We prove uniqueness in the inverse conductivity problem for uniformly elliptic conductivities in $W^{s,p}(Ω)$, where $Ω\subset \mathbb R^n$ is Lipschitz, $3\leq n \leq 6$, and $s$ and $p$ are such that $ W^{s,p}(Ω)\not \subset W^{1,\infty}(Ω)$. In particular, we obtain uniqueness for conductivities in $W^{1,n}(Ω)$ ($n=3,4$). This improves on the result of the author and Tataru, who assumed that the conductivity is Lipschitz.
In this paper we study the inverse conductivity problem with partial data. Moreover, we show that, in dimension $n\geq 3$ the uniqueness of the Calderón problem holds for the $C^{1}\bigcap H^{3/2, 2}$ conductivities.
We show that Nachman's integral equations for the Calderón problem, derived for conductivities in $W^{2,p}(Ω)$, still hold for $L^\infty$ conductivities which are $1$ in a neighborhood of the boundary. We also prove convergence of scattering transforms for smooth approximations to the scattering transform of $L^\infty$ conductivities. We rely on Astala-Päivärinta's formulation of the Calderón problem for a framework in which these convergence results make sense.
We use X^{s,b}-inspired spaces to prove a uniqueness result for Calderon's problem in a Lipschitz domain under the assumption that the conductivity is Lipschitz. For Lipschitz conductivities, we obtain uniqueness for conductivities close to the identity in a suitable sense. We also prove uniqueness for arbitrary C^1 conductivities.
In these notes we prove log-type stability for the Calderón problem with conductivities in $ C^{1,\varepsilon}(\barΩ) $. We follow the lines of a recent work by Haberman and Tataru in which they prove uniqueness for $ C^1(\barΩ) $.
In this note, we study Calderón's problem for certain classes of conductivities in domains with circular symmetry in two and three dimensions. Explicit formulas are obtained for the reconstruction of the conductivity from the Dirichlet-to-Neumann map. As a consequence, we show that the reconstruction is Lipschitz stable.
The unique determination of a measurable conductivity from the Dirichlet-to-Neumann map of the equation $\mathrm{div} (σ\nabla u) = 0$ is the subject of this note. A new strategy, based on Clifford algebras and a higher dimensional analogue of the Beltrami equation, is here proposed. This represents a possible first step for a proof of uniqueness for the Calderón problem in three and higher dimensions in the $L^\infty$ case.
We prove uniqueness for Calderón's problem with Lipschitz conductivities in higher dimensions. Combined with the recent work of Haberman, who treated the three and four dimensional cases, this confirms a conjecture of Uhlmann. Our proof builds on the work of Sylvester and Uhlmann, Brown, and Haberman and Tataru who proved uniqueness for $C^1$ conductivities and Lipschitz conductivities sufficiently close to the identity.
The Calderón Problem for Quasilinear Conductivities of Conformally Transversally Anisotropic Media
Shanghai Center for Mathematical Sciences, Fudan University（复旦大学上海数学中心）；Center for Applied Mathematics, Fudan University（复旦大学应用数学中心）；School of Mathematical Sciences, Fudan University（复旦大学数学科学学院）
This note considers a finite dimensional statistical model for the Calderón problem with piecewise constant conductivities. In this setting it is shown that injectivity of the forward map and its linearisation suffice to prove the invertibility of the information operator, resulting in a Bernstein--von-Mises theorem and optimality guarantees for estimation by Bayesian posterior means.
We extend a global uniqueness result for the Calderón problem with partial data, due to Kenig-Sjöstrand-Uhlmann, to the case of less regular conductivities. Specifically, we show that in dimensions $n\ge 3$, the knowledge of the Diricihlet-to-Neumann map, measured on possibly very small subsets of the boundary, determines uniquely a conductivity having essentially $3/2$ derivatives in an $L^2$ sense.
Determining $W^{1,n}$ conductivities in Calderón's problem for $n\geq4$
Key Laboratory of Algebraic Lie Theory and Analysis, Ministry of Education, School of Mathematics and Statistics, Beijing Institute of Technology（北京理工大学数学与统计学院代数李理论与分析教育部重点实验室）；Department of Mathematics, City University of Hong Kong（香港城市大学数学系）
We study the inverse problem of determining the coefficients of the fractional power of a general second order elliptic operator given in the exterior of an open subset of the Euclidean space. We show the problem can be reduced into determining the coefficients from the boundary Cauchy data of the elliptic operator on the open set, the Calderón problem. As a corollary we establish several new results for nonlocal inverse problems by using the corresponding results for the local inverse problems. In particular the isotropic nonlocal Calderón problem can be resolved completely, assuming some reg…
We show that the inverse problem of Calderon for conductivities in a two-dimensional Lipschitz domain is stable in a class of conductivities that are Dini continuous. This extends previous stability results when the conductivities are known to be Holder continuous.
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…
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 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…
A direct three dimensional EIT reconstruction algorithm based on complex geometrical optics solutions and a nonlinear scattering transform is presented and implemented for spherically symmetric conductivity distributions. The scattering transform is computed both with a Born approximation and from the forward problem for purposes of comparison. Reconstructions are computed for several test problems. A connection to Calderón's linear reconstruction algorithm is established, and reconstructions using both methods are compared.
The Calderón problem is an inverse problem with applications to electrical impedance tomography and geophysical prospection. We prove uniqueness in the Calderón problem in spatial dimension $n \geq 3$ for scalar conductivities in the Sobolev space $W^{1,p}$ with $p \geq n$. This generalizes a result of Haberman who considered the case $p \geq n$ and $n=3$ or $4$. Our method of proof combines a Fourier series approach with an analytic criterion for infinitesimal boundedness of potentials appearing in a Schrödinger equation with respect to the Laplacian.
Let $X$ be a smooth bordered surface in $\real^3$ with smooth boundary and $\hat σ$ a smooth anisotropic conductivity on $X$. If the genus of $X$ is given, then starting from the Dirichlet-to-Neumann operator $Λ_{\hat σ}$ on $\partial X$, we give an explicit procedure to find a unique Riemann surface $Y$ (up to a biholomorphism), an isotropic conductivity $σ$ on $Y$ and the boundary values of a quasiconformal diffeomorphism $F: X \to Y$ which transforms $\hat σ$ into $σ$. As a corollary we obtain the following uniqueness result: if $σ_1, σ_2$ are two smooth anisotropic conductivities on $X$ wi…
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 Calderón problem with partial data in certain admissible geometries, that is, on compact Riemannian manifolds with boundary which are conformally embedded in a product of the Euclidean line and a simple manifold. We show that measuring the Dirichlet-to-Neumann map on roughly half of the boundary determines a conductivity that has essentially 3/2 derivatives. As a corollary, we strengthen a partial data result due to Kenig, Sjöstrand, and Uhlmann.
We are concerned with the Calderón problem of determining an unknown conductivity of a body from the associated boundary measurement. We establish a logarithmic type stability estimate in terms of the Hausdorff distance in determining the support of a convex polygonal inclusion by a single partial boundary measurement. We also derive the uniqueness result in a more general scenario where the conductivities are piecewise constants supported in a nested polygonal geometry. Our methods in establishing the stability and uniqueness results have a significant technical initiative and a strong potent…
Given a general symmetric elliptic operator $$ L\_{a} := \sum\_{k,,j=1}^d \p\_k (a\_{kj} \p\_j) + \sum\_{k=1}^d a\_k \p\_k - \p\_k(\overline{a\_k} .) + a\_0$$we define the associated Dirichlet-to-Neumann (D-t-N) operator with partial data, i.e., data supported in a part of the boundary. We prove positivity, $L^p$-estimates and domination properties for the semigroup associated with this D-t-N operator. Given $L\_a $ and $L\_b$ of the previous type with bounded measurable coefficients $a = \{a\_{kj}, \ a\_k, a\_0 \}$ and $b = \{b\_{kj}, \ b\_k, b\_0 \}$, we prove that if their partial D-t-N ope…
We prove a Plancherel theorem for a nonlinear Fourier transform in two dimensions arising in the Inverse Scattering method for the defocusing Davey-Stewartson II equation. We then use it to prove global well-posedness and scattering in $L^2$ for defocusing DSII. This Plancherel theorem also implies global uniqueness in the inverse boundary value problem of Calderón in dimension $2$, for conductivities $σ>0$ with $\log σ\in \dot H^1$. The proof of the nonlinear Plancherel theorem includes new estimates on classical fractional integrals, as well as a new result on $L^2$-boundedness of pseudo-dif…
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.
In this paper, we address a particular case of Calderón's (or conductivity) inverse problem in dimension two, namely the case of a homogeneous background containing a finite number of cavities (i.e. heterogeneities of infinitely high conductivities). We aim to recover the location and the shape of the cavities from the knowledge of the Dirichlet-to-Neumann (DtN) map of the problem. The proposed reconstruction method is non iterative and uses two main ingredients. First, we show how to compute the so-called generalized Pólia-Szegö tensors (GPST) of the cavities from the DtN of the cavities. Sec…
We discuss the inverse problem of determining the, possibly anisotropic, conductivity of a body $Ω\subset\mathbb{R}^{n}$ when the so--called Dirichlet-to-Neumann map is locally given on a non empty portion $Γ$ of the boundary $\partialΩ$. We extend results of uniqueness and stability at the boundary, obtained by the same authors in SIAM J. Math. Anal. 33 (2001), no. 1, 153--171, where the Dirichlet-to-Neumann map was given on all of $\partialΩ$ instead. We also obtain a pointwise stability result at the boundary among the class of conductivities which are continuous at some point $y\inΓ$. Our …
We consider one-dimensional Calderón's problem for the variable exponent $p(\cdot)$-Laplace equation and find out that more can be seen than in the constant exponent case. The problem is to recover an unknown weight (conductivity) in the weighted $p(\cdot)$-Laplace equation from Dirichlet and Neumann data of solutions. We give a constructive and local uniqueness proof for conductivities in $L^\infty$ restricted to the coarsest sigma-algebra that makes the exponent $p(\cdot)$ measurable.
The purpose of this article is to extend the uniqueness results for the two dimensional Calderón problem to unbounded potentials on general geometric settings. We prove that the Cauchy data sets for Schrödinger equations uniquely determines potentials in $L^{p}$ for $p> 4/3$. In doing so, we first recover singularities of the potential, from which point a $L^2$-based method of stationary phase can be applied. Both steps are done via constructions of complex geometric optic solutions and Carleman estimates.
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 …
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…
In dimensions greater than or equal to three, we establish global uniqueness and obtain reconstruction in the Calderon problem for the Schrodinger equation with certain singular potentials. The potentials considered are conormal of order less than 1-k with respect to submanifolds (of arbitrary codimension k). This gives positive results for (conormal) conductivities which are Holder of any order > 1. A related problem for highly singular potentials is shown to exhibit nonuniqueness.
Electrical Impedance Imaging would suffer a serious obstruction if for two different conductivities the potential and current measured at the boundary were the same. The Calderón's problem is to decide whether the conductivity is indeed uniquely determined by the data at the boundary. In $\mathbb{R}^d$, for $d=5,6$, we show that uniqueness holds when the conductivity is in $W^{1+\frac{d-5}{2p}+, p}(Ω)$, for $d\le p <\infty$. This improves on recent results of Haberman, and of Ham, Kwon and Lee. The main novelty of the proof is an extension of Tao's bilinear Theorem.
Two reconstruction methods of Electrical Impedance Tomography (EIT) are numerically compared for nonsmooth conductivities in the plane based on the use of complex geometrical optics (CGO) solutions to D-bar equations involving the global uniqueness proofs for Calderón problem exposed in [Nachman; Annals of Mathematics 143, 1996] and [Astala and Päivärinta; Annals of Mathematics 163, 2006]: the Astala-Päivärinta theory-based "low-pass transport matrix method" implemented in [Astala et al.; Inverse Problems and Imaging 5, 2011] and the "shortcut method" which considers ingredients of both theori…
This paper is devoted to the problem of recovering a potential $q$ in a domain in $\mathbb{R}^d$ for $d \geq 3$ from the Dirichlet to Neumann map. This problem is related to the inverse Calderón conductivity problem via the Liouville transformation. It is known from the work of Haberman and Tataru [11] and Nachman and Lavine [17] that uniqueness holds for the class of conductivities of one derivative and the class of $W^{2,d/2}$ conductivities respectively. The proof of Haberman and Tataru is based on the construction of complex geometrical optics (CGO) solutions initially suggested by Sylvest…
The unique determination of electrical conductivity is extensively studied for isotropic conductivity ever since Calderon's suggestion of the EIT (Electrical Impedance Tomography) problem. However, it is known that there are many anisotropic conductivities producing the same Dirichlet-to-Neumann map; moreover the anisotropic conductivities giving the same Dirichlet-to-Neumann map are classified using the equivalence relation with respect to the change of variables. The change of variable argument is applied to the theory of near-cloaking: We are under an illusion that the domain of anomaly is …
Uniqueness in the Calderón problem in dimension bigger than two was usually studied under the assumption that conductivity has bounded gradient. For conductivities with unbounded gradients uniqueness results have not been known until recent years. The latest result due to Haberman basically relies on the optimal $L^2$ restriction estimate for hypersurface which is known as the Tomas-Stein restriction theorem. In the course of developments of the Fourier restriction problem bilinear and multilinear generalizations of the (adjoint) restriction estimates under suitable transversality condition be…
We consider the two-dimensional version of Calderòn's problem. When the D-N map is assumed to be known up to an error level $\varepsilon_0$, we investigate how the resolution in the determination of the unknown conductivity deteriorates the farther one goes from the boundary. We provide explicit formulas for the resolution, which apply to conductivities which are perturbations, concentrated near an interior point $q$, of the homogeneous conductivity.
We consider the Gel'fand-Calderón problem for a Schrödinger operator of the form $-(\nabla + iA)^2 + q$, defined on a ball $B$ in $\mathbb R^3$. We assume that the magnetic potential $A$ is small in $W^{s,3}$ for some $s>0$, and that the electric potential $q$ is in $W^{-1,3}$. We show that, under these assumptions, the magnetic field $\operatorname{curl} A$ and the potential $q$ are both determined by the Dirichlet-Neumann relation at the boundary $\partial B$. The assumption on $q$ is critical with respect to homogeneity, and the assumption on $A$ is nearly critical. Previous uniqueness theo…
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.
