In this paper, we consider the direct and inverse problem for time-fractional diffusion in a domain with an impenetrable subregion. Here we assume that on the boundary of the subregion the solution satisfies a generalized impedance boundary condition. This boundary condition is given by a second order spacial differential operator imposed on the boundary. A generalized impedance boundary condition can be used to model corrosion and delimitation. The well-posedness for the direct problem is established where the Laplace Transform is used to study the time dependent boundary value problem. The inverse impedance problem of determining the parameters from the Cauchy data is also studied provided the boundary of the subregion is known. The uniqueness of recovering the boundary parameters from the Neumann to Dirichlet mapping is proven.
The paper considers an inverse source problem for a one-dimensional time-fractional heat equation with the generalized impedance boundary condition. The inverse problem is the time dependent source parameter identification together with the temperature distrubution from the energy measurement. The well-posedness of the inverse problem is shown by applying the Fourier expansion in terms of eigenfunctions of a spectral problem which has the spectral parameter also in the boundary condition and by using the results on Volterra type integral equation with the kernel may have a diagonal singularity…
We consider the problem of determining an unaccessible part of the boundary of a conductor by mean of thermal measurements. We study a problem of corrosion where a Robin type condition is prescribed on the damaged part and we prove logarithmic stability estimate.
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…
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…
We deal with the problem of determining the shape of an inclusion embedded in a homogenous background medium. The multifre-quency electrical impedance tomography is used to image the inclusion. For different frequencies, a current is injected at the boundary and the resulting potential is measured. It turns out that the potential solves an elliptic equation in divergence form with discontinuous leading coefficient. For this inverse problem we aim to establish a logarithmic type stability estimate. The key point in our analysis consists in reducing the original problem to that of determining an…
We consider the reconstruction of the diffusion and absorption coefficients of the diffusion equation from the internal information of the solution obtained from the first step of the inverse photoacoustic tomography (PAT). In practice, the internal information is only partially provided near the boundary due to the high absorption property of the medium and the limitation of the equipment. Our main contribution is to prove a Hölder stability of the inverse problem in a subregion where the internal information is reliably provided based on the stability estimation of a Cauchy problem satisfied…
We deal with the inverse scattering problem of determining the surface impedance of a partially coated obstacle. We prove a stability estimate of logarithmic type for the impedance term by the far field measurements.
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 consider an inverse problem arising in corrosion detection. We prove a stability result of logarithmic type for the determination of the corroded portion of the boundary and impedance by two measurements on the accessible portion of the boundary.
We study an inverse acoustic scattering problem in half-space with a probabilistic impedance boundary value condition. The Robin coefficient (surface impedance) is assumed to be a Gaussian random function $λ= λ(x)$ with a pseudodifferential operator describing the covariance. We measure the amplitude of the backscattered field averaged over the frequency band and assume that the data is generated by a single realization of $λ$. Our main result is to show that under certain conditions the principal symbol of the covariance operator of $λ$ is uniquely determined. Most importantly, no approximati…
We prove a stability estimate of logarithmic type for the inverse problem consisting in the determination of the surface impedance of an obstacle from the scattering amplitude. We present a simple and direct proof which is essentially based on an elliptic Carleman inequality.
This paper presents a mathematical framework for a flexible pressure-sensor model using electrical impedance tomography (EIT). When pressure is applied to a conductive membrane patch with clamped boundary, the pressure-induced surface deformation results in a change in the conductivity distribution. This change can be detected in the current-voltage data ({\it i.e.,} EIT data) measured on the boundary of the membrane patch. Hence, the corresponding inverse problem is to reconstruct the pressure distribution from the data. The 2D apparent conductivity (in terms of EIT data) corresponding to the…
In this paper, we give probabilistic interpretations of both, the forward and the inverse problem of electrical impedance tomography with possibly anisotropic, merely measurable conductivities: Using the theory of symmetric Dirichlet spaces, Feynman-Kac type formulae corresponding to different electrode models on bounded Lipschitz domains are derived. Moreover, we give a probabilistic interpretation of the Calderón inverse conductivity problem in terms of reflecting diffusion processes and their corresponding boundary trace processes.
The multifrequency electrical impedance tomography is considered in order to image a conductivity inclusion inside a homogeneous background medium by injecting one current. An original spectral decomposition of the solution of the forward conductivity problem is used to retrieve the Cauchy data corresponding to the extreme case of perfect conductor. Using results based on the unique continuation we then prove the uniqueness of multifrequency electrical impedance tomography and obtain rigorous stability estimates. Our results in this paper are quite surprising in inverse conductivity problem si…
The inverse problem of electrical impedance tomography is severely ill-posed. In particular, the resolution of images produced by impedance tomography deteriorates as the distance from the measurement boundary increases. Such depth dependence can be quantified by the concept of distinguishability of inclusions. This paper considers the distinguishability of perfectly conducting ball inclusions inside a unit ball domain, extending and improving known two-dimensional results to an arbitrary dimension $d \geq 2$ with the help of Kelvin transformations. The obtained depth-dependent distinguishabil…
Assigning boundary conditions, such as acoustic impedance, to the frequency domain thermoviscous wave equations (TWE), derived from the linearized Navier-Stokes equations (LNSE) poses a Helmholtz problem, solution to which yields a discrete set of complex eigenfunctions and eigenvalue pairs. The proposed method -- the inverse Helmholtz solver (iHS) -- reverses such procedure by returning the value of acoustic impedance at one or more unknown impedance boundaries (IBs) of a given domain, via spatial integration of the TWE for a given real-valued frequency with assigned conditions on other bound…
In this paper, we consider inverse shape problems coming from diffuse optical tomography and inverse scattering. In both problems, our goal is to reconstruct small volume interior regions from measured data on the exterior surface of an object. In order to achieve this, we will derive an asymptotic expansion of the reciprocity gap functional associated with each problem. The reciprocity gap functional takes in the measured Cauchy data on the exterior surface of the object. In diffuse optical tomography, we prove that a MUSIC-type algorithm can be used to recover the unknown subregions. This gi…
In this paper, we use the theory of symmetric Dirichlet forms to derive Feynman-Kac formulae for the forward problem of electrical impedance tomography with possibly anisotropic, merely measurable conductivities corresponding to different electrode models on bounded Lipschitz domains. Subsequently, we employ these Feynman-Kac formulae to rigorously justify stochastic homogenization in the case of a stochastic boundary value problem arising from an inverse anomaly detection problem. Motivated by this theoretical result, we prove an estimate for the speed of convergence of the projected mean-squ…
We consider the unique determinations of impenetrable obstacles or diffraction grating profiles in $\mathbb{R}^3$ by a single far-field measurement within polyhedral geometries. We are particularly interested in the case that the scattering objects are of impedance type. We derive two new unique identifiability results for the inverse scattering problem in the aforementioned two challenging setups. The main technical idea is to exploit certain quantitative geometric properties of the Laplacian eigenfunctions which were initiated in our recent works [8,9]. In this paper, we derive novel geometr…
We discuss the inverse problem of determining the possible presence of an (n-1)-dimensional crack Σin an n-dimensional body Ωwith n > 2 when the so-called Dirichlet-to-Neumann map is given on the boundary of Ω. In combination with quantitative unique continuation techniques, an optimal single-logarithm stability estimate is proven by using the singular solutions method. Our arguments also apply when the Neumann-to-Dirichlet map or the local versions of the D-N and the N-D map are available.
In this paper, we derive a Sampling Method to solve the inverse shape problem of recovering an inclusion with a generalized impedance condition from electrostatic Cauchy data. The generalized impedance condition is a second-order differential operator applied to the boundary of the inclusion. We assume that the Dirichlet-to-Neumann mapping is given from measuring the current on the outer boundary from an imposed voltage. A simple numerical example is given to show the effectiveness of the proposed inversion method for recovering the inclusion. We also consider the inverse impedance problem of …
We give formulas and equations for finding generalized scattering data for the Schrödinger equation in open bounded domain at fixed energy from the impedance boundary map (or Robin-to-Robin map). Combining these results with results of the inverse scattering theory we obtain efficient methods for reconstructing potential from the impedance boundary map.
In our earlier paper [9], it is proved that a homogeneous rigid, traction or impedance condition on one or two intersecting line segments together with a certain zero point-value condition implies that the solution to the Lamé system must be identically zero, which is referred to as the generalized Holmgren principle (GHP). The GHP enables us to solve a longstanding inverse scattering problem of determining a polygonal elastic obstacle of general impedance type by at most a few far-field measurements. In this paper, we include all the possible physical boundary conditions from linear elasticit…
It is proved that a connected polygonal obstacle coated by thin layers together with its surface impedance function can be determined uniquely from the far field pattern of a single incident plane wave. Our proof is based on the Schwarz reflection principle for the Helmholtz equation satisfying the impedance boundary condition on a flat boundary.
Several newly developing hybrid imaging methods (e.g., those combining electrical impedance or optical imaging with acoustics) enable one to obtain some auxiliary interior information (usually some combination of the electrical conductivity and the current) about the parameters of the tissues. This information, in turn, happens to stabilize the exponentially unstable and thus low resolution optical and electrical impedance tomography. Various known instances of this effect have been studied individually. We show that there is a simple general technique (covering all known cases) that shows wha…
The inverse problem of electrical impedance tomography is severely ill-posed, meaning that, only limited information about the conductivity can in practice be recovered from boundary measurements of electric current and voltage. Recently it was shown that a simple monotonicity property of the related Neumann-to-Dirichlet map can be used to characterize shapes of inhomogeneities in a known background conductivity. In this paper we formulate a monotonicity-based shape reconstruction scheme that applies to approximative measurement models, and regularizes against noise and modelling error. We dem…
The purpose of this paper is to propose a non-iterative method for the inverse conductivity problem of recovering multiple small anomalies from the boundary measurements. When small anomalies are buried in a conducting object, the electric potential values inside the object can be expressed by integrals of densities with a common sparse support on the location of anomalies. Based on this integral expression, we formulate the reconstruction problem of small anomalies as a joint sparse recovery and present an efficient non-iterative recovery algorithm of small anomalies. Furthermore, we also pro…
We consider the electrostatic inverse boundary value problem also known as electrical impedance tomography (EIT) for the case where the conductivity is a piecewise linear function on a domain $Ω\subset\mathbb{R}^n$ and we show that a Lipschitz stability estimate for the conductivity in terms of the local Dirichlet-to-Neumann map holds true.
The current paper is the second part of a series of two papers dedicated to 2D problem of diffraction of acoustic waves by a segment bearing impedance boundary conditions. In the first part some preliminary steps were made, namely, the problem was reduced to two matrix Riemann-Hilbert problem. Here the Riemann-Hilbert problems are solved with the help of a novel method of OE-equations. Each Riemann-Hilbert problem is embedded into a family of similar problems with the same coefficient and growth condition, but with some other cuts. The family is indexed by an artificial parameter. It is proven…
Consider the Lamé operator $\mathcal{L}(\mathbf{ u} ) :=μΔ\mathbf{u}+(λ+μ) \nabla(\nabla \cdot \mathbf{ u} )$ that arises in the theory of linear elasticity. This paper studies the geometric properties of the (generalized) Lamé eigenfunction $\mathbf{u}$, namely $-\mathcal{L}(\mathbf{ u} )=κ\mathbf{ u}$ with $κ\in\mathbb{R}_+$ and $\mathbf{ u}\in L^2(Ω)^2$, $Ω\subset\mathbb{R}^2$. We introduce the so-called homogeneous line segments of $\mathbf{u}$ in $Ω$, on which $\mathbf{u}$, its traction or their combination via an impedance parameter is vanishing. We give a comprehensive study on characte…
This paper concerns the reconstruction of a scalar diffusion coefficient $σ(x)$ from redundant functionals of the form $H_i(x)=σ^{2α}(x)|\nabla u_i|^2(x)$ where $α\in\Rm$ and $u_i$ is a solution of the elliptic problem $\nabla\cdot σ\nabla u_i=0$ for $1\leq i\leq I$. The case $α=\frac12$ is used to model measurements obtained from modulating a domain of interest by ultrasound and finds applications in ultrasound modulated electrical impedance tomography (UMEIT) as well as ultrasound modulated optical tomography (UMOT). The case $α=1$ finds applications in Magnetic Resonance Electrical Impedanc…
This work presents a new constructive uniqueness proof for Calderón's inverse problem of electrical impedance tomography, subject to local Cauchy data, for a large class of piecewise constant conductivities that we call "piecewise constant layered conductivities" (PCLC). The resulting reconstruction method only relies on the physically intuitive monotonicity principles of the local Neumann-to-Dirichlet map, and therefore the method lends itself well to efficient numerical implementation and generalization to electrode models. Several direct reconstruction methods exist for the related problem …
This paper is devoted to studying impedance eigenvalues (that is, eigenvalues of a particular Dirichlet-to-Neumann map) for the time harmonic linear elastic wave problem, and their potential use as target-signatures for fluid-solid interaction problems. We first consider several possible families of eigenvalues of the elasticity problem, focusing on certain impedance eigenvalues that are an analogue of Steklov eigenvalues. We show that one of these families arises naturally in inverse scattering. We also analyse their approximation from far field measurements of the scattered pressure field in…
We consider the impedance tomography problem in the plane. Using Bukhgeim's scattering data for the Dirac problem, we prove that the conductivity is uniquely determined by the Dirichlet-to-Neuman map
For a general formulation of linearised hybrid inverse problems in impedance tomography, the qualitative properties of the solutions are analysed. Using an appropriate scalar pseudo-differential formulation, the problems are shown to permit propagating singularities under certain non-elliptic conditions, and the associated directions of propagation are precisely identified relative to the directions in which ellipticity is lost. The same result is found in the setting for the corresponding normal formulation of the scalar pseudo-differential equations. A numerical reconstruction procedure base…
In electrical impedance tomography, we aim to solve the conductivity within a target body through electrical measurements made on the surface of the target. This inverse conductivity problem is severely ill-posed, especially in real applications with only partial boundary data available. Thus regularization has to be introduced. Conventionally regularization promoting smooth features is used, however, the Mumford--Shah regularizer familiar for image segmentation is more appropriate for targets consisting of several distinct objects or materials. It is, however, numerically challenging. We show…
This work considers properties of the logarithm of the Neumann-to-Dirichlet boundary map for the conductivity equation in a Lipschitz domain. It is shown that the mapping from the (logarithm of) the conductivity, i.e. the (logarithm of) the coefficient in the divergence term of the studied elliptic partial differential equation, to the logarithm of the Neumann-to-Dirichlet map is continuously Fréchet differentiable between natural topologies. Moreover, for any essentially bounded perturbation of the conductivity, the Fréchet derivative defines a bounded linear operator on the space of square i…
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 …
We analyze the factorization method (introduced by Kirsch in 1998 to solve inverse scattering problems at fixed frequency from the far field operator) for a general class of boundary conditions that generalizes impedance boundary conditions. For instance, when the surface impedance operator is of pseudo-differential type, our main result stipulates that the factorization method works if the order of this operator is different from one and the operator is Fredholm of index zero with non negative imaginary part. We also provide some validating numerical examples for boundary operators of second …
An inverse boundary value problem for the Helmholtz equation in a bounded domain is considered. The problem is to extract information about an unknown obstacle embedded in the domain with unknown impedance boundary condition (the Robin condition) from the associated Dirichlet-to-Neumann map. The main result is a characterization of the unknown obstacle via the sequences that are constructed by the Dirichlet-to-Neumann map, under smallness conditions on the wave number and the upper bound of the impedance. Moreover two alternative simple proofs of a previous result of Cheng-Liu-Nakamura which a…
This paper concerns the reconstruction of an anisotropic conductivity tensor in an elliptic second-order equation from knowledge of the so-called power density functionals. This problem finds applications in several coupled-physics medical imaging modalities such as ultrasound modulated electrical impedance tomography and impedance-acoustic tomography. We consider the linearization of the nonlinear hybrid inverse problem. We find sufficient conditions for the linearized problem, a system of partial differential equations, to be elliptic and for the system to be injective. Such conditions are f…
We present a general framework to study uniqueness, stability and reconstruction for infinite-dimensional inverse problems when only a finite-dimensional approximation of the measurements is available. For a large class of inverse problems satisfying Lipschitz stability we show that the same estimate holds even with a finite number of measurements. We also derive a globally convergent reconstruction algorithm based on the Landweber iteration. This theory applies to nonlinear ill-posed problems such as electrical impedance tomography, inverse scattering and quantitative photoacoustic tomography…
A generalized variant of the Calderón problem from electrical impedance tomography with partial data for anisotropic Lipschitz conductivities is considered in an arbitrary space dimension $n \geq 2$. The following two results are shown: (i) The selfadjoint Dirichlet operator associated with an elliptic differential expression on a bounded Lipschitz domain is determined uniquely up to unitary equivalence by the knowledge of the Dirichlet-to-Neumann map on an open subset of the boundary, and (ii) the Dirichlet operator can be reconstructed from the residuals of the Dirichlet-to-Neumann map on th…
We consider the partial data inverse boundary problem for the Schrödinger operator at a frequency $k>0$ on a bounded domain in $\mathbb{R}^n$, $n\ge 3$, with impedance boundary conditions. Assuming that the potential is known in a neighborhood of the boundary, we first show that the knowledge of the partial Robin-to-Dirichlet map at the fixed frequency $k>0$ along an arbitrarily small portion of the boundary, determines the potential in a logarithmically stable way. We prove, as the principal result of this work, that the logarithmic stability can be improved to the one of Hölder type in the h…
