John Bondestam Malmberg

Reconstruction of shapes and refractive indices from backscattering experimental data using the adaptivity

We consider the inverse problem of the reconstruction of the spatially distributed dielectric constant epsilon(r)(x), x is an element of R-3, which is an unknown coefficient in the Maxwells equations, from time-dependent backscattering experimental radar data associated with a single source of electric pulses. The refractive index is n(x) = root epsilon(r)(x). The coefficient epsilon(r)(x) is reconstructed using a two-stage reconstruction procedure. In the first stage an approximately globally convergent method proposed is applied to get a good first approximation of the exact solution. In the second stage a locally convergent adaptive finite element method is applied, taking the solution of the first stage as the starting point of the minimization of the Tikhonov functional. This functional is minimized on a sequence of locally refined meshes. It is shown here that all three components of interest of targets can be simultaneously accurately imaged: refractive indices, shapes and locations.

Globally convergent and adaptive finite element methods in imaging of buried objects from experimental backscattering radar measurements

We consider a two-stage numerical procedure for imaging of objects buried in dry sand using time-dependent backscattering experimental radar measurements. These measurements are generated by a single point source of electric pulses and are collected using a microwave scattering facility which was built at the University of North Carolina at Charlotte. Our imaging problem is formulated as the inverse problem of the reconstruction of the spatially distributed dielectric constant e r ( x ) , x ? R 3 , which is an unknown coefficient in Maxwells equations.On the first stage the globally convergent method of Beilina and Klibanov (2012) is applied to get a good first approximation for the exact solution. Results of this stage were presented in Thi?nh et?al. (2014). On the second stage the locally convergent adaptive finite element method of Beilina (2011) is applied to refine the solution obtained on the first stage. The two-stage numerical procedure results in accurate imaging of all three components of interest of targets: shapes, locations and refractive indices. In this paper we briefly describe methods and present new reconstruction results for both stages.

A Posteriori Error Estimate in the Lagrangian Setting for an Inverse Problem Based on a New Formulation of Maxwell’s System

In this paper we consider an inverse problem of determination of a dielectric permittivity function from a backscattered electromagnetic wave. The inverse problem is formulated as an optimal control problem for a certain partial differential equation derived from Maxwell’s system. We study a solution method based on finite element approximation and provide a posteriori error estimate for the use in an adaptive algorithm.

Iterative regularization and adaptivity for an electromagnetic coefficient inverse problem

We study how the choice of the regularization parameter affects the quality of the reconstruction of the dielectric permittivity for an inhomogeneous medium, with data consisting of boundary observations of the electric field. Our method is based on the minimization of a Tikhonov functional and uses a finite element method for computations of the electric field. We conclude that the choice of the regularization parameter does not affect the quality of the reconstruction significantly in the studied cases, and can even be removed with results not significantly different from those with regularization.

Adaptive finite element method for the solution of electromagnetic inverse problem using limited observations

We consider the problem of determination of a dielectric permittivity function from boundary observations of an electric field. We summarize our recent theoretical analysis of an adaptive finite element method for this problem, and present new numerical examples.

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.

Approximate globally convergent algorithm with applications in electrical prospecting

In this paper we present at the first time an approximate globally convergent method for the reconstruction of an unknown conductivity function from backscattered electric field measured at the boundary of geological medium under assumptions that dielectric permittivity and magnetic permeability functions are known. This is the typical case of an coefficient inverse problem in electrical prospecting. We consider a simplified mathematical model assuming that geological medium is isotropic and non-dispersive.