Nguyen Trung Thành

Reconstruction of the Refractive Index from Experimental Backscattering Data Using a Globally Convergent Inverse Method

The problem to be studied in this work is within the context of coefficient identification problems for the wave equation. More precisely, we consider the problem of reconstruction of the refractive index (or equivalently, the dielectric constant) of an inhomogeneous medium using one backscattering boundary measurement. The goal of this paper is to analyze the performance of the globally convergent algorithm of Beilina and Klibanov on experimental data collected using a microwave scattering facility at the University of North Carolina at Charlotte. The main challenge in working with experimental data is the huge misfit between these data and computationally simulated data. We present data preprocessing steps to make the former somehow look similar to the latter. Results of both nonblind and blind targets are shown that indicate good reconstructions even for high contrasts between the targets and the background medium.

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.

Infrared Thermography for Buried Landmine Detection: Inverse Problem Setting

This paper deals with an inverse problem arising in infrared (IR) thermography for buried landmine detection. It is aimed at using a thermal model and measured IR images to detect the presence of buried objects and characterize them in terms of thermal and geometrical properties. The inverse problem is mathematically stated as an optimization one using the well-known least-square approach. The main difficulty in solving this problem comes from the fact that it is severely ill posed due to lack of information in measured data. A two-step algorithm is proposed for solving it. The performance of the algorithm is illustrated using some simulated and real experimental data. The sensitivity of the proposed algorithm to various factors is analyzed. A data processing chain including anomaly detection and characterization is also introduced and discussed.

Reconstruction from blind experimental data for an inverse problem for a hyperbolic equation

We consider the problem of the reconstruction of dielectrics from blind backscattered experimental data. The reconstruction is done from time domain data, as opposed to a more conventional case of frequency domain data. Experimental data were collected using a microwave scattering facility which was built at the University of North Carolina at Charlotte. This system sends electromagnetic pulses into the medium and collects the time-resolved backscattered data on a part of a plane. The spatially distributed dielectric constant epsilon(r)(x), x is an element of R-3 is the unknown coefficient of a wave-like PDE. This coefficient is reconstructed from those data in blind cases. To do this, a globally convergent numerical method is used.

Imaging of buried objects from experimental backscattering time dependent measurements using a globally convergent inverse algorithm

We consider the problem of imaging of objects buried under the ground using experimental backscattering time-dependent measurements generated by a single point source or one incident plane wave. In particular, we estimate dielectric constants of these objects using the globally convergent inverse algorithm of Beilina and Klibanov. Our algorithm is tested on experimental data collected using a microwave scattering facility at the University of North Carolina at Charlotte. There are two main challenges in working with this type of experimental data: (i) there is a huge misfit between these data and computationally simulated data, and (ii) the signals scattered from the targets may overlap with and be dominated by the reflection from the grounds surface. To overcome these two challenges, we propose new data preprocessing steps to make the experimental data look similar to the simulated data, as well as to remove the reflection from the grounds surface. Results of a total of 25 data sets of both nonblind an...

RECOVERING DIELECTRIC CONSTANTS OF EXPLOSIVES VIA A GLOBALLY STRICTLY CONVEX COST FUNCTIONAL

The inverse problem of estimating dielectric constants of explosives using boundary measurements of one component of the scattered electric field is addressed. It is formulated as a coefficient inverse problem for a hyperbolic differential equation. After applying the Laplace transform, a new cost functional is constructed and a variational problem is formulated. The key feature of this functional is the presence of the Carleman weight function for the Laplacian. The strict convexity of this functional on a bounded set in a Hilbert space of an arbitrary size is proven. This allows for establishing the global convergence of the gradient descent method. Some results of numerical experiments are presented.

Finite-Difference Methods and Validity of a Thermal Model for Landmine Detection With Soil Property Estimation

In this paper, we introduce and validate a 3-D linear thermal model for landmine detection. A finite-difference approximation of generalized solutions to the model is proposed, and its convergence properties are proved. An efficient numerical algorithm based on splitting methods is suggested for solving the discretized problem. Moreover, we introduce methods to estimate the (bare) soil and air-soil interface thermal properties. These parameters depend strongly on weather, environmental conditions, and soil type; their accuracy affects strongly the thermal modeling. The validity of the thermal model with the estimated soil properties is verified by comparing the simulations with data sets acquired in outdoor minefields

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.

Detection of point-like scatterers using one type of scattered elastic waves

In this paper, we are concerned with the detection of point-like obstacles using elastic waves. We show that one type of waves, either the P or the S scattered waves, is enough for localizing the points. We also show how the use of S incident waves gives better resolution than the P waves. These affirmations are demonstrated by several numerical examples using a MUSIC type algorithm.

An analysis of the accuracy of the linear sampling method for an acoustic inverse obstacle scattering problem

We investigate the accuracy of the linear sampling method for a two-dimensional acoustic inverse obstacle scattering problem with a Dirichlet boundary condition using asymptotic analysis of the so-called indicator function around the boundary of the obstacle. An asymptotic expansion of the limit, as the noise level and the regularization parameter tend to zero, of the indicator function is obtained. The theoretical results show the dependence of the blowup rate of this limit on the geometrical properties of the obstacle. This partly (up to the above limit) explains the dependence of the accuracy of the linear sampling method on the obstacles geometry. Some numerical results are given to verify the theoretical results.