Some Results on Inverse Scattering

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 Schr\"odinger 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 theory, 9) Born inversion as an ill-posed problem, 10) Inverse obstacle scattering with fixed-frequency data, 11) Inverse scattering with data at a fixed energy and a fixed incident direction, 12) Creating materials with a desired refraction coefficient and wave-focusing properties.Comment: 24p

Inverse obstacle problem for the non-stationary wave equation with an unknown background

We consider boundary measurements for the wave equation on a bounded domain $M \subset \R^2$ or on a compact Riemannian surface, and introduce a method to locate a discontinuity in the wave speed. Assuming that the wave speed consist of an inclusion in a known smooth background, the method can determine the distance from any boundary point to the inclusion. In the case of a known constant background wave speed, the method reconstructs a set contained in the convex hull of the inclusion and containing the inclusion. Even if the background wave speed is unknown, the method can reconstruct the distance from each boundary point to the inclusion assuming that the Riemannian metric tensor determined by the wave speed gives simple geometry in $M$. The method is based on reconstruction of volumes of domains of influence by solving a sequence of linear equations. For \tau \in C(\p M) the domain of influence $M(\tau)$ is the set of those points on the manifold from which the distance to some boundary point $x$ is less than $\tau(x)$.Comment: 4 figure

A hybrid method for inverse scattering for shape and impedance

We present a hybrid method to numerically solve the inverse scattering problem for shape and impedance, given the far-field pattern for one incident direction. This method combines ideas of both iterative and decomposition methods, inheriting the advantages of each of them, such as getting good reconstructions and not needing a forward solver at each step. An optimization problem is presented as the theoretical background of the method and numerical results show its feasibility.FC

Numerical Approximation of Asymptotically Disappearing Solutions of Maxwell's Equations

This work is on the numerical approximation of incoming solutions to Maxwell's equations with dissipative boundary conditions whose energy decays exponentially with time. Such solutions are called asymptotically disappearing (ADS) and they play an importarnt role in inverse back-scatering problems. The existence of ADS is a difficult mathematical problem. For the exterior of a sphere, such solutions have been constructed analytically by Colombini, Petkov and Rauch [7] by specifying appropriate initial conditions. However, for general domains of practical interest (such as Lipschitz polyhedra), the existence of such solutions is not evident. This paper considers a finite-element approximation of Maxwell's equations in the exterior of a polyhedron, whose boundary approximates the sphere. Standard Nedelec-Raviart-Thomas elements are used with a Crank-Nicholson scheme to approximate the electric and magnetic fields. Discrete initial conditions interpolating the ones chosen in [7] are modified so that they are (weakly) divergence-free. We prove that with such initial conditions, the approximation to the electric field is weakly divergence-free for all time. Finally, we show numerically that the finite-element approximations of the ADS also decay exponentially with time when the mesh size and the time step become small.Comment: 15 pages, 3 figure