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We consider the 3D inverse scattering problem with non-over-determined scattering data. The data are the scattering amplitude $A(\beta, \alpha_0, k)$ for all $\beta \in S_\beta^2$, where $S_\beta^2$ is an open subset of the unit sphere…

Numerical Analysis · Mathematics 2017-02-02 C. Van

In this paper we consider the inverse scattering problem for the Schr{\"o}dinger operator with short-range electric potential. We prove in dimension n $\geq$ 2 that the knowledge of the scattering operator determines the electric potential…

Analysis of PDEs · Mathematics 2018-09-07 Luc Robbiano , Mourad Bellassoued

Consider the scattering of the two- or three-dimensional Helmholtz equation where the source of the electric current density is assumed to be compactly supported in a ball. This paper concerns the stability analysis of the inverse source…

Analysis of PDEs · Mathematics 2016-07-26 Peijun Li , Ganghua Yuan

We obtain a nonperturbative, analytical solution to integral equation of scattering theory by assuming the field within the scattering object is a spherical wave with a scattering amplitude equal to that of the far field. This approximation…

Classical Physics · Physics 2018-09-26 Brian Slovick , Srini Krishnamurthy

In this paper we study the continuous dependence with respect to obstacles for obstacle problems with measure data. This is deeply investigated introducing a suitable type of convergence, which gives stability under very general hypotheses.…

Functional Analysis · Mathematics 2007-05-23 Paolo Dall'Aglio

We consider an inverse scattering problem and its near-field approximation at high frequencies. We first prove, for both problems, Lipschitz stability results for determining the low-frequency component of the potential. Then we show that,…

Analysis of PDEs · Mathematics 2012-05-31 Habib Ammari , Hajer Bahouri , David Dos Santos Ferreira , Isabelle Gallagher

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…

Analysis of PDEs · Mathematics 2013-08-28 Guanghui Hu , Hongyu Liu

Let $A_q(\alpha',\alpha,k)$ 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, $\alpha',\alpha\in S^2$ are unit vectors, $S^2$ is the unit sphere in…

Mathematical Physics · Physics 2016-09-07 A. G. Ramm

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…

Analysis of PDEs · Mathematics 2021-04-20 Genqian Liu

We consider the inverse scattering problem at fixed and sufficiently large energy for the nonrelativistic and relativistic Newton equation in $\R^n$, $n \ge 2$, with a smooth and short range electromagnetic field $(V,B)$. Using results of…

Mathematical Physics · Physics 2012-10-25 Alexandre Jollivet

This paper considers the inverse problem of scattering of time-harmonic acoustic and electromagnetic plane waves by a bounded, inhomogeneous, penetrable obstacle with embedded objects inside. A new method is proposed to prove that the…

Analysis of PDEs · Mathematics 2017-06-14 Jiaqing Yang , Bo Zhang , Haiwen Zhang

In this paper, we will study increasing stability in the inverse source problem for the Helmholtz equation in the plane when the source term is assumed to be compactly supported in a bounded domain $\Omega$ with sufficiently smooth…

Analysis of PDEs · Mathematics 2018-04-18 Mozhgan Nora Entekhabi , Victor Isakov

We develop two inverse scattering schemes for locating multiple electromagnetic (EM) scatterers by the electric far-field measurement corresponding to a single incident/detecting plane wave. The first scheme is for locating scatterers of…

Analysis of PDEs · Mathematics 2013-01-28 Jingzhi Li , Hongyu Liu , Zaijiu Shang , Hongpeng Sun

This paper is mainly concerned with the inverse scattering problem of determining the unknown potential for the classical Schr\"odinger equation in two and three dimensions. We establish the increasing stability of the inverse scattering…

Analysis of PDEs · Mathematics 2023-06-21 Jian Zhai , Yue Zhao

In this work we study the increasing resolution of linear inverse scattering problems at a large fixed frequency. We consider the problem of recovering the density of a Herglotz wave function, and the linearized inverse scattering problem…

Analysis of PDEs · Mathematics 2025-03-07 Pu-Zhao Kow , Mikko Salo , Sen Zou

We study an inverse random obstacle scattering problems in $\mathbb{R}^2$ where the scatterer is formulated by a Gaussian process defined on the angular parameter domain. Equipped with a modified covariance function which is mathematically…

Numerical Analysis · Mathematics 2026-02-02 Zhiqi Sun , Xiang Xu , Yiwen Lin

We analyze the inverse problem to reconstruct the shape of a three dimensional homogeneous dielectric obstacle from the knowledge of noisy far field data. The forward problem is solved by a system of second kind boundary integral equations.…

Numerical Analysis · Mathematics 2020-06-22 Thorsten Hohage , Frédérique Le Louër

We consider the scattering problem governed by the Helmholtz equation with inhomogeneity in both `conductivity' in the divergence form and `potential' in the lower order term. The support of the inhomogeneity is assumed to contain a convex…

Analysis of PDEs · Mathematics 2020-09-14 Fioralba Cakoni , Jingni Xiao

It is proved that a general polyhedral perfect conducting obstacle in $\mathbb{R}^3$, possibly consisting of finitely many solid polyhedra, is uniquely determined by the far-field pattern corresponding to a single incident wave. This…

Analysis of PDEs · Mathematics 2009-11-13 Hongyu Liu

This paper is devoted to the algorithmic development of inverse elastic scattering problems. We focus on reconstructing the locations and shapes of elastic scatterers with known dictionary data for the nearly incompressible materials. The…

Analysis of PDEs · Mathematics 2017-11-02 Li Jingzhi , Liu Hongyu , Sun Hongpeng