Inverse scattering at fixed energy on surfaces with Euclidean ends

On a fixed Riemann surface $(M_0,g_0)$ with $N$ Euclidean ends and genus $g$, we show that, under a topological condition, the scattering matrix S_V(\la) at frequency \la > 0 for the operator $\Delta+V$ determines the potential $V$ if $V\in C^{1,\alpha}(M_0)\cap e^{-\gamma d(\cdot,z_0)^j}L^\infty(M_0)$ for all $\gamma>0$ and for some $j\in\{1,2\}$, where $d(z,z_0)$ denotes the distance from $z$ to a fixed point $z_0\in M_0$. The topological condition is given by $N\geq\max(2g+1,2)$ for $j=1$ and by $N\geq g+1$ if $j=2$. In \rr^2 this implies that the operator S_V(\la) determines any $C^{1,\alpha}$ potential $V$ such that $V(z)=O(e^{-\gamma|z|^2})$ for all $\gamma>0$.Comment: 21 page

Limiting Carleman weights and anisotropic inverse problems

In this article we consider the anisotropic Calderon problem and related inverse problems. The approach is based on limiting Carleman weights, introduced in Kenig-Sjoestrand-Uhlmann (Ann. of Math. 2007) in the Euclidean case. We characterize those Riemannian manifolds which admit limiting Carleman weights, and give a complex geometrical optics construction for a class of such manifolds. This is used to prove uniqueness results for anisotropic inverse problems, via the attenuated geodesic X-ray transform. Earlier results in dimension $n \geq 3$ were restricted to real-analytic metrics.Comment: 58 page

Methods of quantitative reconstruction of shapes and refractive indices from experimental data

In this chapter we summarize results of [5, 6, 14] and present new results of reconstruction of refractive indices and shapes of objects placed in the air from blind backscattered experimental data using two-stage numerical procedure of [4]. Data are collected using a microwave scattering facility which was built at the University of North Carolina at Charlotte.</br></br> On the first stage the approximately globally convergent method of [4] is applied to get a good first approximation for the exact solution. Results of this stage are presented in [5, 14]. On the second stage the local adaptive finite element method of [1] is applied to refine the solution obtained on the first stage. In this chapter we briefly describe methods and present new results for both stages

Electrical Impedance tomography

Electrical Impedance Tomography (EIT) is the recovery of the conductivity (or conductivity and permittivity) of the interior ofa body from a knowledge of currents and voltages applied to its surface