A. G. Ramm
Let $q(x)$ be real-valued compactly supported sufficiently smooth function. It is proved that the scattering data $A(β,α_0,k)$ $\forall β\in S^2$, $\forall k>0,$ determine $q$ uniquely. Here $α_0\in S^2$ is a fixed direction of the incident plane wave.
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In this paper, we give a positive answer to a longstanding open problem for determining the shape of an obstacle from the knowledge of the far field pattern for the scattering of time-harmonic elastic wave. We show that the elastic far field pattern by an incoming plane wave with a fixed frequency, a fixed incident direction and a fixed polarization determines the obstacle $D$ and the boundary condition on $\partial D$ uniquely. The boundary condition on $\partial D$ is either the Dirichlet, or the Neumann, or the Robin one.
In this paper, we give a positive answer to a challenging open problem for recovering unknown obstacle (which is usually referred to as a scatterer) by acoustic wave probe associated to the Helmholtz equation. We show that the acoustic scattering amplitude $A(β, α_0, k_0)$, known for all $β\in {\mathbb S}^2$, where ${\mathbb {S}}^2$ is the unit sphere in ${\mathbb{R}}^3$, ${α}_0\in {\mathbb{S}}^2$ is fixed, $k_0>0$ is fixed, determines the obstacle $D$ and the boundary condition on $\partial D$ uniquely (The boundary condition on $\partial D$ is either the Dirichlet, or Neumann, or the impedan…
In this paper, we solve a longstanding open problem for determining the shape of an obstacle from the knowledge of the electric (or magnetic) far field pattern for the scattering of time-harmonic electromagnetic field. We show that the electric (or magnetic) far field patten ${\mathbf{E}}^\infty(\boldsymbolβ, {\boldsymbolα}_0, k_0)$ (or ${\mathbf{H}}^\infty (\boldsymbolβ, {\boldsymbolα}_0, k_0)$), known for all $\boldsymbolβ\in {\mathbb S}^2$, where ${\mathbb {S}}^2$ is the unit sphere in ${\mathbb{R}}^3$, ${\boldsymbolα}_0\in {\mathbb{S}}^2$ is fixed, $k_0>0$ is fixed, determines the obstacle…
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.
Uniqueness is proven for two 3-d inverse problems of the determination of the spatially distributed sound speed in the frequency dependent acoustic PDE. The main new point is the assumption that only the modulus of the scattered complex valued wave field is measured on a certain set.
We prove uniqueness in inverse acoustic scattering in the case the density of the medium has an unbounded gradient across $Σ\subseteqΓ=\partialΩ$, where $Ω$ is a bounded open subset of $\mathbb{R}^{3}$ with a Lipschitz boundary. This follows from a uniqueness result in inverse scattering for Schrödinger operators with singular $δ$-type potential supported on the surface $Γ$ and of strength $α\in L^{p}(Γ)$, $p>2$.
Uniqueness of mass-conserving self-similar solutions to Smoluchowski's coagulation equation is shown when the coagulation kernel $K$ is given by $K(x,x\_*)=2(x x\_*)^{-α}$, $(x,x\_*)\in (0,\infty)^2$, for some $α>0$.
In this paper we prove a stable determination of the coefficients of the time-harmonic Maxwell equations from local boundary data. The argument --due to Isakov-- requires some restrictions on the domain.
An inverse problem of finding an obstacle and the boundary condition on its surface from the fixed-energy scattering data is studied. A new method is developed for a proof of the uniqueness results. The method does not use the discreteness of the spectrum of the corresponding Laplacian in a bounded domain. Proof of the uniqueness results is based on the fact that the Hilbert space of square integrable functions is separable.
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 boundary value problems for elliptic equations of second order of determining coefficients by Dirichlet-to-Neumann map on subboundaries, that is, the mapping from Dirichlet data supported on $\partialΩ\setminus Γ_-$ to Neumann data on $\partialΩ\setminus Γ_+$. First we prove uniqueness results in three dimensions under some conditions such as $\bar{Γ_+ \cup Γ_-} = \partialΩ$. Next we survey uniqueness results in two dimensions for various elliptic systems for arbitrarily given $Γ_- = Γ_+$. Our proof is based on complex geometric optics solutions which are constructed by a C…
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.
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.
This paper gives a remark on the Enclosure Method by considering inverse obstacle scattering problems with a single incident wave whose governing equation is given by the Helmholtz equation in two dimensions. It is concerned with the indicator function in the Enclosure Method. The previous indicator function is essentially real-valued since only its absolute value is used. In this paper, another method for the use of the indicator function is introduced. The method employs the logarithmic differential with respect to the independent variable of the indicator function and yields directly the co…
We transform an inverse scattering problem to be an interior transmission problem. We find an inverse uniqueness on the scatterer with a knowledge of a fixed interior transmission eigenvalue. By examining the solution in a series of spherical harmonics at far fields, we can decide the perturbation uniquely for the radially symmetric perturbations.
An inverse scattering problems for the 3-D generalized Helmholtz equation is considered. Only the modulus of the complex valued scattered wave field is assumed to be measured and the phase is not measured. Uniqueness theorem is proved.
This is a review paper of the role of Carleman estimates in the theory of Multidimensional Coefficient Inverse Problems since the first inception of this idea in 1981.
We consider phaseless inverse scattering for the Schrödinger equation with compactly supported potential in dimension $d\ge 2$. We give explicit formulas for solving this problem from appropriate data at high energies. As a corollary, we give also a global uniqueness result for this problem with appropriate data on a fixed energy neighborhood.
This paper is addressed to an inverse stochastic hyperbolic equation with three unknowns, i.e., a source term, an initial displacement and an initial velocity. The global uniqueness is proved by a new global Carleman estimate for the stochastic hyperbolic equation. It is found that both the formulation of stochastic inverse problems and the tools to solve them differ considerably from their deterministic counterpart.
This paper is concerned with uniqueness results in inverse acoustic and electromagnetic scattering problems with phaseless total-field data at a fixed frequency. Motivated by our previous work ({\em SIAM J. Appl. Math. \bf78} (2018), 1737-1753), where uniqueness results were proved for inverse acoustic scattering with phaseless far-field data generated by superpositions of two plane waves as the incident waves at a fixed frequency, in this paper, we use superpositions of two point sources as the incident fields at a fixed frequency and measure the modulus of the acoustic total-field (called ph…
Let $q(x)$ be real-valued compactly supported sufficiently smooth function. It is proved that the scattering data $A(-β,β,k)$ $\forall β\in S^2$, $\forall k>0,$ determine $q$ uniquely.
After publishing his recent paper in SIAM J. Appl. Math, 74, 392-410, 2014 the author has realized that actually he has addressed in that paper, for the first time, a long standing open question being unaware about this. This question is about the uniqueness of a 3-d inverse scattering problem without the phase information. Thus, it makes sense in the current paper to explicitly make the latter statement and to formulate corresponding uniqueness theorems.
We present two uniqueness results for the inverse problem of determining an index of refraction by the corresponding acoustic far-field measurement encoded into the scattering amplitude. The first one is a local uniqueness in determining a variable index of refraction by the fixed incident-direction scattering amplitude. The inverse problem is formally posed with such measurement data. The second one is a global uniqueness in determining a constant refractive index by a single far-field measurement. The arguments are based on the study of certain nonlinear and non-selfadjoint interior transmis…
We consider inverse potential scattering problems where the source of the incident waves is located on a smooth closed surface outside of the inhomogeneity of the media. The scattered waves are measured on the same surface at a fixed value of the energy. We show that this data determines the bounded potential uniquely.
We show that the knowledge of the set of the Cauchy data on the boundary of a bounded open set in $\R^n$, $n\ge 3$, for the magnetic Schrödinger operator with $L^\infty$ magnetic and electric potentials determines the magnetic field and electric potential inside the set uniquely. The proof is based on a Carleman estimate for the magnetic Schrödinger operator with a gain of two derivatives.
We use a new method to prove uniqueness theorem for a coefficient inverse scattering problem without the phase information for the 3-D Helmholtz equation. We consider the case when only the modulus of the scattered wave field is measured and the phase is not measured. The spatially distributed refractive index is the subject of the interest in this problem. Applications of this problem are in imaging of nanostructures and biological cells.
We prove that the electromagnetic material parameters are uniquely determined by boundary measurements for the time-harmonic Maxwell equations in certain anisotropic settings. We give a uniqueness result in the inverse problem for Maxwell equations on an admissible Riemannian manifold, and a uniqueness result for Maxwell equations in Euclidean space with admissible matrix coefficients. The proofs are based on a new Fourier analytic construction of complex geometrical optics solutions on admissible manifolds, and involve a proper notion of uniqueness for such solutions.
We establish various uniqueness results for inverse spectral problems of Sturm-Liouville operators with a finite number of discontinuities at interior points at which we impose the usual transmission conditions. We consider both the case of classical Robin and of eigenparameter dependent boundary conditions.
Let $q(x)$ be real-valued compactly supported sufficiently smooth function, $q\in H^\ell_0(B_a)$, $B_a:=\{x: |x|\leq a, x\in R^3$ . It is proved that the scattering data $A(-β,β,k)$ $\forall β\in S^2$, $\forall k>0$ determine $q$ uniquely. here $A(β,α,k)$ is the scattering amplitude, corresponding to the potential $q$.
Uniqueness of solutions in high-energy x-ray based `eigenstrain tomography' and other inverse eigenstrain problems: Counter examples and necessary conditions for well-posedness
本文研究了高能X射线残差应变断层成像中解的唯一性问题，通过反例和条件分析，证明了在特定条件下残余应力场的唯一重建可能性。
A 1-2 model configuration is a subset of edges of the hexagonal lattice such that each vertex is incident to one or two edges. We prove that for any translation-invariant Gibbs measure of 1-2 model, almost surely the infinite homogeneous cluster is unique.
This paper is concerned with uniqueness in inverse acoustic scattering with phaseless far-field data at a fixed frequency. The main difficulty of this problem is the so-called translation invariance property of the modulus of the far-field pattern generated by one plane wave as the incident field. Based on our previous work (J. Comput. Phys. 345 (2017), 58-73), the translation invariance property of the phaseless far-field pattern can be broken by using infinitely many sets of superpositions of two plane waves as the incident fields at a fixed frequency. In this paper, we prove that the obstac…
It is shown that the operator algebraic setting of local quantum physics leads to a uniqueness proof for the inverse scattering problem. The important mathematical tool is the thermal KMS aspect of wedge-localized operator algebras and its strengthening by the requirement of crossing symmetry for generalized formfactors. The theorem extends properties which were previously seen in d=1+1 factorizing models.
Let $A_q(α',α,k)$ be the scattering amplitude, corresponding to a local potential $q(x)$, $x\in\R^3$, $q(x)=0$ for $|x|>a$, where $a>0$ is a fixed number, $α',α\in S^2$ are unit vectors, $S^2$ is the unit sphere in $\R^3$, $α$ is the direction of the incident wave, $k^2>0$ is the energy. We prove that given an arbitrary function $f(α')\in L^2(S^2)$, an arbitrary fixed $α_0\in S^2$, an arbitrary fixed $k>0$, and an arbitrary small $\ve>0$, there exists a potential $q(x)\in L^2(D)$, where $D\subset R^3$ is a bounded domain such that \bee \|A_q(α',α_0,k)-f(α')\|_{L^2(S^2)}<\ve. \tag{$\ast$}\eee T…
We establish a global uniqueness result for an inverse boundary problem with partial data for the magnetic Schrödinger operator with a magnetic potential of class $W^{1,n}\cap L^\infty$, and an electric potential of class $L^n$. Our result is an extension, in terms of the regularity of the potentials, of the results [16] and [25]. As a consequence, we also show global uniqueness for a partial data inverse boundary problem for the advection-diffusion operator with the advection term of class $W^{1,n}\cap L^\infty$.
Inverse characterization and non-uniqueness of the Gaussian configurational partition function
该研究针对统计力学中构形配分函数的逆问题，证明单阱势下简谐振子是唯一与高斯构形配分函数相容的光滑对称势，多阱势下该唯一性不成立，为构形统计提供了新几何视角。
The inverse Henderson problem of statistical mechanics concerns classical particles in continuous space which interact according to a pair potential depending on the distance of the particles. Roughly stated, it asks for the interaction potential given the equilibrium pair correlation function of the system. In 1974 Henderson proved that this potential is uniquely determined in a canonical ensemble and he claimed the same result for the thermodynamical limit of the physical system. Here we provide a rigorous proof of a slightly more general version of the latter statement using Georgii's versi…
Consider the Schrödinger operator $-\nabla^2+q$ $ $q$, $q=q(x), x \in \mathbf{R}^3$. Let $A(β,α, k)$ be the corresponding scattering amplitude, $k^2$ be the energy, $α\in S^2$ be the incident direction, $β\in S^2$ be the direction of scattered wave, $S^2$ be the unit sphere in $\mathbf{R}^3$. Assume that $k=k_0 >0$ is fixed, and $α=α_0$ is fixed. Then the scattering data are $A(β)= A(β,α_0, k_0)=A_q(β)$ is a function on $S^2$. The following invers$ \textit{IP: Given an arbitrary $f \in L^2(S^2)$ and an arbitrary small number $$ $q \in C_0^{\infty}(D)$, where $D \in \mathbf{R}^3$ is an arbitrar…
A uniqueness result for the inverse problem of identifying boundaries from weighted Radon transform
本文研究了奇数维欧氏空间中的加权Radon变换反问题，证明了确定被积函数不连续曲面的唯一性。
Simultaneous uniqueness and numerical inversion for an inverse problem in the time-domain diffuse optical tomography with fluorescence
本文研究了含荧光的时域漫射光学断层成像中多系数逆问题，通过时间相关边界测量同时恢复多个系数分布，建立了唯一性定理，并引入加速Landweber迭代算法进行数值反演验证。
Uniqueness of Inverse Spectral Problem of Non-Local Sturm-Liouville Operators on Star Graph
本文研究了星形图上带非局部势的 Sturm-Liouville 算子逆谱问题，在顶点引入冻结参数条件模拟网络控制中心，证明了系统在非局部边界条件下的逆唯一性与可解性，并给出了 Weyl 型谱渐近性质。
Uniqueness of MAP estimates for inverse problems under information field theory
本文在信息场论框架下推导反问题最大后验估计的唯一性条件，通过引入模型信任度参数识别模型形式误差，并以泊松方程为例验证了正确模型导致无限信任、误差模型导致信任度降低的规律。
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