Mikhail Isaev, Roman Novikov
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.
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We study the impedance boundary map (or Robin-to-Robin map) for the Schrodinger equation in open bounded demain at fixed energy in multidimensions. We give global stability estimates for determining potential from these boundary data and, as corollary, from the Cauchy data set. Our results include also, in particular, an extension of the Alessandrini identity to the case of the impedance boundary map.
Using a conjecture that allows to approach separable-variables conductivity functions, the elements of the Modern Pseudoanalytic Function Theory are used, for the first time, to numerically solve the Dirichlet boundary value problem of the two-dimensional Electrical Impedance Equation, when the conductivity function arises from geometrical figures, located within bounded domains.
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…
M-dissipative generalized impedance boundary conditions, discrete spectra, and pointwise multipliers between fractional Sobolev spaces
本文研究了有界Lipschitz域中m-耗散广义阻抗边界条件下声学算子的性质，证明了其紧预解算子与阻抗算子紧性等价，并应用于谱离散性及波动方程。
Based upon elements of the modern Pseudoanalytic Function Theory, we analyse a new method for numerically approaching the solution of the Dirichlet boundary value problem, corresponding to the two-dimensional Electrical Impedance Equation. The analysis is performed by interpolating piecewise separable-variables conductivity functions, that are eventually used in the numerical calculations in order to obtain finite sets of orthonormal functions, whose linear combinations succeed to approach the imposed boundary conditions. To warrant the effectiveness of the numerical method, we study six diffe…
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.
In electrical impedance tomography, algorithms based on minimizing a linearized residual functional have been widely used due to their flexibility and good performance in practice. However, no rigorous convergence results have been available in the literature yet, and reconstructions tend to contain ringing artifacts. In this work, we shall minimize the linearized residual functional under a linear constraint defined by a monotonicity test, which plays a role of a special regularizer. Global convergence is then established to guarantee that this method is stable under the effects of noise. Mor…
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 …
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…
Maxwell's equations with mixed impedance boundary conditions
本文研究有界Lipschitz域上带矩阵值阻抗边界条件的时谐麦克斯韦方程组，推导Fredholm二择一定理，证明非共振频率下任意源的弱解存在性，并涵盖奇异阻抗系数情况。
The reconstruction problem in electrical impedance tomography is highly ill-posed, and it is often observed numerically that reconstructions have poor resolution far away from the measurement boundary but better resolution near the measurement boundary. The observation can be quantified by the concept of distinguishability of inclusions. This paper provides mathematically rigorous results supporting the intuition. Indeed, for a model problem lower and upper bounds on the distinguishability of an inclusion are derived in terms of the boundary data. These bounds depend explicitly on the distance…
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.
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 i…
On the simultaneous recovery of boundary impedance and internal conductivity
本文研究EIT模型中利用局部内部测量数据同时恢复边界阻抗与内部电导率的反问题，证明了解的唯一性，并提出结合磨光化与Tikhonov正则化的重建方案及误差估计。
Monotonicity based shape reconstruction in electrical impedance tomography
本文基于单调性关系，提出了一种利用电流-电压测量进行形状重建的新方法，通过比较测量值与测试区域的差异来确定区域的外边界。
We study stability of reconstruction in current density impedance imaging (CDII), that is, the inverse problem of recovering the conductivity of a body from the measurement of the magnitude of the current density vector field in the interior of the object. Our results show that CDII is stable with respect to errors in interior measurements of the current density vector field, and confirm the stability of reconstruction which was previously observed in numerical simulations, and was long believed to be the case.
In this paper we consider the electromagnetic scattering problem by an obstacle characterised by a Generalized Impedance Boundary Condition in the harmonic regime. These boundary conditions are well known to provide accurate models for thin layers or imperfectly conducting bodies. We give two different formulations of the scattering problem and we provide some general assumptions on the boundary condition under which the scattering problem has at most one solution. We also prove that it is well-posed for three different boundary conditions which involve second order surface differential operat…
Helmholtz transmission problem and intrinsic impedance scattering problem on extension domains
本文研究了扩展领域中亥姆霍兹方程的传输问题与阻抗散射问题，推广了相关边界算子并讨论了其在不同边界条件下的适定性。
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…
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 …
Simultaneous reconstruction of outer boundary shape and admittivity distribution in electrical impedance tomography
针对电阻抗成像中几何建模不准确的问题，提出一种基于牛顿型输出最小二乘法的算法，利用电流-电压映射关于边界形状的Fréchet导数，同时重建导纳分布和物体形状。
A class of evolutionary operator equations is studied. As an application the equations of linear acoustics are considered with complex material laws. A dynamic boundary condition is imposed which in the time-harmonic case corresponds to an impedance or Robin boundary condition. Memory and delay effects in the interior and also on the boundary are built into the problem class.
On some Impedance Boundary Conditions for a Thermo-Piezo-Electromagnetic System
本文基于已有见解扩展了压电电磁阻抗边界条件，通过引入新方程和边界项计入热影响，并证明了热-压电-电磁系统在该条件下的演化适定性。
The inverse elastic scattering problem in the cavity with impedance boundary
本文研究二维阻抗边界腔体的弹性逆散射问题，证明形状与阻抗参数可由内曲线散射场唯一确定，并用线性采样法和最小二乘法数值重构。
Immersed Boundary Method for the Complete Electrode Model in Electrical Impedance Tomography
本文提出一种求解电阻抗断层成像中完整电极模型的浸入边界格式，该方法无需精确离散边界即可使用笛卡尔网格，适用于复杂或变化的边界，并证明了其收敛性且通过二维正逆问题验证了有效性。
Objective: To present the first 3D CGO-based absolute EIT reconstructions from experimental tank data. Approach: CGO-based methods for absolute EIT imaging are compared to traditional TV regularized non-linear least squares reconstruction methods. Additional robustness testing is performed by considering incorrect model\textbf{}ing of domain shape. Main Results: The CGO-based methods are fast, and show strong robustness to incorrect domain modeling comparable to classic difference EIT imaging and fewer boundary artefacts than the TV regularized non-linear least squares reference reconstruction…
The monotonicity-based approach has become one of the fundamental methods for reconstructing inclusions in the inverse problem of electrical impedance tomography. Thus far the method has not been proven to be able to handle extreme inclusions that correspond to some parts of the studied domain becoming either perfectly conducting or perfectly insulating. The main obstacle has arguably been establishing suitable monotonicity principles for the corresponding Neumann-to-Dirichlet boundary maps. In this work, we tackle this shortcoming by first giving a convergence result in the operator norm for …
A Direct Reconstruction Method for Anisotropic Electrical Impedance Tomography
本文提出一种非迭代、抗噪的直接重建方法，用于解决平面各向异性反导数问题，通过绕过边界上等温坐标值的不稳定重建步骤，利用等温坐标变形恢复唯一的各向同性导电率，为各向异性EIT的正则化铺平道路。
For Maxwell operators $(E,H) \to (i ε^{-1} \nabla\times H, -i μ^{-1} \nabla \times E)$ in Lipschitz domains, we describe all m-dissipative boundary conditions and apply this result to generalized impedance and Leontovich boundary conditions including the cases of singular, degenerate, and randomized impedance coefficients. To this end we construct Riesz bases in the trace spaces associated with the curl-operator and introduce a modified version of boundary triple adapted for the specifics of Maxwell equations, namely, to the mixed-order duality of the related trace spaces. This provides a tran…
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.
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…
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.
We prove quantitative norm bounds for a family of operators involving impedance boundary conditions on convex, polygonal domains. A robust numerical construction of Helmholtz scattering solutions in variable media via the Dirichlet-to-Neumann operator involves a decomposition of the domain into a sequence of rectangles of varying scales and constructing impedance-to-impedance boundary operators on each subdomain. Our estimates in particular ensure the invertibility, with quantitative bounds in the frequency, of the merge operators required to reconstruct the original Dirichlet-to-Neumann opera…
We treat the stability of determining the boundary impedance of an obstacle by scattering data, with a single incident field. A previous result by Sincich (SIAM J. Math. Anal. 38, (2006), 434-451) showed a log stability when the boundary of the obstacle is assumed to be C^{1,1}-smooth. We prove that, when the obstacle boundary is merely Lipschitz, a log-log type stability still holds.
This paper proves the asymptotic stability of the multidimensional wave equation posed on a bounded open Lipschitz set, coupled with various classes of positive-real impedance boundary conditions, chosen for their physical relevance: time-delayed, standard diffusive (which includes the Riemann-Liouville fractional integral) and extended diffusive (which includes the Caputo fractional derivative). The method of proof consists in formulating an abstract Cauchy problem on an extended state space using a dissipative realization of the impedance operator, be it finite or infinite-dimensional. The a…
Wavenumber-explicit analytic regularity of the heterogeneous Maxwell equations with impedance boundary conditions
研究在非零波数下，具有阻抗边界条件的异质Maxwell方程的解析正则性，证明弱解在域内分段解析且导数增长可显式控制。
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…
The present work deals with the resolution of the Linearized Navier-Stokes problem in a domain made of an array that consists into a repetition of elongated resonators connected to an half-space. We provide and justify a limit equivalent model which takes into account the presence of resonators array as an equivalent boundary condition. Our approach combines the method of matched asymptotic expansions and the method of periodic surface homogenization adapted to more than two scales, and a complete justification is included in the paper.
This paper considers the time-harmonic Maxwell equations with impedance boundary condition.We present $H^2$-norm bound and other high-order norm bounds for strong solutions. The $H^2$-estimate have been derived in [M. Dauge, M. Costabel and S. Nicaise, Tech. Rep. 10-09, IRMAR (2010)] for the case with homogeneous boundary condition. Unfortunately, their method can not be applied to the inhomogeneous case. The main novelty of this paper is that we follow the spirit of the $H^1$-estimate in [R. Hiptmair, A. Moiola and I. Perugia, Math. Models Methods Appl. Sci., 21(2011), pp. 2263-2287] and modi…
We present Helmholtz or Helmholtz like equations for the approximation of the time-harmonic wave propagation in gases with small viscosity, which are completed with local boundary conditions on rigid walls. We derived approximative models based on the method of multiple scales for the pressure and the velocity separately, both up to order 2. Approximations to the pressure are described by the Helmholtz equations with impedance boundary conditions, which relate its normal derivative to the pressure itself. The boundary conditions from first order on are of Wentzell type and include a second tan…
Generalized impedance boundary conditions with vanishing or sign-changing impedance
研究带有消失或变号阻抗的广义阻抗边界条件的Laplace问题，证明适定性取决于参数α，α∈[0,1)时Fredholm指标为零，α=1时不适定，并给出数值验证。
Wavenumber-explicit $hp$-FEM analysis of Maxwell's equations with impedance boundary conditions in piecewise smooth media
本文研究了在分段光滑介质中，使用hp-FEM对具有阻抗边界条件的时谐Maxwell方程进行频数显式分析，证明了在满足特定条件时，基于Nédélec元素的伽辽金离散化是准最优的。
Deep D-bar: Real time Electrical Impedance Tomography Imaging with Deep Neural Networks
本文提出利用深度神经网络对电阻抗断层成像进行后处理，以提升图像清晰度和可靠性，通过模拟数据训练网络并应用于实验数据，实现高非线性逆问题的有效解决。
Explicit Robin Green's Functions, Resonance Spectra, and Impedance Recovery on Annuli and Spherical Shells
本文为环形和球壳域上的Helmholtz方程构造显式Robin格林函数，利用加法定理简化共振谱，并通过双线性特征行列式实现阻抗恢复，同时证明谱性质并给出渐近与数值验证。
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…
Two boundary value problems for the Helmholtz equation in a semi-infinite strip are considered. The main feature of these problems is that, in addition to the function and its normal derivative on the boundary, the functionals of the boundary conditions possess tangential derivatives of the second and fourth orders. Also, the setting of the problems is complimented by certain edge conditions at the two vertices of the semi-strip. The problems model wave propagation in a semi-infinite waveguide with membrane and plate walls. A technique for the exact solution of these fluid-structure interactio…
Continuity of the linearized forward map of electrical impedance tomography from square-integrable perturbations to Hilbert-Schmidt operators
本文证明二维电阻抗断层成像正演映射的Fréchet导数在常背景电导率、$C^{1,α}$边界单连通区域条件下，从平方可积扰动空间到边界希尔伯特-施密特算子空间有界，为无穷维线性化一步重建算法提供理论支撑。
Stable determination of an impedance obstacle by a single far-field measurement
该论文针对二维空间中一般多边形阻抗障碍物，利用单个远场测量建立了对数型稳定性估计，首次解决了该逆散射问题，并提出了微局部分析和误差传播等新技术。
Graph convolutional networks enable fast hemorrhagic stroke monitoring with electrical impedance tomography
本研究提出用graph U-net后处理线性差分成像的电阻抗断层成像重建方法，可在极低计算成本下达到甚至优于耗时非线性方法的图像质量，且能以2D训练处理3D数据，为出血性卒中在线监测提供可行方案。
A boundary integral equation method for the complete electrode model in electrical impedance tomography with tests on experimental data
本文提出一种基于边界积分方程的数值方法，用于求解完整电极模型下的二维静电势问题，并开发了以此为核心的电阻抗断层成像逆求解器，在实验数据上验证了其准确性与适用性。
Breast Cancer Detection through Electrical Impedance Tomography and Optimal Control Theory: Theoretical and Computational Analysis
本文研究了利用电阻抗断层成像和最优控制理论检测乳腺癌的问题，提出了一种基于Besov空间的PDE约束最优控制框架，通过数值分析验证了最优控制的存在性和Fréchet可微性。
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…
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…
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.
Interpolation of missing electrode data in electrical impedance tomography
本文提出了一种新的插值方法，用于从更可靠的电流自由电极测量中预测电流驱动电极上的电压，以在差分电阻抗断层成像设置中恢复导电性变化。该方法利用导电性变化的上界，以与差分电阻抗断层成像数据的特殊几何光滑性一致的方式进行插值，且计算成本低，适合实时应用。
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 …
In this paper, we will discuss the use of a Sampling Method to reconstruct impenetrable inclusions from Electrostatic Cauchy data. We consider the case of a perfectly conducting and impedance inclusion. In either case, we show that the Dirichlet to Neumann mapping can be used to reconstruct impenetrable sub-regions via a sampling method. We also propose a non-iterative method based on boundary integral equations to reconstruct the impedance parameter using the reconstructed boundary of the inclusion from the knowledge of multiple Cauchy pairs which can be computed from Dirichlet to Neumann map…
This paper is devoted to the study of the generalized impedance boundary conditions (GIBCs) for a strongly absorbing obstacle in the {\bf time} regime in two and three dimensions. The GIBCs in the time domain are heuristically derived from the corresponding conditions in the time harmonic regime. The latters are frequency dependent except the one of order 0; hence the formers are non-local in time in general. The error estimates in the time regime can be derived from the ones in the time harmonic regime when the frequency dependence is well-controlled. This idea is originally due to Nguyen and…
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…
A variational Lippmann-Schwinger-type approach for the Helmholtz impedance problem on bounded domains
本文针对有界区域上带阻抗边界条件的 Helmholtz 反边值问题，推导了变分 Lippmann-Schwinger 型方程，证明参数到状态映射的弱到强序列连续性，并据此建立降阶模型优化与波形反演方法中最小化元的存在性。
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