Allan Greenleaf, Matti Lassas, Gunther Uhlmann
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.
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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.
This paper consists of two parts. In the first part we describe the recent works on the inverse problems for the wave equation in $(n+1)$-dimensional space equipped with pseudo-Riemannian metric with Lorentz signature. We study the conditions of the existence of black (or white) holes for these wave equations. In the second part we prove energy type estimates on a finite time interval in the presence of black or white holes. We use these estimates to prove the nonuniqueness of the inverse problems.
The inverse problem of finding the velocity profile from the surface measurements of acoustic field is considered. The dimensionalities of the data and of the unknown velocity profile are equal, both are equal 3. Analytical explicit examples are given showing vast non-uniqueness of the solution to this inverse 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…
This paper studies uniqueness and nonuniqueness for potential reconstruction from one boundary measurement in quantum fields, associated with the steady state Schrödinger equation. A uniqueness theorem of the inverse problem is established. In the meanwhile, a nonuniqueness theorem is also given when different potential and shape are considered. Finally, Tikhonov regularization method is applied to solve the reconstruction problem, and some numerical examples are presented to confirm the theoretical results and the effectiveness of the proposed method.
Calderón's inverse problem via Vekua theory
本文通过维库亚理论在克利福德分析框架下证明了卡勒顿反问题的唯一性结果。
We prove a global logarithmic stability estimate for the Gel'fand-Calderon inverse problem on a two-dimensional domain.
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.
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 $W^{α,p}$ $α>0, 1<p<\infty$.
Uniqueness and nonuniqueness in the disjoint data inverse problem: a sharp spectral threshold
该研究探讨了从波观测确定流形的反问题，证明了在次临界谱下界条件下唯一性成立，并构造了临界端点处唯一性失效的反例。
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 new global stability estimates for the Gel'fand-Calderon inverse problem in 3D. For sufficiently regular potentials this result of the present work is a principal improvement of the result of [G. Alessandrini, Stable determination of conductivity by boundary measurements, Appl. Anal. 27 (1988), 153-172].
We prove a global logarithmic stability estimate for the multi-channel Gel'fand-Calderón inverse problem on a two-dimensional bounded domain, i.e. the inverse boundary value problem for the equation $-Δψ+ v\, ψ= 0$ on $D$, where $v$ is a smooth matrix-valued potential defined on a bounded planar domain $D$.
We prove a local Lipschitz stability estimate for Gel'fand-Calderón's inverse problem for the Schrödinger equation. The main novelty is that only a finite number of boundary input data is available, and those are independent of the unknown potential, provided it belongs to a known finite-dimensional subspace of $L^\infty$. A similar result for Calderón's problem is obtained as a corollary. This improves upon two previous results of the authors on several aspects, namely the number of measurements and the stability with respect to mismodeling errors. A new iterative reconstruction scheme based …
In this paper we prove log log type stability estimates for inverse boundary value problems on admissible Riemannian manifolds of dimension $n \geq 3$. The stability estimates correspond to a couple of uniqueness results by Dos Santos Ferrera, Kenig, Salo and Uhlmann. These inverse problems arise naturally when studying the anisotropic Calderón problem.
We prove that an $L^\infty$ potential in the Schrödinger equation in three and higher dimensions can be uniquely determined from a finite number of boundary measurements, provided it belongs to a known finite dimensional subspace $\mathcal W$. As a corollary, we obtain a similar result for Calderón's inverse conductivity problem. Lipschitz stability estimates and a globally convergent nonlinear reconstruction algorithm for both inverse problems are also presented. These are the first results on global uniqueness, stability and reconstruction for nonlinear inverse boundary value problems with f…
A Calderón type inverse problem for the active scalar equations with fractional dissipation
针对环面上带分数阶耗散的主动标量方程反问题，采用二阶线性化关联线性分数阶扩散方程，利用分数阶Laplacian唯一延拓性等非局部性质求解，揭示无散结构的关键作用。
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…
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。
By developing the d-bar approach to global "inverse scattering" at zero energy we give a principal effectivization of the global reconstruction method for the Gel'fand-Calderon inverse boundary value problem in three dimensions. This work goes back to results published by the author in 1987, 1988 and proceeds from recent progress in the d-bar approach to inverse scattering in 3D published by the author in 2005, 2006.
We prove that uniqueness for the Calderón problem on a Riemannian manifold with boundary follows from a hypothetical unique continuation property for the elliptic operator $Δ+V+(Λ^{1}_{t}-q)\otimes (Λ^{2}_{t}-q)$ defined on $\partial\mathcal{M}^{2}\times [0,1]$ where $V$ and $q$ are potentials and $Λ^{i}_{t}$ is a Dirichlet-Neumann operator at depth $t$. This is done by showing that the difference of two Dirichlet-Neumann maps is equal to the Neumann boundary values of the solution to an inhomogeneous equation for said operator, where the source term is a measure supported on the diagonal of $…
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…
On the interplay between inverse scattering for asymptotically hyperbolic manifolds and the Calderón problem for the Conformal Laplacian
本文通过渐近双曲流形与Conformal Laplacian的关系，证明了特定能量下的散射矩阵能确定边界度量的jet结构。
We are concerned with the Calderón inverse inclusion problem, where one intends to recover the shape of an inhomogeneous conductive inclusion embedded in a homogeneous conductivity by the associated boundary measurements. We consider the highly challenging case with a single partial boundary measurement, which constitutes a long-standing open problem in the literature. It is shown in several existing works that corner singularities can help to resolve the uniqueness and stability issues for this inverse problem. In this paper, we show that the corner singularity can be relaxed to be a certain …
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 survey recent results on Calderon's inverse problem with partial data, focusing on three and higher dimensions.
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…
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 investigate inverse problems in the determination of leading coefficients for nonlocal parabolic operators, by knowing the corresponding Cauchy data in the exterior space-time domain. The key contribution is that we reduce nonlocal parabolic inverse problems to the corresponding local inverse problems with the lateral boundary Cauchy data. In addition, we derive a new equation and offer a novel proof of the unique continuation property for this new equation. We also build both uniqueness and non-uniqueness results for both nonlocal isotropic and anisotropic parabolic Calderón problems, resp…
On a fixed smooth compact Riemann surface with boundary $(M_0,g)$, we show that the Cauchy data space (or Dirichlet-to-Neumann map $\mc{N}$) of the Schrödinger operator $Δ+V$ with $V\in C^2(M_0)$ determines uniquely the potential $V$. We also discuss briefly the corresponding consequences for potential scattering at 0 frequency on Riemann surfaces with asymptotically Euclidean or asymptotically hyperbolic ends.
Boundary Control and Calderón type Inverse Problems in Non-local heat equation
本文研究了非局部热方程的边界控制与Calderón型反问题，利用Pohozaev恒等式分析解的性质及势函数的恢复。
We study the multi-channel Gel'fand-Calderón inverse problem in two dimensions, i.e. the inverse boundary value problem for the equation $-Δψ+ v(x) ψ= 0$, $x\in D$, where $v$ is a smooth matrix-valued potential defined on a bounded planar domain $D$. We give an exact global reconstruction method for finding $v$ from the associated Dirichlet-to-Neumann operator. This also yields a global uniqueness results: if two smooth matrix-valued potentials defined on a bounded planar domain have the same Dirichlet-to-Neumann operator then they coincide.
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。
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 study the inverse problem of recovering a tree graph together with the weights on its edges (equivalently a metric tree) from the knowledge of the Dirichlet-to-Neumann matrix associated with the Laplacian. We prove an explicit formula which relates this matrix to the pairwise weighted distances of the leaves of the tree and, thus, allows to recover the weighted tree. This result can be viewed as a counterpart of the Calderón problem in the analysis of PDEs. In contrast to earlier results on inverse problems for metric graphs, we only assume knowledge of the Dirichlet-to-Neumann matrix for a…
In this paper, we use the theory of symmetric Dirichlet forms to give a probabilistic interpretation of Calderón's inverse conductivity problem in terms of reflecting diffusion processes and their corresponding boundary trace processes.
Inverse problems and invisibility cloaking for FEM models and resistor networks
本文研究了电阻网络和FEM模型的反问题，探讨了离散逆问题的非唯一性及隐身伪装技术的应用。
We show global uniqueness in an inverse problem for the fractional Schrödinger equation: an unknown potential in a bounded domain is uniquely determined by exterior measurements of solutions. We also show global uniqueness in the partial data problem where the measurements are taken in arbitrary open, possibly disjoint, subsets of the exterior. The results apply in any dimension $\geq 2$ and are based on a strong approximation property of the fractional equation that extends earlier work. This special feature of the nonlocal equation renders the analysis of related inverse problems radically d…
On a fixed smooth compact Riemann surface with boundary $(M_0,g)$, we show that for the Schrödinger operator $Δ+V$ with potential $V\in C^{1,α}(M_0)$ for some $α>0$, the Dirichlet-to-Neumann map $N|_Γ$ measured on an open set $Γ\subset \partial M_0$ determines uniquely the potential $V$. We also discuss briefly the corresponding consequences for potential scattering at 0 frequency on Riemann surfaces with asymptotically Euclidean or asymptotically hyperbolic ends.
Remarks on the anisotropic Calderón problem
本文研究各向异性Calderón问题，证明了横截各向异性流形上该问题的唯一性，给出一般黎曼流形上Dirichlet-to-Neumann映射值域在零势附近的凸性，并得到半线性椭圆方程对应的Calderón型反问题的相关结果。
The partial data Calderón problem in dimension three
针对三维稳态Schrödinger方程，证明边界点附近的局部Dirichlet-to-Neumann映射唯一确定该点邻域内的势函数，通过加权X射线变换将问题归结为积分几何。
The inverse problem for the fractional conductivity equation: a survey
本文综述了分数阶Calderón问题，重点收集了与其电导率方程表述相关的最新研究成果。
Entanglement principle and fractional Calderón problem for nonlocal parabolic operators
本文针对变系数非局部抛物算子的反问题，提出全新纠缠原理并证明可通过狄利克雷-诺伊曼映射唯一确定低阶扰动，还推导修正版纠缠原理解决直接应用时解正则性不足的问题。
Reconstruction of cracks in Calderón's inverse conductivity problem using energy comparisons
针对Calderón反电导率问题中的Lipschitz超曲面型裂纹，提出两种重构方法：一种可通过算子不等式收缩上界得到精确位置，能同时处理两类裂纹；另一种针对单一类型裂纹生成保证为未知裂纹子集的近似结果，填补了相关严格重构方法的空白。
In this paper, we introduce an inverse problem of a Schrödinger type variable nonlocal elliptic operator $(-\nabla\cdot(A(x)\nabla))^{s}+q)$, for $0<s<1$. We determine the unknown bounded potential $q$ from the exterior partial measurements associated with the nonlocal Dirichlet-to-Neumann map for any dimension $n\geq2$. Our results generalize the recent initiative [16] of introducing and solving inverse problem for fractional Schrödinger operator $((-Δ)^{s}+q)$ for $0<s<1$. We also prove some regularity results of the direct problem corresponding to the variable coefficients fractional differ…
The Calderón's problem via DeepONets
针对电导率反问题中的Dirichlet-to-Neumann算子及正反卡尔德隆映射，研究证明其可通过深度学习模型DeepONets严格逼近，为无限维映射求解提供新途径。
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…
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…
In this article we consider direct and inverse problems for $α$-stable, elliptic nonlocal operators whose kernels are possibly only supported on cones and which satisfy the structural condition of \emph{directional antilocality} as introduced in \cite{I86}. We consider the Dirichlet problem for these operators on the "domain of dependence of the operator" and in several, adapted function spaces. This formulation allows one to avoid natural "gauges" which would else have to be considered in the study of the associated inverse problems. Exploiting the directional antilocality of these operators …
We prove a new global stability estimate for the Gel'fand-Calderón inverse problem on a two-dimensional bounded domain or, more precisely, the inverse boundary value problem for the equation $-Δψ+ v\, ψ= 0$ on $D$, where $v$ is a smooth real-valued potential of conductivity type defined on a bounded planar domain $D$. The principal feature of this estimate is that it shows that the more a potential is smooth, the more its reconstruction is stable, and the stability varies exponentially with respect to the smoothness (in a sense to be made precise). As a corollary we obtain a similar estimate f…
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.
We outline an approach to the inverse problem of Calderón that highlights the role of microlocal normal forms and propagation of singularities and extends a number of earlier results also in the anisotropic case. The main result states that from the boundary measurements it is possible to recover integrals of the unknown coefficient over certain two-dimensional manifolds called good bicharacteristic leaves. This reduces the Calderón problem into solving a linear integral geometry problem (inversion of a bicharacteristic leaf transform).
The Born approximation for the fixed energy Calderón problem
本文研究了固定能量下Calderón问题的Born近似，证明其作为径向分布的良定义性，并展示了其在恢复势中的应用及高能极限下的收敛性。
Local Hölder Stability in the Inverse Steklov and Calderón Problems for Radial Schrödinger operators and Quantified Resonances
本文针对单位球上特殊 $L^2$ 径向势的薛定谔算子，利用 Steklov 频谱差的 Laplace 变换公式与矩稳定性估计，获得了逆 Steklov 与 Calderón 问题的局部 Hölder 稳定性，改进了先前的对数稳定性结果。
In this paper we show uniqueness of the conductivity for the quasilinear Calderón's inverse problem. The nonlinear conductivity depends, in a nonlinear fashion, of the potential itself and its gradient. Under some structural assumptions on the direct problem, a real-valued conductivity allowing a small analytic continuation to the complex plane induces a unique Dirichlet-to-Neumann (DN) map. The method of proof considers some complex-valued, linear test functions based on a point of the boundary of the domain, and a linearization of the DN map placed at these particular set of solutions.
The Calderón problem for the fractional wave equation: Uniqueness and optimal stability
通过时空域外部测量，利用分数阶Laplacian的唯一延拓性质，研究了带势分数阶波动方程反问题的唯一性与最优对数型稳定性。
We prove uniqueness and stability for the inverse boundary value problem of the two dimensional Schrödinger equation. We do not assume the potentials to be continuous or even bounded. Instead, we assume that some of their positive fractional derivatives are in a specific Lorentz space. These spaces are a natural generalization to the usual fractional Sobolev spaces. The thesis consists of two parts. In the first part, we define the generalized fractional Sobolev spaces and prove some of their properties including embeddings and interpolation identities. In particular we sharpen the usual Sobol…
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 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.
This work derives explicit series reversions for the solution of Calderón's problem. The governing elliptic partial differential equation is $\nabla\cdot(A\nabla u)=0$ in a bounded Lipschitz domain and with a matrix-valued coefficient. The corresponding forward map sends $A$ to a projected version of a local Neumann-to-Dirichlet operator, allowing for the use of partial boundary data and finitely many measurements. It is first shown that the forward map is analytic, and subsequently reversions of its Taylor series up to specified orders lead to a family of numerical methods for solving the inv…
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