problem in the frame of time-harmonic Maxwell systems without TE or TM polarization. An orthogonal relation for two refractive indices is obtained, and then inspired by Kirsch's idea, the refractive index can be identified by utilizing the eigenvalues and eigenfunctions of a quasi-periodic Sturm-Liouville eigenvalue problem.
This paper is concerned with uniqueness in inverse electromagnetic scattering with phaseless far-field pattern at a fixed frequency. In our previous work [{\em SIAM J. Appl. Math.} {\bf 78} (2018), 3024-3039], by adding a known reference ball into the acoustic scattering system, it was proved that the impenetrable obstacle and the index of refraction of an inhomogeneous medium can be uniquely determined by the acoustic phaseless far-field patterns generated by infinitely many sets of superpositions of two plane waves with different directions at a fixed frequency. In this paper, we extend thes…
This paper concerns the inverse source problems for the time-harmonic elastic and electromagnetic wave equations. The goal is to determine the external force and the electric current density from boundary measurements of the radiated wave field, respectively. The problems are challenging due to the ill-posedness and complex model systems. Uniqueness and stability are established for both of the inverse source problems. Based on either continuous or discrete multi-frequency data, a unified increasing stability theory is developed. The stability estimates consist of two parts: the Lipschitz type…
In this paper we study the uniqueness and the increasing stability in the inverse source problem for electromagnetic waves in homogeneous and inhomogeneous media from boundary data at multiple wave numbers. For the unique determination of sources, we consider inhomogeneous media and use tangential components of the electric field and magnetic field at the boundary of the reference domain. The proof relies on the Fourier transform with respect to the wave numbers and the unique continuation theorems. To study the increasing stability in the source identification, we consider homogeneous media a…
This paper considers the inverse problem of scattering of time-harmonic acoustic and electromagnetic plane waves by a bounded, inhomogeneous, penetrable obstacle with embedded objects inside. A new method is proposed to prove that the inhomogeneous penetrable obstacle can be uniquely determined from the far-field pattern at a fixed frequency, disregarding its contents. Our method is based on constructing a well-posed interior transmission problem in a small domain associated with the Helmholtz or modified Helmholtz equation and the Maxwell or modified Maxwell equations. A key role is played by…
In the current paper we consider an inverse boundary value problem of electromagnetism with nonlinear Second Harmonic Generation (SHG) process. We show the unique determination of the electromagnetic material parameters and the SHG susceptibility parameter of the medium by making electromagnetic measurements on the boundary. We are interested in the case when a frequency is fixed.
A recent problem of interest in inverse problems has been the study of eigenvalue problems arising from scattering theory and their potential use as target signatures in nondestructive testing of materials. Towards this pursuit we introduce a new eigenvalue problem related to Maxwell's equations that is generated from a comparison of measured scattering data to that of a non-standard auxiliary scattering problem. This choice of auxiliary problem permits the application of regularity results for Maxwell's equations in order to show that a related interior transmission problem possesses the Fred…
This paper is concerned with the problem of scattering of time-harmonic electromagnetic waves from an impenetrable obstacle in a piecewise homogeneous medium. The well-posedness of the direct problem is established, employing the integral equation method. Inspired by a novel idea developed by Hahner [11], we prove that the penetrable interface between layers can be uniquely determined from a knowledge of the electric far field pattern for incident plane waves. Then, using the idea developed by Liu and Zhang [21], a new mixed reciprocity relation is obtained and used to show that the impenetrab…
This paper is dedicated to design a direct sampling method of inverse electromagnetic scattering problems, which uses multi-frequency sparse backscattering far field data for reconstructing the boundary of perfectly conducting obstacles. We show that a smallest strip containing the unknown object can be approximately determined by the multi-frequency backscattering far field data at two opposite observation directions. The proof is based on the Kirchhoff approximation and Fourier transform. Such a strip is then reconstructed by an indicator, which is the absolute value of an integral of the pr…
In the current paper we consider an inverse boundary value problem of electromagnetism in a nonlinear Kerr medium. We show the unique determination of the electromagnetic material parameters and the nonlinear susceptibility parameters of the medium by making electromagnetic measurements on the boundary. We are interested in the case of the time-harmonic Maxwell equations.
This paper is concerned with the scattering problem for time-harmonic electromagnetic waves, due to the presence of scatterers and of inhomogeneities in the medium. We prove a sharp stability result for the solutions to the direct electromagnetic scattering problem, with respect to variations of the scatterer and of the inhomogeneity, under minimal regularity assumptions for both of them. The stability result leads to uniform bounds on solutions to the scattering problems for an extremely general class of admissible scatterers and inhomogeneities. The uniform bounds are a key step to tackle th…
We consider the inverse problem of determining the time and space dependent electromagnetic potential of the Schrödinger equation in a bounded domain of $\mathbb R^n$, $n\geq 2$, by boundary observation of the solution over the entire time span. Assuming that the divergence of the magnetic potential is fixed, we prove that the electric potential and the magnetic potential can be Hölder stably retrieved from these data, whereas stability estimates for inverse time-dependent coefficients problems of evolution partial differential equations are usually of logarithmic type.
The inverse problem of determining the right-hand side of the subdiffusion equation with the fractional Caputo derivative is considered. The right-hand side of the equation has the form $f(x)g(t)$ and the unknown is function $f(x)$. The condition $ u (x,t_0)= ψ(x) $ is taken as the over-determination condition, where $t_0$ is some interior point of the considering domain and $ψ(x) $ is a given function. It is proved by the Fourier method that under certain conditions on the functions $g(t)$ and $ψ(x) $ the solution of the inverse problem exists and is unique. An example is given showing the vi…
In this paper we consider the inverse electromagnetic scattering for a cavity surrounded by an inhomogeneous medium in three dimensions. The measurements are scattered wave fields measured on some surface inside the cavity, where such scattered wave fields are due to sources emitted on the same surface. We first prove that the measurements uniquely determine the shape of the cavity, where we make use of a boundary value problem called the exterior transmission problem. We then complete the inverse scattering problem by designing the linear sampling method to reconstruct the cavity. Numerical e…
This paper is concerned with analysis of electromagnetic wave scattering by an obstacle which is embedded in a two-layered lossy medium separated by an unbounded rough surface. Given a dipole point source, the direct problem is to determine the electromagnetic wave field for the given obstacle and unbounded rough surface; the inverse problem is to reconstruct simultaneously the obstacle and unbounded rough surface from the electromagnetic field measured on a plane surface above the obstacle. For the direct problem, a new boundary integral equation is proposed and its well-posedness is establis…
We study boundary determination for an inverse problem associated to the time-harmonic Maxwell equations and another associated to the isotropic elasticity system. We identify the electromagnetic parameters and the Lamé moduli for these two systems from the corresponding boundary measurements. In a first step we reconstruct Lipschitz magnetic permeability, electric permittivity and conductivity on the surface from the ideal boundary measurements. Then, we study inverse problems for Maxwell equations and the isotropic elasticity system assuming that the data contains measurement errors. For bot…
We consider the inverse problem of determining an unknown vectorial source current distribution associated with the homogeneous Maxwell system. We propose a novel non-iterative reconstruction method for solving the aforementioned inverse problem from far-field measurements. The method is based on recovering the Fourier coefficients of the unknown source. A key ingredient of the method is to establish the relationship between the Fourier coefficients and the multi-frequency far-field data. Uniqueness and stability results are established for the proposed reconstruction method. Numerical experim…
The purpose of this licentiate thesis is to present Bukhgeim's result of 2007, which solves the inverse boundary value problem of the Schrödinger equation in the plane. The thesis is mainly based on Bukhgeim's paper and Kari Astala's seminar talk, which he gave the 11th and 18th September of 2008 at the University of Helsinki. Section 3 is devoted to the history and past results concerning some related problems: notably the inverse problem of the Schrödinger and conductivity equations in different settings. We also describe why some of the past methods do not work in the general case in a plan…
This paper is concerned with uniqueness, phase retrieval and shape reconstruction methods for solving inverse electromagnetic source scattering problems with multi-frequency sparse phased or phaseless far field data. With the phased data, we show that the smallest strip containing the source support with the observation direction as the normal can be uniquely determined by the multi-frequency far field pattern at a single observation direction. A phased direct sampling method is also proposed to reconstruct the strip. The phaseless far field data is closely related to the outward energy flux, …
It is proved that a convex polyhedral scatterer of impedance type can be uniquely determined by the electric far-field pattern of a non-vanishing incident field. The incoming wave is allowed to bean electromagnetic plane wave, a vector Herglotz wave function or a point source wave incited by some magnetic dipole. Our proof relies on the reflection principle for Maxwell's equations with the impedance (or Leontovich) boundary condition enforcing on a hyper-plane. We prove that it is impossible to analytically extend the total field across any vertex of the scatterer. This leads to a data-driven …
In this paper, a time domain enclosure method for an inverse obstacle scattering problem of electromagnetic wave is introduced. The wave as a solution of Maxwell's equations is generated by an applied volumetric current having an {\it orientation} and supported outside an unknown obstacle and observed on the same support over a finite time interval. It is assumed that the obstacle is a perfect conductor. Two types of analytical formulae which employ a {\it single} observed wave and explicitly contain information about the geometry of the obstacle are given. In particular, an effect of the orie…
We consider an inverse boundary value problem for Maxwell's equations, which aims to recover the electromagnetic material properties of a body from measurements on the boundary. We show that a Lipschitz continuous conductivity, electric permittivity, and magnetic permeability are uniquely determined by knowledge of all tangential electric and magnetic fields on the boundary of the body at a fixed frequency.
We consider the inverse problem of recovering both an unknown electric current and the surrounding electromagnetic parameters of a medium from boundary measurements. This inverse problem arises in brain imaging. We show that under generic conditions one can recover both the source and the electromagnetic parameters if these are piecewise constant and the current source is invariant in a fixed direction or a harmonic function.
We consider the inverse problem of determining the isotropic inhomogeneous electromagnetic coefficients of the non-stationary Maxwell equations in a bounded domain of R^3, from a finite number of boundary measurements. Our main result is a Hölder stability estimate for the inverse problem, where the measurements are exerted only in some boundary components. For it, we prove a global Carleman estimate for the heterogeneous Maxwell's system with boundary conditions.
We consider two inverse boundary value problems for the time-harmonic Maxwell equations in an infinite slab. Assuming that tangential boundary data for the electric and magnetic fields at a fixed frequency is available either on subsets of one boundary hyperplane, or on subsets of different boundary hyperplanes, we show that the electromagnetic material parameters, the conductivity, electric permittivity, and magnetic permeability, are uniquely determined by these partial measurements.
Solving the multi-frequency electromagnetic inverse source problem by the Fourier method
School of Mathematics, Jilin University（吉林大学数学学院）；Department of Mathematics, Harbin Institute of Technology（哈尔滨工业大学数学系）；Department of Mathematics, Southern University of Science and Technology（南方科技大学数学系）
This paper is concerned with uniqueness results in inverse acoustic and electromagnetic scattering problems with phaseless total-field data at a fixed frequency. Motivated by our previous work ({\em SIAM J. Appl. Math. \bf78} (2018), 1737-1753), where uniqueness results were proved for inverse acoustic scattering with phaseless far-field data generated by superpositions of two plane waves as the incident waves at a fixed frequency, in this paper, we use superpositions of two point sources as the incident fields at a fixed frequency and measure the modulus of the acoustic total-field (called ph…
This article introduces the Zeffiro interface (ZI) version 2.2 for brain imaging. ZI aims to provide a simple, accessible and multimodal open source platform for finite element method (FEM) based and graphics processing unit (GPU) accelerated forward and inverse computations in the Matlab environment. It allows one to (1) generate a given multi-compartment head model, (2) to evaluate a lead field matrix as well as (3) to invert and analyze a given set of measurements. GPU acceleration is applied in each of the processing stages (1)-(3). In its current configuration, ZI includes forward solvers…
This article is concerned with the time-harmonic electromagnetic (EM) scattering from a generic inhomogeneous medium. It is shown that if there is a right corner on the support of the medium, then it scatters every pair of incident EM fields, excluding a possible class of EM fields which are of very particular forms. That is, for every pair of admissible incident EM fields, the corresponding scattered wave fields associated to the medium scatterer cannot be identically vanishing outside the support of the medium. Indeed, we achieve the corner scattering result by establishing a stronger result…
We consider the dynamical inverse problem for the Maxwell system on a Riemannian 3-manifold with boundary in a time-optimal set-up. Using BC-method we show that the data of the inverse problem (electromagnetic measurements on the boundary) determine a $C$*-algebra, which has a spectrum homeomorphic to a part of the manifold. This part depends on the duration of measurements.
This paper is concerned with the inverse electromagnetic scattering problem for anisotropic media. We use the interior resonant modes to develop an inverse scattering scheme for imaging the scatterer. The whole procedure consists of three phases. First, we determine the interior Maxwell transmission eigenvalues of the scatterer from a family of far-field data by the mechanism of the linear sampling method. Next, we determine the corresponding transmission eigenfunctions by solving a constrained optimization problem. Finally, based on both global and local geometric properties of the transmissi…
