Giovanni Alessandrini, Michele Di Cristo
We deal with the problem of determining an inclusion within an electrical conductor from electrical boundary measurements. Under mild a priori assumptions we establish an optimal stability estimate.
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Using a distributed representation formula of the Gateaux derivative of the Dirichlet to Neumann map with respect to movements of a polygonal conductivity inclusion, [11], we extend the results obtained in [8] proving global Lipschitz stability for the determination of a polygonal conductivity inclusion embedded in a layered medium from knowledge of the Dirichlet to Neumann map.
We consider the problem of determining a polyhedral conductivity inclusion embedded in a homogeneous isotropic medium from boundary measurements. We prove global Lipschitz stability for the polyhedral inclusion from the local Dirichlet-to-Neumann map extending in a highly nontrivial way the results obtained in [18] and [19] in the two-dimensional case to the three-dimensional setting.
Lipschitz Stability in the Simultaneous Determination of Polygonal Inclusions and Constant Conductivities
研究通过边界测量建立未知多边形包含物和常数导率的Lipschitz稳定性结果，利用Dirichlet到Neumann映射进行识别。
In this paper we consider the inverse problem of determining a rigid inclusion inside a thin plate by applying a couple field at the boundary and by measuring the induced transversal displacement and its normal derivative at the boundary of the plate. The plate is made by non-homogeneous linearly elastic material belonging to a general class of anisotropy. For this severely ill-posed problem, under suitable a priori regularity assumptions on the boundary of the inclusion, we prove a stability estimate of log-log type.
In this paper we consider the stability issue for the inverse problem of determining an unknown inclusion contained in an elastic body by all the pairs of measurements of displacement and traction taken at the boundary of the body. Both the body and the inclusion are made by inhomogeneous linearly elastic isotropic material. Under mild a priori assumptions about the smoothness of the inclusion and the regularity of the coefficients, we show that the logarithmic stability estimate proved in \cite{ADiCMR14} in the case of piecewise constant coefficients continues to hold in the inhomogeneous cas…
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…
We consider the inverse problem of identifying an unknown inclusion contained in an elastic body by the Dirichlet-to-Neumann map. The body is made by linearly elastic, homogeneous and isotropic material. The Lamé moduli of the inclusion are constant and different from those of the surrounding material. Under mild a-priori regularity assumptions on the unknown defect, we establish a logarithmic stability estimate. For the proof, we extend the approach used for electrical and thermal conductors in a novel way. Main tools are propagation of smallness arguments based on three-spheres inequality fo…
We deal with the problem of determining a time varying inclusion within a thermal conductor. In particular we study the continuous dependance of the inclusion from the Dirichlet-to-Neumann map. Under a priori regularity assumptions on the unknown defect we establish logarithmic stability estimates.
We derive an exact formula for the volume fraction of an inclusion in a body when the inclusion and the body are linearly elastic materials with the same shear modulus. Our formula depends on an appropriate measurement of the displacement and traction around the boundary of the body. In particular, the boundary conditions around the boundary of the body must be such that they mimic the body being placed in an infinite medium with an appropriate displacement applied at infinity.
Stable determination of an anisotropic inclusion in the Schrödinger equation from local Cauchy data
本文研究利用局部柯西数据反演非均匀各向异性介质中薛定谔方程夹杂物的问题，建立了对数稳定性估计，并给出了基于失配泛函的稳定性结果。
We prove logarithmic stability in the determination of the time-dependent scalar potential in a 1-periodic quantum cylindrical waveguide, from the boundary measurements of the solution to the dynamic Schrödinger equation.
We deal with the inverse scattering problem of determining the surface impedance of a partially coated obstacle. We prove a stability estimate of logarithmic type for the impedance term by the far field measurements.
We prove Hölder type stability estimates near generic simple Riemannian metrics for the inverse problem of recovering such metrics from the Dirichlet-to-Neumann map associated to the wave equation for the Laplace-Beltrami operator.
Stable determination of the initial data in an IBVP for the wave equation outside a non-trapping obstacle
本文研究了非俘获障碍物外波动方程初边值问题中初始数据的确定问题，通过两次局部测量建立了一个双对数稳定性不等式。
We establish a Lipschitz stability inequality for the problem of determining the nonlinear term in a quasilinear elliptic equation by boundary measurements. We give a proof based on a linearization procedure together with special solutions constructed from the fundamental solution of the linearized problem.
We consider the inverse problem of determining the time independent scalar potential of the dynamic Schrödinger equation in an infinite cylindrical domain from one boundary Neumann observation of the solution. We prove Hölder stability by choosing the Dirichlet boundary condition suitably.
We consider the inverse problem of the detection of a single body, immersed in a bounded container filled with a fluid which obeys the Stokes equations, from a single measurement of force and velocity on a portion of the boundary. We obtain an estimate of stability of log-log type.
We study an inverse boundary value problem for the Helmholtz equation using the Dirichlet-to-Neumann map as the data. We consider piecewise constant wavespeeds on an unknown tetrahedral partition and prove a Lipschitz stability estimate in terms of the Hausdorff distance between partitions.
In this paper, we consider the inverse boundary value problem for the polyharmonic operator. We prove that the second order perturbations are uniquely determined by the corresponding Dirichlet to Neumann map. More precisely, we show in dimension $n \geq 3$, a logarithmic type stability estimate for the inverse problem under consideration.
We consider the inverse problem of the detection of a single body, immersed in a bounded container filled with a fluid which obeys the stationary Navier-Stokes equations, from a single measurement of force and velocity on a portion of the boundary. We obtain an estimate of stability of log-log type.
In this work, we study the stable determination of four space-dependent coefficients appearing in a coupled semilinear parabolic system with variable diffusion matrices subject to dynamic boundary conditions which couple intern-boundary phenomena. We prove a Lipschitz stability result for interior and boundary potentials by means of only one observation component, localized in any arbitrary open subset of the physical domain. The proof mainly relies on some new Carleman estimates for dynamic boundary conditions of surface diffusion type.
In this paper we show that the shear modulus $μ$ of an isotropic elastic body can be stably recovered by the knowledge of one single displacement field whose boundary data can be assigned independently of the unknown elasticity tensor.
Stable determination of a rigid scatterer in elastodynamics
针对时谐情形下刚性散射体形状确定的弹性逆散射问题，证明了单次远场测量下的log log型局部稳定性估计，且散射体接近度的先验条件可由Friedrichs不等式的通用常数估计，为该类逆问题的稳定求解提供理论支撑。
In this paper we review some recent results concerning inverse problems for thin elastic plates. The plate is assumed to be made by non-homogeneous linearly elastic material belonging to a general class of anisotropy. A first group of results concerns uniqueness and stability for the determination of unknown boundaries, including the cases of cavities and rigid inclusions. In the second group of results, we consider upper and lower estimates of the area of unknown inclusions given in terms of the work exerted by a couple field applied at the boundary of the plate. In particular, we extend prev…
The multifrequency electrical impedance tomography is considered in order to image a conductivity inclusion inside a homogeneous background medium by injecting one current. An original spectral decomposition of the solution of the forward conductivity problem is used to retrieve the Cauchy data corresponding to the extreme case of perfect conductor. Using results based on the unique continuation we then prove the uniqueness of multifrequency electrical impedance tomography and obtain rigorous stability estimates. Our results in this paper are quite surprising in inverse conductivity problem si…
Stable determination of a time-dependent matrix potential for a wave equation in an infinite waveguide
本文研究无限波导中波动方程含时矩阵势反问题的稳定性，利用几何光学解和光线变换，从输入-输出映射导出了稳定性估计。
We deal with the stability issue for the determination of outgoing time-harmonic acoustic waves from their far-field patterns. We are especially interested in keeping as explicit as possible the dependence of our stability estimates on the wavenumber of the corresponding Helmholtz equation and in understanding the high wavenumber, that is frequency, asymptotics. Applications include stability results for the determination from far-field data of solutions of direct scattering problems with sound-soft obstacles and an instability analysis for the corresponding inverse obstacle problem. The key t…
Stable determination of the potential for the Helmholtz equation in the high frequency limit from boundary measurements
本文在高频极限下通过边界测量稳定确定Helmholtz方程的势，并展示了内部阻抗问题的三对数稳定性。
In this paper, we treat the inverse problem of determining two time-dependent coefficients appearing in a dissipative wave equation, from measured Neumann boundary observations. We establish in dimension $n\geq 2$, stability estimates with respect to the Dirichlet-to-Neumann map of these coefficients provided that are known outside a cloaking regions. Moreover, we prove that it can be stably recovered in larger subsets of the domain by enlarging the set of data
We consider the inverse problem of determining some class of nonlinear terms appearing in an elliptic equation from boundary measurements. More precisely, we study the stability issue for this class of inverse problems. Under suitable assumptions, we prove a Lipschitz and a Hölder stability estimate associated with the determination of quasilinear and semilinear terms appearing in this class of elliptic equations from measurements restricted to an arbitrary parts of the boundary of the domain. Besides their mathematical interest, our stability estimates can be useful for improving numerical re…
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.
In this paper we consider the inverse problem of determining on a compact Riemannian manifold the electric potential or the magnetic field in a Schrödinger equation with Dirichlet data from measured Neumann boundary observations. This information is enclosed in the dynamical Dirichlet-to-Neumann map associated to the magnetic Schrödinger equation. We prove that the knowledge of the Dirichlet-to-Neumann map for the Schrödinger equation uniquely determines the magnetic field and the electric potential and we establish Hölder-type stability.
Stable determination of time-dependent collision kernel in the nonlinear Boltzmann equation
本文研究了维数 $n\ge 2$ 下非线性Boltzmann方程中时间相关碰撞核的逆问题，通过二阶线性化、多元有限差分和光线变换的稳定性，建立了碰撞核的对数型稳定性结果并导出唯一性。
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 …
Stable determination of the first order perturbation of the biharmonic operator from partial data
研究双调和算子在三维及以上领域的一阶扰动的逆边界值问题，通过部分狄利克雷到神经元映射建立对数型稳定性估计。
This paper deals with an inverse problem for a non-self-adjoint Schrödinger equation on a compact Riemannian manifold. Our goal is to stably determine a real vector field from the dynamical Dirichlet-to Neumann map. We establish in dimension n greater than 2, an Hölder type stability estimate for the inverse problem under study. The proof is mainly based on the reduction to an equivalent problem for an electro-magnetic Schrödinger equation and the use of a Carleman estimate designed for elliptic operators.
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.
In this paper we are interested in establishing stability estimates in the inverse problem of determining on a compact Riemannian manifold the electric potential or the conformal factor in a Schrödinger equation with Dirichlet data from measured Neumann boundary observations. This information is enclosed in the dynamical Dirichlet-to-Neumann map associated to the Schrödinger equation. We prove in dimension n bigger than 2 that the knowledge of the Dirichlet-to-Neumann map for the Schrödinger equation uniquely determines the electric potential and we establish Hölder-type stability estimates in…
We consider the Dirichlet Laplacian $A_q=-Δ+q$ in a bounded domain $Ω\subset \mathbb{R}^d$, $d \ge 3$, with real-valued perturbation $q \in L^{\max(2 , 3 d / 5)}(Ω)$. We examine the stability issue in the inverse problem of determining the electric potential $q$ from the asymptotic behavior of the eigenvalues of $A_q$. Assuming that the boundary measurement of the normal derivative of the eigenfunctions is a square summable sequence in $L^2(\partial Ω)$, we prove that $q$ can be Hölder stably retrieved through knowledge of the asymptotics of the eigenvalues
Stable determination of the Winkler subgrade coefficient in a nanoplate
本文研究了边界固定的弹性基础纳米板中Winkler系数的反问题，基于简化应变梯度弹性理论，通过单次内部挠度测量证明了基床系数的全局Hölder稳定性估计。
We consider compact smooth Riemmanian manifolds with boundary of dimension greater than or equal to two. For the initial-boundary value problem for the wave equation with a lower order term $q(t,x)$, we can recover the X-ray transform of time dependent potentials $q(t,x)$ from the dynamical Dirichlet-to-Neumann map in a stable way. We derive Hölder stability estimates for the X-ray transform of $q(t,x)$ which do not require measurements on the entirety of the boundary. The essential technique involved is the Gaussian beam Ansatz, and the proofs are done with the minimal assumptions on the geom…
Simultaneous stable determination of quasilinear terms for parabolic equations
该研究针对抛物方程两类拟线性项的同时恢复反问题，结合线性化、奇异解构造等方法推导了Lipschitz与Hölder稳定性估计，可为数值重构收敛速率分析提供支撑。
The goal of this paper is to prove a stable determination of the coefficients for the time-harmonic Maxwell equations, in a Lipschitz domain, by boundary measurements.
Stable determination of damping and potential coefficients in a semilinear wave equation
针对半线性波动方程，通过二阶线性化与几何光学方法，首次建立了从Dirichlet-to-Neumann映射同时稳定恢复阻尼、线性势和非线性势系数的Hölder型估计。
Stable boundary determination of a complex anisotropic admittivity and its derivatives from a local Neumann-to-Dirichlet map
该研究针对椭圆方程div(σ∇u)=0对应的经典各向异性Calderón问题，从局部Neumann-to-Dirichlet映射建立复各向异性电导率及其任意阶导数的边界稳定性估计，推广了2009年的相关研究成果。
We give new stability estimates for the Gel'fand-Calderon inverse boundary value problem
Stable determination of coefficients for the nonlinear third-order acoustic equations
本文针对广义Jordan-Moore-Gibson-Thompson方程，利用Dirichlet-to-Neumann映射和有限差分线性化及几何光学方法，同时稳定确定摩擦、阻尼、势和非线性系数，并获Hölder型稳定性估计。
The inverse problem of antiplane elasticity on determination of the profiles of $n$ uniformly stressed inclusions is studied. The inclusions are in ideal contact with the surrounding matrix, the stress field inside the inclusions is uniform, and at infinity the body is subjected to antiplane uniform shear. The exterior of the inclusions, an $n$-connected domain, is treated as the image by a conformal map of an $n$-connected slit domain with the slits lying in the same line. The inverse problem is solved by quadratures by reducing it to two Riemann-Hilbert problems on a Riemann surface of genus…
Stable Determination of Coefficients in Nonlinear Dynamical Schrödinger Equations by Carleman Estimates
研究一类含局部解析非线性项的动力学薛定谔方程稳态系数反问题，利用线性薛定谔方程的高阶线性化和卡尔曼估计，在已知边界附近系数及解的诺伊曼数据时，实现系数的稳定唯一确定，还讨论不同边界测量情况。
We investigate a generalization of Calderón's problem of recovering the conductivity coefficient in a conductivity equation from boundary measurements. As a model equation we consider the p-conductivity equation with p strictly between one and infinity, which reduces to the standard conductivity equation when p equals two, and to the p-Laplace equation when the conductivity is constant. The thesis consists of results on the direct problem, boundary determination and detecting inclusions. We formulate the equation as a variational problem also when the conductivity may be zero or infinity in la…
Stable determination of the nonlinear parameter in the non-diffusive Westervelt equation from the Dirichlet-to-Neumann map
该研究针对非扩散Westervelt方程，证明在声速接近参考值且满足几何条件时，可通过Dirichlet-to-Neumann映射稳定恢复非线性参数与声速，并通过数值实验验证，为超声成像等领域的重建方法提供理论支撑。
We are concerned with the time-harmonic elastic scattering due to an inhomogeneous elastic material inclusion located inside a uniformly homogeneous isotropic medium. We establish a sharp stability estimate of logarithmic type in determining the support of the elastic scatterer, independent of its material content, by a single far-field measurement when the support is a convex polyhedral domain in $\mathbb{R}^n$, $n=2,3$. Our argument in establishing the stability result is localized around a corner of the medium scatterer. This enables us to further establish a byproduct result by proving tha…
The aim of the paper is to establish optimal stability estimates for the determination of sound-hard polyhedral scatterers in $\mathbb{R}^N$, $N \geq 2$, by a minimal number of far-field measurements. This work is a significant and highly nontrivial extension of the stability estimates for the determination of sound-soft polyhedral scatterers by far-field measurements, proved by one of the authors, to the much more challenging sound-hard case. The admissible polyhedral scatterers satisfy minimal a priori assumptions of Lipschitz type and may include at the same time solid obstacles and screen-…
The size estimation problem in electrical impedance tomography is considered when the conductivity is a complex number and the body is two-dimensional. Upper and lower bounds on the volume fraction of the unknown inclusion embedded in the body are derived in terms of two pairs of voltage and current data measured on the boundary of the body. These bounds are derived using the translation method. We also provide numerical examples to show that these bounds are quite tight and stable under measurement noise.
We consider the Cauchy problem and the source problem for normally hyperbolic operators on the Minkowski spacetime, and study the determination of solutions from their integrals along null geodesics. For the Cauchy problem, we give a new proof of the stable determination result obtained in Vasy and Wang [12]. For the source problem, we obtain stable determination for sources with space-like singularities. Our proof is based on the microlocal analysis of the normal operator of the light ray transform composed with the parametrix for strictly hyperbolic operators.
Stable determination of time-dependent coefficients in a reaction-diffusion-convection system
本文针对R^{1+n}（n≥2）有界域内反应-扩散-对流系统的逆边值问题，结合Carleman估计与几何光学解，通过部分边界测量得到时变对流系数的双对数稳定性估计，进而得到矩阵值势的三对数稳定性估计。
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。
In this paper, we establish two sharp quantitative results for the direct and inverse time-harmonic acoustic wave scattering. The first one is concerned with the recovery of the support of an inhomogeneous medium, independent of its contents, by a single far-field measurement. For this challenging inverse scattering problem, we establish a sharp stability estimate of logarithmic type when the medium support is a polyhedral domain in $\mathbb{R}^n$, $n=2,3$. The second one is concerned with the stability for corner scattering. More precisely if an inhomogeneous scatterer, whose support has a co…
Stable Determination and Reconstruction of a Quasilinear Term in an Elliptic Equation
本文研究从边界测量确定椭圆方程中拟线性项的反问题，推导了多维和一维情形下的Hölder稳定性估计，并通过数值重构验证了含噪数据下拟线性项的稳定恢复能力。
In this paper, we study an inverse problem of determining the cross section of an infinitely long cylindrical-like material structure from the transverse electromagnetic scattering measurement. We establish a sharp logarithmic stability result in determining a polygonal scatterer by a single far-field measurement. The argument in establishing the stability result is localized around a corner and can be as well used to produce two highly intriguing implications for invisibility and transmission resonance in the wave scattering theory. In fact, we show that if a generic medium scatterer possesse…
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