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 arbitrary fixed domai$ $||A_q(β)-f(β)||_{L^2(S^2)} < ε$?} A positive answer to this question is given. A method for constructing such a $q$ is proposed. There are infinitely many such $q$, not necessarily real-valued.
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.
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…
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.
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…
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…
This work is concerned with a direct sampling method (DSM) for inverse acoustic scattering problems using far-field data. The method characterizes some unknown obstacles, inhomogeneous media or cracks, directly through an indicator function computed from the measured data. Using one or very few incident waves, the DSM provides quite reasonable profiles of scatterers in time-harmonic inverse acoustic scattering without a priori knowledge of either the physical properties or the number of disconnected components of the scatterer. We shall first derive the DSM using far-field data, then carry out…
Assume a time-harmonic elastic wave is incident onto a penetrable anisotropic body embedded into a homogeneous isotropic background medium. We propose an equivalent variational formulation in a truncated bounded domain and show the uniqueness and existence of weak solutions by applying the Fredholm alternative and using properties of the Dirichlet-to-Neumann map in both two and three dimensions. The Fréchet derivative of the near-field solution operator with respect to the boundary of the scatterer is derived. As an application, we design a descent algorithm for recovering the interface from t…
A review of some of the author's results in the area of inverse scattering is given. The following topics are discussed: 1) Property $C$ and applications, 2) Stable inversion of fixed-energy 3D scattering data and its error estimate, 3) Inverse scattering with ''incomplete`` data, 4) Inverse scattering for inhomogeneous Schrödinger equation, 5) Krein's inverse scattering method, 6) Invertibility of the steps in Gel'fand-Levitan, Marchenko, and Krein inversion methods, 7) The Newton-Sabatier and Cox-Thompson procedures are not inversion methods, 8) Resonances: existence, location, perturbation …
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…
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.
The 3-d inverse scattering problem of the reconstruction of the unknown dielectric permittivity in the generalized Helmholtz equation is considered. The main difference with the conventional inverse scattering problems is that only the modulus of the scattering wave field is measured. The phase is not measured. The initializing wave field is the incident plane wave. On the other hand, in the previous recent works of the authors about the "phaseless topic" the case of the point source was considered [20,21,22]. Two reconstruction procedures are developed for a linearized case. However, the line…
We study a question arising in inverse scattering theory: given a penetrable obstacle, does there exist an incident wave that does not scatter? We show that every penetrable obstacle with real-analytic boundary admits such an incident wave. At zero frequency, we use quadrature domains to show that there are also obstacles with inward cusps having this property. In the converse direction, under a nonvanishing condition for the incident wave, we show that there is a dichotomy for boundary points of any penetrable obstacle having this property: either the boundary is regular, or the complement of…
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…
Wave-based analog computing in the forms of inverse-designed metastructures and the meshes of Mach-Zehnder interferometers (MZI) have recently received considerable attention due to their capability in emulating linear operators, performing vector-matrix multiplication, inverting matrices, and solving integral and differential equations, via electromagnetic wave interaction and manipulation in such structures. Here, we combine these two platforms to propose a wave-based metadevice that can compute scattered fields in electromagnetic forward scattering problems. The proposed device consists of …
The inverse fixed angle problem for operator $Δ^2 u + V(x,|u|) u$ is considered in dimensions $n=2,3$. We prove that the difference between an inverse fixed angle Born approximation and the function $V(\cdot,1)$ is smoother than the function $V$ itself in some Sobolev scale. This allows us to conclude that the main singularities of the perturbation $V$ can be reconstructed from the knowledge of the scattering amplitude with some fixed incident angle.
It is proved that a convex polyhedral scatterer of impedance type can be uniquely determined by the electric far-field pattern of a non-vanishing incident field. The incoming wave is allowed to bean electromagnetic plane wave, a vector Herglotz wave function or a point source wave incited by some magnetic dipole. Our proof relies on the reflection principle for Maxwell's equations with the impedance (or Leontovich) boundary condition enforcing on a hyper-plane. We prove that it is impossible to analytically extend the total field across any vertex of the scatterer. This leads to a data-driven …
