This paper is concerned with an inverse random source problem for the three-dimensional time-harmonic Maxwell equations. The source is assumed to be a centered complex-valued Gaussian vector field with correlated components, and its covariance operator is a pseudo-differential operator. The well-posedness of the direct source scattering problem is established and the regularity of the electromagnetic field is given. For the inverse source scattering problem, the micro-correlation strength matrix of the covariance operator is shown to be uniquely determined by the high frequency limit of the expectation of the electric field measured in an open bounded domain disjoint with the support of the source. In particular, we show that the diagonal entries of the strength matrix can be uniquely determined by only using the amplitude of the electric field. Moreover, this result is extended to the almost surely sense by deducing an ergodic relation for the electric field over the frequencies.
We prove a uniqueness theorem for an inverse boundary value problem for the Maxwell system with boundary data assumed known only in part of the bound- ary. We assume that the inaccessible part of the boundary is either part of a plane, or part of a sphere. This work generalizes the results obtained by Isakov for the Schrödinger equation to Maxwell equations.
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.
In this article we consider an inverse boundary value problem for the time-harmonic Maxwell equations. We show that the electromagnetic material parameters are determined by boundary measurements where part of the boundary data is measured on a possibly very small set. This is an extension of earlier scalar results of Bukhgeim-Uhlmann and Kenig-Sjöstrand-Uhlmann to the Maxwell system. The main contribution is to show that the Carleman estimate approach to scalar partial data inverse problems introduced in those works can be carried over to the Maxwell system.
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.
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.
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.
This work is concerned with an inverse electromagnetic scattering problem in two dimensions. We prove that in the TE polarization case, the knowledge of the electric far-field pattern incited by a single incoming wave is sufficient to uniquely determine the shape of a penetrable scatterer of rectangular type. As a by-product, the uniqueness is also confirmed to inverse transmission problems modelled by scalar Helmholtz equations with discontinuous normal derivatives at the scattering interface. Keywords: Uniqueness, inverse medium scattering, Maxwell equations, one incoming wave, shape identif…
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.
We consider Calderón's inverse boundary value problems for a class of nonlinear Helmholtz Schrödinger equations and Maxwell's equations in a bounded domain in $\R^n$. The main method is the higher-order linearization of the Dirichlet-to-Neumann map of the corresponding equations. The local uniqueness of the linearized partial data Calderón's inverse problem is obtained following \cite{DKSU}. The Runge approximation properties and unique continuation principle allow us to extend to global situations. Simultaneous recovery of some unknown cavity$/$boundary and coefficients are given as some appl…
This paper concerns the quantitative step of the medical imaging modality Thermo-acoustic Tomography (TAT). We model the radiation propagation by a system of Maxwell's equations. We show that the index of refraction of light and the absorption coefficient (conductivity) can be uniquely and stably reconstructed from a sufficiently large number of TAT measurements. Our method is based on verifying that the linearization of the inverse problem forms a redundant elliptic system of equations. We also observe that the reconstructions are qualitatively quite different from the setting where radiation…
Pride (1994, Phys. Rev. B 50 15678-96) derived the governing model of electroseismic conversion, in which Maxwell's equations are coupled with Biot's equations through an electrokinetic mobility parameter. The inverse problem of electroseismic conversion was first studied by Chen and Yang (2013, Inverse Problem 29 115006). By following the construction of Complex Geometrical Optics (CGO) solutions to a matrix \Schrodinger equation introduced by Ola and Somersalo (1996, SIAM J. Appl. Math. 56 No. 4 1129-1145), we analyze the reconstruction of conductivity, permittivity and the electrokinetic mo…
The article deals with electrodynamics in the presence of anisotropic materials having scalar wave impedance. Maxwell's equations written for differential forms over a 3-manifold are analysed. The system is extended to a Dirac type first order elliptic system on the Grassmannian bundle over the manifold. The second part of the article deals with the dynamical inverse boundary value problem of determining the electromagnetic material parameters from boundary measurements. By using the boundary control method, it is proved that the dynamical boundary data determines the electromagnetic travel ti…
This paper considers a class of nonlinear time harmonic Maxwell systems at fixed frequency, with nonlinear terms taking the form $\mathscr{X}(x,|\vec E(x)|^2)\vec E(x)$, $\mathscr{Y}(x,|\vec H(x)|^2)\vec H(x)$, such that $\mathscr{X}(x,s)$, $\mathscr{Y}(x,s)$ are both real analytic in $s$. Such nonlinear terms appear in nonlinear optics theoretical models. Under certain regularity conditions, it can be shown that boundary measurements of tangent components of the electric and magnetic fields determine the electric permittivity and magnetic permeability functions as well as the form of the nonl…
We consider a linearised inverse conductivity problem for electromagnetic waves in a three dimensional bounded domain at a high time-harmonic frequency. Increasing stability bounds for the conductivity coefficient in the full Maxwell system and in a simplified transverse electric mode are derived. These bounds contain a Lipschitz term with a factor growing polynomially in terms of the frequency, a Holder term, and a logarithmic term which decays with respect to the frequency as a power. To validate this increasing stability numerically, we propose a reconstruction algorithm aiming at the recov…
We consider the time-harmonic Maxwell system in a domain with a generalized impedance edge-corner, namely the presence of two generalized impedance planes that intersect at an edge. The impedance parameter can be $0, \infty$ or a finite non-identically vanishing variable function. We establish an accurate relationship between the vanishing order of the solutions to the Maxwell system and the dihedral angle of the edge-corner. In particular, if the angle is irrational, the vanishing order is infinity, i.e. strong unique continuation holds from the edge-corner. The establishment of those new qua…
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…
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 study Maxwell's equations in time domain in an anisotropic medium. The goal of the paper is to solve an inverse boundary value problem for anisotropies characterized by scalar impedance $α$. This means that the material is conformal, i.e., the electric permittivity $ε$ and magnetic permeability $μ$ are tensors satisfying $μ=α^2ε$. This condition is equivalent to a single propagation speed of waves with different polarizations which uniquely defines an underlying Riemannian structure. The analysis is based on an invariant formulation of the system of electrodynamics as a Dirac type first ord…
We prove that the electromagnetic material parameters are uniquely determined by boundary measurements for the time-harmonic Maxwell equations in certain anisotropic settings. We give a uniqueness result in the inverse problem for Maxwell equations on an admissible Riemannian manifold, and a uniqueness result for Maxwell equations in Euclidean space with admissible matrix coefficients. The proofs are based on a new Fourier analytic construction of complex geometrical optics solutions on admissible manifolds, and involve a proper notion of uniqueness for such solutions.
When a porous rock is saturated with an electrolyte, electrical fields are coupled with seismic waves via the electro-seismic conversion. Pride derived the governing models, in which Maxwell equations are coupled with Biot equations through the electro-kinetic mobility parameter. The inverse problem of the linearized electro-seismic conversion consists in two step, namely the inversion of Biot equations and the inversion of Maxwell equations. We analyze the reconstruction of conductivity and electro-kinetic mobility parameter in Maxwell equations with internal measurements, while the internal …
We consider for the full time-dependent Maxwell's equations the inverse problem of identifying locations and certain properties of small electromagnetic inhomogeneities in a homogeneous background medium from dynamic boundary measurements on the boundary for a finite time interval.
We study the homogenization of elliptic systems of equations in divergence form where the coefficients are compositions of periodic functions with a random diffeomorphism with stationary gradient. This is done in the spirit of scalar stochastic homogenization by Blanc, Le Bris and P.-L. Lions. An application of the abstract result is given for Maxwell's equations in random dissipative bianisotropic media.
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…
We study inverse problems for the Einstein-Maxwell equations. We prove that it is possible to generate gravitational waves from the nonlinear interactions of electromagnetic waves. By sending electromagnetic waves from a neighborhood of a freely falling observer and taking measurements of the gravitational perturbations in the same neighborhood, one can determine the vacuum space-time structure up to diffeomorphisms in the largest region where these waves can travel to from the observer and return.
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…
Inverse scattering problem is discussed for the Maxwell's equations. A reduction of the Maxwell's system to a new Fredholm second-kind integral equation with a {\it scalar weakly singular kernel} is given for electromagnetic (EM) wave scattering. This equation allows one to derive a formula for the scattering amplitude in which only a scalar function is present. If this function is small (an assumption that validates a Born-type approximation), then formulas for the solution to the inverse problem are obtained from the scattering data: the complex permittivity $\ep'(x)$ in a bounded region $D\…
The aim of this manuscript is to study the influence of the vorticity on the existence time in fluid systems for which global smoothness and decay is known in the case of small irrotational data. We focus on two examples: the Euler-Korteweg system and the two-fluid Euler Maxwell system. We prove that the lower bound of the lifespan of these systems is no less than the inverse of the $H^s$ $(s>5/2)$ norm of the rotational part of the initial velocity. Our approach is based on energy estimates and the fast time decay results of global solutions to these systems with small irrotational initial da…
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…
We obtain explicit estimates on the stability of the unique continuation for a linear system of hyperbolic equations. In particular our result applies to the elasticity system and also the Maxwell system. As an application, we study the kinematic inverse rupture problem of determining the jump in displacement and the friction force at the rupture surface, and we obtain new features on the stable unique continuation up to the rupture surface.
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…
A uniqueness result for the recovery of the electric and magnetic coefficients in the time-harmonic Maxwell equations from local boundary measurements is proven. No special geometrical condition is imposed on the inaccessible part of the boundary of the domain, apart from imposing that the boundary of the domain is $C^{1,1}$. The coefficients are assumed to coincide on a neighbourhood of the boundary, a natural property in applications.
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.
We shall give conditions on the illuminations $φ_{i}$ such that the solutions to Maxwell's equations \[ \left\{ \begin{array}{l} {\rm curl} E^{i}=iωμH^{i}\qquad\text{in }Ω,\\ {\rm curl} H^{i}=-i(ω\varepsilon+iσ)E^{i}\qquad\text{in }Ω,\\ E^{i}\timesν=φ_{i}\timesν\qquad\text{on }\partialΩ, \end{array}\right. \] satisfy certain non-zero qualitative properties inside the domain $Ω$, provided that a finite number of frequencies $ω$ are chosen in a fixed range. The illuminations are explicitly constructed. This theory finds applications in several hybrid imaging problems, where unknown parameters ha…
In this paper we prove uniqueness for an inverse boundary value problem (IBVP) arising in electrodynamics. We assume that the electromagnetic properties of the medium, namely the magnetic permeability, the electric permittivity and the conductivity, are described by continuously differentiable functions.
In this paper, we are concerned with the inverse electromagnetic scattering problem of recovering a complex scatterer by the corresponding electric far-field data. The complex scatterer consists of an inhomogeneous medium and a possibly embedded perfectly electric conducting (PEC) obstacle. The far-field data are collected corresponding to incident plane waves with a fixed incident direction and a fixed polarisation, but frequencies from an open interval. It is shown that the embedded obstacle can be uniquely recovered by the aforementioned far-field data, independent of the surrounding medium…
In this paper we prove a stable determination of the coefficients of the time-harmonic Maxwell equations from local boundary data. The argument --due to Isakov-- requires some restrictions on the domain.
We introduce and discuss spatially homogeneous Maxwell-type models of the nonlinear Boltzmann equation undergoing binary collisions with a random component. The random contribution to collisions is such that the usual collisional invariants of mass, momentum and energy do not hold pointwise, even if they all hold in the mean. Under this assumption it is shown that, while the Boltzmann equation has the usual conserved quantities, it possesses a steady state with power-like tails for certain random variables. A similar situation occurs in kinetic models of economy recently considered by two of t…
Consider the one-dimensional stochastic Helmholtz equation where the source is assumed to be driven by the white noise. This paper concerns the stability analysis of the inverse random source problem which is to reconstruct the statistical properties of the source such as the mean and variance. Our results show that increasing stability can be obtained for the inverse problem by using suitable boundary data with multi-frequencies.
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 the direct and inverse random source scattering problems for elastic waves where the source is assumed to be driven by an additive white noise. Given the source, the direct problem is to determine the displacement of the random wave field. The inverse problem is to reconstruct the mean and variance of the random source from the boundary measurement of the wave field at multiple frequencies. The direct problem is shown to have a unique mild solution by using a constructive proof. Based on the explicit mild solution, Fredholm integral equations of the first kind are …
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.
The aim of this chapter is to make a review of the recent results using the Enclosure Method on inverse obstacle problems governed by the wave equation and the Maxwell system in time domain. We also describe some of unsolved problems related to further possibility of the Enclosure Method itself.
In this paper we provide a mathematical model for imaging an anisotropic, orthotropic medium with Polarization-Sensitive Optical Coherence Tomography (PS-OCT). The imaging problem is formulated as an inverse scattering problem in three dimensions for reconstructing the electrical susceptibility of the medium using Maxwell's equations. Our reconstruction method is based on the second-order Born-approximation of the electric field.
In this paper we introduce the randomised stability constant for abstract inverse problems, as a generalisation of the randomised observability constant, which was studied in the context of observability inequalities for the linear wave equation. We study the main properties of the randomised stability constant and discuss the implications for the practical inversion, which are not straightforward.
In recent decades qualitative inverse scattering methods with eigenvalues as target signatures received much attention. To understand those methods a knowledge on the properties of the related eigenvalue problems is essential. However, even the existence of eigenvalues for such (nonselfadjoint) problems is a challenging question and existing results for absorbing media are usually established under unrealistic assumptions or a smoothing of the eigenvalue problem. We present a technique to prove the existence of infinitely many eigenvalues for such problems under realistic assumptions. In parti…
We address the following inverse problem in quantum statistical physics: does the quantum free energy (von Neumann entropy + kinetic energy) admit a unique minimizer among the density operators having a given local density $n(x)$? We give a positive answer to that question, in dimension one. This enables to define rigourously the notion of local quantum equilibrium, or quantum Maxwellian, which is at the basis of recently derived quantum hydrodynamic models and quantum drift-diffusion models. We also characterize this unique minimizer, which takes the form of a global thermodynamic equilibrium…
