Loc Hoang Nguyen

Bounds on the volume fraction of 2-phase, 2-dimensional elastic bodies and on (stress, strain) pairs in composites

Abstract Bounds are obtained on the volume fraction in a two-dimensional body containing two elastically isotropic materials with known bulk and shear moduli. These bounds use information about the average stress and strain fields, energy, determinant of the stress, and determinant of the displacement gradient, which can be determined from measurements of the traction and displacement at the boundary. The bounds are sharp if in each phase certain displacement gradient field components are constant. The inequalities we obtain also directly give bounds on the possible (average stress, average strain) pairs in a two-phase, two-dimensional, periodic or statistically homogeneous composite.

Numerical solution of a coefficient inverse problem with multi-frequency experimental raw data by a globally convergent algorithm

Abstract We analyze in this paper the performance of a newly developed globally convergent numerical method for a coefficient inverse problem for the case of multi-frequency experimental backscatter data associated to a single incident wave. These data were collected using a microwave scattering facility at the University of North Carolina at Charlotte. The challenges for the inverse problem under the consideration are not only from its high nonlinearity and severe ill-posedness but also from the facts that the amount of the measured data is minimal and that these raw data are contaminated by a significant amount of noise, due to a non-ideal experimental setup. This setup is motivated by our target application in detecting and identifying explosives. We show in this paper how the raw data can be preprocessed and successfully inverted using our inversion method. More precisely, we are able to reconstruct the dielectric constants and the locations of the scattering objects with a good accuracy, without using any advanced a priori knowledge of their physical and geometrical properties.

Reconstruction of a piecewise smooth absorption coefficient by an acousto-optic process ∗

The aim of this paper is to tackle the nonlinear optical reconstruction problem. Given a set of acousto-optic measurements, we develop a mathematical framework for the reconstruction problem in the case where the optical absorption distribution is supposed to be a perturbation of a piecewise constant function. Analyzing the acousto-optic measurements, we prove that the optical absorption coefficient satisfies, in the sense of distributions, a new equation. For doing so, we introduce a weak Helmholtz decomposition and interpret in a weak sense the cross-correlation measurements using the spherical Radon transform. We next show how to find an initial guess for the unknown coefficient. Finally, in order to construct the true coefficient we provide a Landweber type iteration and prove that the resulting sequence converges to the solution of the system constituted by the optical diffusion equation and the new equation mentioned above. Our results in this paper generalize the acousto-optic process proposed in [5] for piecewise smooth optical absorption distributions.

A reconstruction algorithm for ultrasound-modulated diffuse optical tomography

The aim of this paper is to develop an efficient reconstruction algorithm for ultrasound-modulated diffuse optical tomography. In diffuse optical imaging, the resolution is in general low. By mechanically perturbing the medium, we show that it is possible to achieve a significant resolution enhancement. When a spherical acoustic wave is propagating inside the medium, the optical parameter of the medium is perturbed. Using cross-correlations of the boundary measurements of the intensity of the light propagating in the perturbed medium and in the unperturbed one, we provide an iterative algorithm for reconstructing the optical absorption coefficient. Using a spherical Radon transform inversion, we first establish an equation that the optical absorption satisfies. This equation together with the diffusion model constitutes a nonlinear system. Then, solving iteratively such a nonlinear coupled system, we obtain the true absorption parameter. We prove the convergence of the algorithm and present numerical results to illustrate its resolution and stability performances.

Imaging of buried objects from multi-frequency experimental data using a globally convergent inversion method

Abstract This paper is concerned with the numerical solution to a three-dimensional coefficient inverse problem for buried objects with multi-frequency experimental data. The measured data, which are associated with a single direction of an incident plane wave, are backscatter data for targets buried in a sandbox. These raw scattering data were collected using a microwave scattering facility at the University of North Carolina at Charlotte. We develop a data preprocessing procedure and exploit a newly developed globally convergent inversion method for solving the inverse problem with these preprocessed data. It is shown that dielectric constants of the buried targets as well as their locations are reconstructed with a very good accuracy. We also prove a new analytical result which rigorously justifies an important step of the so-called “data propagation” procedure.