Robert McOwen, Bindu K Veetel
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.
机构、总结、中文摘要均由AI分析而来，可能存在错误，请以论文正文为准。
继续探索
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 discuss the inverse problem of determining the, possibly anisotropic, conductivity of a body $Ω\subset\mathbb{R}^{n}$ when the so-called Neumann-to-Dirichlet map is locally given on a non empty curved portion $Σ$ of the boundary $\partialΩ$. We prove that anisotropic conductivities that are \textit{a-priori} known to be piecewise constant matrices on a given partition of $Ω$ with curved interfaces can be uniquely determined in the interior from the knowledge of the local Neumann-to-Dirichlet map.
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 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.
The Calderón inverse problem for isotropic quasilinear conductivities
该研究证明了高维有界区域上一般拟线性各向同性电导率方程的Calderón逆问题全局唯一性，通过高阶线性化和复几何光学解实现。
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.
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 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
We consider inverse problems for $p$-Laplace type equations under monotonicity assumptions. In two dimensions, we show that any two conductivities satisfying $σ_1 \geq σ_2$ and having the same nonlinear Dirichlet-to-Neumann map must be identical. The proof is based on a monotonicity inequality and the unique continuation principle for $p$-Laplace type equations. In higher dimensions, where unique continuation is not known, we obtain a similar result for conductivities close to constant.
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 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.
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.
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…
Partial data stability for the inverse fractional conductivity problem
针对逆分数阶电导率问题的部分数据稳定性，利用分数阶 Liouville 约化，在电导率先验一致且差有紧支撑时获得对数-对数稳定性，在外部整体一致时获得更精确的对数稳定性。
The classical $\overline \partial$-method has been generalized recently [lnv], [lnv2] to be used in the presence of exceptional points. We apply this generalization to solve Dirac inverse scattering problem with weak assumptions on smoothness of potentials. As a consequence, this provides an effective method of reconstruction of complex-valued one time differentiable conductivities in the inverse impedance tomography problem.
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 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.
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…
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…
We find a complete characterization for sets of isotropic conductivities with stable recovery in the $L^2$ norm when the data of the Calderón Inverse Conductivity Problem is obtained in the boundary of a disk and the conductivities are constant in a neighborhood of its boundary. To obtain this result, we present minimal a priori assumptions which turn to be sufficient for sets of conductivities to have stable recovery in a bounded and rough domain. The condition is presented in terms of the modulus of continuity of the coefficients involved and their ellipticity bound.
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 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.
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…
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…
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 …
Relative formalization in Isabelle/HOL of a result in inverse problems
本研究利用Isabelle/HOL对arXiv:2606.15977中的反问题结果开展自动形式化实验，基于熟知的外部结果完成证明，相关文件已在GitHub仓库公开。
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.
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 $W^{α,p}$ $α>0, 1<p<\infty$.
Poincar{é}-Steklov operator and Calder{ó}n's problem on extension domains
本文在包含非Lipschitz和分形形状的Sobolev延拓域上推广Poincaré-Steklov算子，证明正问题稳定性，并利用边界稳定性在n≥3时识别Lipschitz电导率及边界附近为常数的W^{2,∞}电导率。
Infinite-dimensional Lipschitz stability in the Calderón problem and general Zernike bases
该研究证明了在特定条件下，无限维导电性类中存在Lipschitz稳定性，并通过构造Zernike基展示了其应用。
We investigate the continuity of boundary operators, such as the Neumann-to-Dirichlet map, with respect to the coefficient matrices of the underlying elliptic equations. We show that for nonsmooth coefficients the correct notion of convergence is the one provided by $H$-convergence (or $G$-convergence for symmetric matrices). We prove existence results for minimum problems associated to variational methods used to solve the so-called inverse conductivity problem, at least if we allow the conductivities to be anisotropic. In the case of isotropic conductivities we show that on certain occasions…
Researchers familiar with the state of the art are aware that the development of close-formed solutions for the EIT problem was not able to overpass the case of once-time differentiable conductivities beside the well known particular Astala-Päivärinta result for zero frequency. In this paper, we introduce some new techniques for the inverse conductivity problem combined with a transmission problem and achieve a reconstruction result based on an adaptation of the scattering data. The idea for these techniques, in particular the concept of admissible points is coming from E. Lakshtanov and B. Va…
Low regularity theory for the inverse fractional conductivity problem
本文研究低正则性反分数阶电导率问题，在所有维度刻画 $H^{s,n/s}$ 假设下的部分数据唯一性，构造关键反例，并给出不依赖 Runge 逼近的新证明。
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…
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…
Fractional Calderón problem with $L^{\infty}$-conductivities
针对具有一致椭圆型电导率的分数阶电导率方程逆问题，研究人员证明有界开区域外任意非空开子集上的局部外狄利克雷-诺伊曼映射可唯一确定电导率几乎处处的值，无需连续性、索伯列夫正则性或无穷远处指定行为，证明结合了外确定、加权解等式与分数阶拉普拉斯算子唯一延拓原理。
Extension and neural operator approximation of the electrical impedance tomography inverse map
本文提出通过扩展反演算子到Hilbert空间并利用神经算子近似，提升电阻抗断层成像在噪声环境下的逆问题求解能力。
Counterexamples to uniqueness in the inverse fractional conductivity problem with partial data
本文证明分数阶电导率方程的部分外部数据无法唯一确定电导率，并在一般有界及单方向有界区域上构造满足正则性的反例。
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 $Σ$.
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.
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 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…
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…
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…
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.
Hölder Stability from Exact Uniqueness for Finite-Dimensional Analytic Inverse Problems
本文证明了有限维分析反问题的Hölder稳定性定理，通过希尔伯特-施密特算子方程和洛乔斯维茨距离不等式，展示精确唯一性如何推导出Hölder稳定性估计。
Stability estimates for the inverse fractional conductivity problem
本文研究有界光滑区域上分数阶电导率方程的反问题，结合分数阶 Schrödinger 反问题、外部确定问题及 Alessandrini 思想的非局部类比，建立并证明了最优对数稳定性估计。
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 …
Characterization of Dirichlet-to-Neumann maps via the Born approximation
本文通过证明径向电导率在单位球中的DtN映射可表示为唯一确定的可积函数的线性化DtN映射，为Calderón反问题的数值求解提供理论基础，并探讨了Born近似在不同情况下的鲁棒性。
The Born approximation in the three-dimensional Calderón problem II: Numerical reconstruction in the radial case
本文通过数值实验研究三维Calderón问题中Born近似的性质，提出从Born近似重构径向电导率的新算法，并揭示其深度相关唯一性及半径增大时的散射极限。
Recovery of an Anisotropic Conductivity from the Neumann-to-Dirichlet Map in a Semilinear Elliptic Equation
研究通过线性化方法证明从半线性椭圆方程的Neumann到Dirichlet映射中恢复各向异性导电率的唯一性。
We consider abstract inverse problems between infinite-dimensional Banach spaces. These inverse problems are typically nonlinear and ill-posed, making the inversion with limited and noisy measurements a delicate process. In this work, we assume that the unknown belongs to a finite-dimensional manifold: this assumption arises in many real-world scenarios where natural objects have a low intrinsic dimension and belong to a certain submanifold of a much larger ambient space. We prove uniqueness and Hölder and Lipschitz stability results in this general setting, also in the case when only a finite…
Fractional Calderón problems and Poincaré inequalities on unbounded domains
本文将分数阶 Calderón 问题的唯一性理论推广至具非空外部的无界区域，刻画部分数据反问题的唯一性与非唯一性，并证明 Poincaré 不等式、Runge 逼近及外部确定结果。
We consider the inverse conductivity problem with discontinuous conductivities. We show in a rigorous way, by a convergence analysis, that one can construct a completely discrete minimization problem whose solution is a good approximation of a solution to the inverse problem. The minimization problem contains a regularization term which is given by a total variation penalization and is characterized by a regularization parameter. The discretization involves at the same time the boundary measurements, by the use of the complete electrode model, the unknown conductivity and the solution to the d…
扫码添加微信好友，提出您的宝贵建议 👇
💡 备注请填写：网站反馈
发表机构由 AI 从论文文本抽取，如发现错误，欢迎反馈修正建议。