Uniqueness and factorization method for inverse elastic scattering with a single incoming wave

The first part of this paper is concerned with the uniqueness to inverse time-harmonic elastic scattering from bounded rigid obstacles in two dimensions. It is proved that a connected polygonal obstacle can be uniquely identified by the far-field pattern over all observation directions corresponding to a single incident plane wave. Our approach is based on a new reflection principle for the first boundary value problem of the Navier equation. In the second part, we propose a revisited factorization method to recover a rigid elastic body with a single far-field pattern

The work of Vladimir Maz'ya on integral and pseudodifferential operators

The paper presents an outline of Vladimir Maz'ya's important and influential contributions to the solvability theory of integral and pseudodifferential equations

An optimization method in inverse elastic scattering for one-dimensional grating profiles

Consider the inverse diffraction problem to determine a two-dimensional periodic structure from scattered elastic waves measured above the structure. We formulate the inverse problem as a least squares optimization problem, following the two-step algorithm by G. Bruckner and J. Elschner (Inverse Problems (2003) 19, 315-329) for electromagnetic diffraction gratings. Such a method is based on the Kirsch-Kress optimization scheme and consists of two parts: a linear severely ill-posed problem and a nonlinear well-posed one. We apply this method to both smooth ($C^2$) and piecewise linear gratings for the Dirichlet boundary value problem of the Navier equation. Numerical reconstructions from exact and noisy data illustrate the feasibility of the method

Inverse elastic scattering from rigid scatterers with a single incoming wave

The first part of this paper is concerned with uniqueness to inverse time-harmonic elastic scattering from bounded rigid obstacles in two dimensions. It is proved that a connected polygonal obstacle can be uniquely identified by the far-field pattern corresponding to a single elastic plane wave. Our approach is based on a new reflection principle for the first boundary value problem of the Navier equation. In the second part, we propose a revisited factorization method to recover a rigid elastic body with a single far-field pattern

Inverse scattering for periodic structures: Stability of polygonal interfaces

We consider the two-dimensional TE and TM diffraction problems for a time harmonic plane wave incident on a periodic grating structure. An inverse diffraction problem is to determine the grating profile from measured reflected and transmitted waves away from the structure. We present a new approach to this problem which is based on the material derivative with respect to the variation of the dielectric coefficient. This leads to local stability estimates in the case of interfaces with corner points

Uniqueness in determining polygonal periodic structures

We consider the inverse problem of recovering a two-dimensional perfectly reflecting diffraction grating from scattered waves measured above the structure. We establish the uniqueness within the class of general polygonal grating profiles by a minimal number of incoming plane waves, without excluding Rayleigh frequencies and further geometric constraints on the profile. This extends and improves the uniqueness results of [10]

Uniqueness in determining polygonal sound-hard obstacles with a single incoming wave

We consider the two dimensional inverse scattering problem of determining a sound-hard obstacle by the far field pattern. We establish the uniqueness within the class of polygonal domains by a single incoming plane wave

Elastic scattering by unbounded rough surfaces

We consider the two-dimensional time-harmonic elastic wave scattering problem for an unbounded rough surface, due to an inhomogeneous source term whose support lies within a finite distance above the surface. The rough surface is supposed to be the graph of a bounded and uniformly Lipschitz continuous function, on which the elastic displacement vanishes. We propose an upward propagating radiation condition (angular spectrum representation) for solutions of the Navier equation in the upper half-space above the rough surface, and establish an equivalent variational formulation. Existence and uniqueness of solutions at arbitrary frequency is proved by applying a priori estimates for the Navier equation and perturbation arguments for semi-Fredholm operators

Uniqueness results for an inverse periodic transmission problem

The paper is devoted to the inverse problem of recovering a 2D periodic structure from scattered waves measured above and below the structure. We show that measurements corresponding to a finite number of refractive indices above or below the grating profile, uniquely determine the periodic interface in the inverse TE transmission problem. If a priori information on the height of the diffraction grating is available, then we also obtain upper bounds of the required number of wavenumbers by using the Courant-Weyl min-max principle for a fourth-order elliptic problem. This extends uniqueness results by Hettlich and Kirsch [11] to the inverse transmission problem

A two-step algorithm for the reconstruction of perfectly reflecting periodic profiles

We consider the inverse problem of recovering a 2D periodic structure from scattered waves measured above the structure. First, following [5], the inverse problem is reformulated as an optimization problem which consists of two parts: a linear severely ill-posed problem and a nonlinear well--posed one. Then, contrary to [5], here the two problems are solved separately to diminish the computational effort by exploiting their special properties. Numerical results for exact and noisy data demonstrate the practicability of the inversion algorithm