We show how to eliminate the error caused by an incorrectly modeled boundary in electrical impedance tomography (EIT). In practical measurements, one usually lacks the exact knowledge of the boundary. Because of this the numerical reconstruction from the measured EIT data is done using a model domain that represents the best guess for the true domain. However, it has been noticed that the inaccurate model of the boundary causes severe errors for the reconstructions. We introduce a new algorithm to find a deformed image of the original isotropic conductivity based on the theory of Teichmuller spaces and implement it numerically.
We consider the inverse conductivity problem in a strictly convex domain whose boundary is not known. Usually the numerical reconstruction from the measured current and voltage data is done assuming the domain has a known fixed geometry. However, in practical applications the geometry of the domain is usually not known. This introduces an error, and effectively changes the problem into an anisotropic one. The main result of this paper is a uniqueness result characterizing the isotropic conductivities on convex domains in terms of measurements done on a different domain, which we call the model…
A direct reconstruction algorithm based on Calderón's linearization method for the reconstruction of isotropic conductivities is proposed for anisotropic conductivities in two-dimensions. To overcome the non-uniqueness of the anisotropic inverse conductivity problem, the entries of the unperturbed anisotropic tensors are assumed known \emph{a priori}, and it remains to reconstruct the multiplicative scalar field. The quasi-conformal map in the plane facilitates the Calderón-based approach for anisotropic conductivities. The method is demonstrated on discontinuous radially symmetric conductivit…
This work deals with the problem of determining a non-homogeneous heat conductivity profile in a steady-state heat conduction boundary-value problem with mixed Dirichlet-Neumann boundary conditions over a bounded domain in $\mathbb{R}^n$, from the knowledge of the state over the whole domain. We develop a method based on a variational approach leading to an optimality equation which is then projected into a finite dimensional space. Discretization yields a linear although severely ill-posed equation which is then regularized via appropriate ad-hoc penalizers resulting a in a generalized Tikhon…
This paper concerns the reconstruction of a scalar coefficient of a second-order elliptic equation in divergence form posed on a bounded domain from internal data. This theory finds applications in multi-wave imaging, greedy methods to approximate parameter-dependent elliptic problems, and image treatment with partial differential equations. We first show that the inverse problem for smooth coefficients can be rewritten as a linear transport equation. Assuming that the coefficient is known near the boundary, we study the well-posedness of associated transport equation as well as its numerical …
We study inverse conductivity problem for an anisotropic conductivity in $L^\infty$ in bounded and unbounded domains. Also, we give applications of the results in the case when Dirichlet-to-Neumann and Neumann-to-Dirichlet maps are given only on a part of the boundary.
We study the inverse conductivity problem with discontinuous conductivities. We consider, simultaneously, a regularisation and a discretisation for a variational approach to solve the inverse problem. We show that, under suitable choices of the regularisation and discretisation parameters, the discrete regularised solutions converge, as the noise level on the measurements goes to zero, to the looked for solution of the inverse problem.
This paper shows global uniqueness in two inverse problems for a fractional conductivity equation: an unknown conductivity in a bounded domain is uniquely determined by measurements of solutions taken in arbitrary open, possibly disjoint subsets of the exterior. Both the cases of infinitely many measurements and a single measurement are addressed. The results are based on a reduction from the fractional conductivity equation to the fractional Schrödinger equation, and as such represent extensions of previous works. Moreover, a simple application is shown in which the fractional conductivity eq…
This paper concerns the inverse problem of determining a planar conductivity inclusion. Our aim is to analytically recover from the generalized polarization tensors (GPTs), which can be obtained from exterior measurements, a homogeneous inclusion with arbitrary constant conductivity. The primary outcome of recovering a homogeneous inclusion is an inversion formula in terms of the GPTs for conformal mapping coefficients associated with the inclusion. To prove the formula, we establish matrix factorizations for the GPTs.
It is proved that, in two dimensions, the Calderón inverse conductivity problem in Lipschitz domains is stable in the $L^p$ sense when the conductivities are uniformly bounded in any fractional Sobolev space $W^{α,p}$ $α>0, 1<p<\infty$.
We consider the stability issue of the inverse conductivity problem for a conformal class of anisotropic conductivities in terms of the local Dirichlet-to-Neumann map. We extend here the stability result obtained by Alessandrini and Vessella in Advances in Applied Mathematics 35:207-241, where the authors considered the piecewise constant isotropic case.
We prove uniqueness of the inverse conductivity problem in three dimensions for complex conductivities in $W^{1,\infty}$. We apply quaternionic analysis to transform the inverse problem into an inverse Dirac scattering problem, as established in two dimensions by Brown and Uhlmann. This is a novel methodology that allows to extend the uniqueness result from once-differentiable real conductivities to complex ones.
For the heat equation in a bounded domain we give a stability result for a smooth diffusion coefficient. The key ingredients are a global Carleman-type estimate, a Poincaré-type estimate and an energy estimate with a single observation acting on a part of the boundary.
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
In this work, a novel approach for the solution of the inverse conductivity problem from one and multiple boundary measurements has been developed on the basis of the implication of the framework of BV - functions. The space of the functions of bounded variation is here recommended as the most appropriate functional space hosting the conductivity profile under reconstruction. For the numerical solution, we propose and implement a suitable minimization scheme of an enriched - constructed herein - functional, by exploiting the inner structure of BV - space. Finally, we validate and illustrate ou…
We show that the knowledge of the Dirichlet-to-Neumann maps given on an arbitrary open non-empty portion of the boundary of a smooth domain in $\mathbb{R}^n$, $n\ge 2$, for classes of semilinear and quasilinear conductivity equations, determines the nonlinear conductivities uniquely. The main ingredient in the proof is a certain $L^1$-density result involving sums of products of gradients of harmonic functions which vanish on a closed proper subset of the boundary.
We prove \emph{global} uniqueness for an inverse problem for the fractional conductivity equation on domains that are bounded in one direction. The conductivities are assumed to be isotropic and nontrivial in the exterior of the domain, while the data is given in the form of partial Dirichlet-to-Neumann (DN) maps measured in nondisjoint open subsets of the exterior. This can be seen as the fractional counterpart of the classical inverse conductivity problem. The proof is based on a unique continuation property (UCP) for the DN maps and an exterior determination method from the partial exterior…
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…
An electrical potential U on a bordered Riemann surface X with conductivity function sigma>0 satisfies equation d(sigma d^cU)=0. The problem of effective reconstruction of sigma is studied. We extend to the case of Riemann surfaces the reconstruction scheme given, firstly, by R.Novikov (1988) for simply connected X. We apply for this new kernels for dbar on affine algebraic Riemann surfaces constructed in Henkin, arXiv:0804.3761
The aim of the paper is twofold. Firstly, we would like to derive quantitative uniqueness estimates for solutions of the general complex conductivity equation. It is still unknown whether the \emph{strong} unique continuation property holds for such equations. Nonetheless, in this paper, we show that the unique continuation property in the form of three-ball inequalities is satisfied for the complex conductivity equation under only Lipschitz assumption on the leading coefficients. The derivation of such estimates relies on a delicate Carleman estimate. Secondly, we study the problem of estimat…
Let $\hat Ω\subset \mathbb R^2$ be a bounded domain with smooth boundary and $\hat σ$ a smooth anisotropic conductivity on $\hat Ω$. Starting from the Dirichlet-to-Neumann operator $Λ_{\hat σ}$ on $\partial \hat Ω$, we give an explicit procedure to find a unique domain $Ω$, an isotropic conductivity $σ$ on $Ω$ and the boundary values of a quasiconformal diffeomorphism $F:\hat Ω\to Ω$ which transforms $\hat σ$ into $σ$.
We consider the inverse problem of determining the unknown function $α: \mathbb{R} \rightarrow \mathbb{R}$ from the DN map associated to the operator $\mbox{div}(A(x',α(x\_3))\nabla \cdot)$ acting in the infinite straight cylindrical waveguide $Ω=ω\times \mathbb{R}$, where $ω$ is a bounded domain of $\mathbb{R}^2$. Here $A=(A\_{ij}(x))$, $x=(x',x\_3) \in Ω$, is a matrix-valued metric on $Ω$ obtained by straightening a twisted waveguide. This inverse anisotropic conductivity problem remains generally open, unless the unknown function $α$ is assumed to be constant. In this case we prove Lipschit…
We consider an inverse problem for electrically conductive material occupying a domain $Ω$ in $\Bbb R^2$. Let $γ$ be the conductivity of $Ω$, and $D$ a subdomain of $Ω$. We assume that $γ$ is a positive constant $k$ on $D$, $k\not=1$ and is $1$ on $Ω\setminus D$; both $D$ and $k$ are unknown. The problem is to find a reconstruction formula of $D$ from the Cauchy data on $\partialΩ$ of a non-constant solution $u$ of the equation $\nabla\cdotγ\nabla u=0$ in $Ω$. We prove that if $D$ is known to be a convex polygon such that $\text{diam}\,D<\text{dist}\,(D,\partialΩ)$, there are two formulae for …
The ill-posedness of Calderón's inverse conductivity problem, responsible for the poor spatial resolution of Electrical Impedance Tomography (EIT), has been an impetus for the development of hybrid imaging techniques, which compensate for this lack of resolution by coupling with a second type of physical wave, typically modeled by a hyperbolic PDE. Here we show how, using EIT data alone, to efficiently detect interior jumps and other singularities of the conductivity. Analysis of the complex geometrical optics solutions of Astala and Päivärinta [\emph{Ann. Math.}, {\bf 163} (2006)] in 2D makes…
This paper considers the inverse problem of recovering both the unknown, spatially-dependent conductivity $a(x)$ and the nonlinear reaction term $f(u)$ in a reaction-diffusion equation from overposed data. These measurements can consist of: the value of two different solution measurements taken at a later time $T$; time-trace profiles from two solutions; or both final time and time-trace measurements from a single forwards solve data run. We prove both uniqueness results and the convergence of iteration schemes designed to recover these coefficients. The last section of the paper shows numeric…
This paper concerns the reconstruction of an anisotropic conductivity tensor $γ$ from internal current densities of the form $J = γ\nabla u$, where $u$ solves a second-order elliptic equation $\nabla\cdot(γ\nabla u) = 0$ on a bounded domain $X$ with prescribed boundary conditions. A minimum number of such functionals equal to $n + 2$, where $n$ is the spatial dimension, is sufficient to guarantee a local reconstruction. We show that $γ$ can be uniquely reconstructed with a loss of one derivative compared to errors in the measurement of $J$. In the special case where $γ$ is scalar, it can be re…
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…
This paper is devoted to the investigation of inverse problems related to stationary drift-diffusion equations modeling semiconductor devices. In this context we analyze several identification problems corresponding to different types of measurements, where the parameter to be reconstructed is an inhomogeneity in the PDE model (doping profile). For a particular type of measurement (related to the voltage-current map) we consider special cases of drift-diffusion equations, where the inverse problems reduces to a classical inverse conductivity problem. A numerical experiment is presented for one…
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.
We investigate the problem of reconstructing a fully anisotropic conductivity tensor $γ$ from internal functionals of the form $\nabla u\cdotγ\nabla u$ where $u$ solves $\nabla\cdot(γ\nabla u) = 0$ over a given bounded domain $X$ with prescribed Dirichlet boundary condition. This work motivated by hybrid medical imaging methods covers the case $n\ge 3$, following the previously published case $n=2$ \cite{Monard2011}. Under knowledge of enough such functionals, and writing $γ= τ\tilde γ$ ($\det \tildeγ= 1$) with $τ$ a positive scalar function, we show that all of $γ$ can be explicitely and loca…
In this paper, we consider the inverse scattering problem associated with an inhomogeneous media with a conductive boundary. First, we discuss the inverse conductivity problem of reconstructing the conductivity parameter from scattering data. Next, we consider the corresponding interior transmission eigenvalue problem. This is a new class of eigenvalue problem that is not elliptic, not self-adjoint, and non-linear, which gives the possibility of complex eigenvalues. We investigate the convergence of the eigenvalues as the conductivity parameter tends to zero as well as prove existence and disc…
We consider the multidimensional inverse problem of determining the conductivity coefficient of a hyperbolic equation in an infinite cylindrical domain, from a single boundary observation of the solution. We prove H{ö}lder stability with the aid of a Carleman estimate specifically designed for hyperbolic waveguides.
We consider the determination of a conductivity function in a two-dimensional domain from the Cauchy data of the solutions of the conductivity equation on the boundary. We prove uniqueness results for this inverse problem, posed by Calderon, for conductivities that are degenerate, that is, they may not be bounded from above or below. In particular, for scalar conductivities we solve the inverse problem in a class which is larger than $L^\infty$. Also, we give new counterexamples for the uniqueness of the inverse conductivity problem. We say that a conductivity is visible if the inverse problem…
We consider the inverse problem of recovering an isotropic quasilinear conductivity from the Dirichlet-to-Neumann map when the conductivity depends on the solution and its gradient. We show that the conductivity can be recovered on an open subset of small gradients, hence extending a partial result to all real analytic conductivities. We also recover non-analytic conductivities with additional growth assumptions along large gradients. Moreover, the results hold for non-homogeneous conductivities if the non-homogeneous part is assumed known.
We consider the inverse problem of determining, the possibly anisotropic, conductivity of a body by means of the so called local Neumann to Dirichlet map on a curved portion $Σ$ of the boundary. Motivated by the uniqueness result for piecewise constant anisotropic conductivities proved in \cite{Al-dH-G}, we provide a Hölder stability estimate on $Σ$ when the conductivity is a priori known to be a constant matrix near $Σ$.
We study the inverse conductivity problem of how to reconstruct an isotropic electrical conductivity distribution $γ$ in an object from static electrical measurements on the boundary of the object. We give an exact reconstruction algorithm for the conductivity $γ\in C^{1+ε}(\ol \Om)$ in the plane domain $Ω$ from the associated Dirichlet to Neumann map on $\partial \Om.$ Hence we improve earlier reconstruction results. The method used relies on a well-known reduction to a first order system, for which the $\ol\partial$-method of inverse scattering theory can be applied.
We prove uniqueness for minimizers of the weighted least gradient problem \[\inf \left\lbrace \int_Ω a|Du|: \ \ u\in BV(Ω), \ \ u|_{\partial Ω}=f \right\rbrace.\] The weight function $a$ is assumed to be continuous and it is allowed to vanish in certain subsets of $Ω$. Existence is assumed a priori. Our approach is motivated by the hybrid inverse problem of imaging electric conductivity from interior knowledge (obtainable by MRI) of the magnitude of one current density vector field.
In this paper, we consider an inverse conductivity problem on a bounded domain $Ω\subset\mathbb{R}^n$, $n\geq2$, also known as Electrical Impedance Tomography (EIT), for the case where unknown impenetrable obstacles are embedded into $Ω$. We show that a piecewise-constant conductivity function and embedded obstacles can be simultaneously recovered in terms of the local Dirichlet-to-Neumann map defined on an arbitrary small open subset of the boundary of the domain $Ω$. The method depends on the well-posedness of a coupled PDE-system constructed for the conductivity equations in the $H^1$-space…
In this work we establish log-type stability estimates for the inverse potential and conductivity problems with partial Dirichlet-to-Neumann map, where the Dirichlet data is homogeneous on the inaccessible part. This result, to some extent, improves our former result on the partial data problem in which log-log-type estimates were derived.
We construct anisotropic conductivities with the same Dirichlet-to-Neumann map as a homogeneous isotropic conductivity. These conductivities are singular close to a surface inside the body.
We consider a time-independent variable coefficients fractional porous medium equation and formulate an associated inverse problem. We determine both the conductivity and the absorption coefficient from exterior partial measurements of the Dirichlet-to-Neumann map. Our approach relies on a time-integral transform technique as well as the unique continuation property of the fractional operator.
