arXiv subjects

Loc H. Nguyen

Publications and source records attributed to Loc H. Nguyen.

42 records · Page 3Linked to original sources

arXiv2016

A globally convergent numerical method for a 3D coefficient inverse problem with a single measurement of multi-frequency data

The goal of this paper is to reconstruct spatially distributed dielectric constants from complex-valued scattered wave field by solving a 3D coefficient inverse problem for the Helmholtz equation at multi-frequencies. The data are generated by only a single direction of the incident plane wave. To solve this inverse problem, a globally convergent algorithm is analytically developed. We prove that this algorithm provides a good approximation for the exact coefficient without any \textit{a priori} knowledge of any point in a small neighborhood of that coefficient. This is the main advantage of our method, compared with classical approaches using optimization schemes. Numerical results are presented for both computationally simulated data and experimental data. Potential applications of this problem are in detection and identification of explosive-like targets.

math.NA↗

arXiv2016

Numerical Solution of a Coefficient Inverse Problem with Multi-Frequency Experimental Raw Data by a Globally Convergent Algorithm

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 \emph{a priori} knowledge of their physical and geometrical properties.

math.NA↗

arXiv2016

Single measurement experimental data for an inverse medium problem inverted by a multi-frequency globally convergent numerical method

The recently developed globally convergent numerical method for an inverse medium problem for the Helmholtz equation is tested on experimental data. The data were originally collected in the time domain, whereas the method works in the frequency domain with the multi-frequency data. Due to a huge discrepancy between the collected and computationally simulated data, the straightforward Fourier transform of the experimental data does not work. Hence, it is necessary to develop a heuristic data preprocessing procedure. This procedure is described. The preprocessed data are used as the input for the inversion algorithm. Numerical results demonstrate good accuracy in the reconstruction of both refracive indices and locations of targets. Furthermore, the reconstruction errors for refractive indices of dielectric targets are significantly less than errors of a posteriori direct measurements.

math.NA↗

arXiv2016

A globally convergent method for a 3-D inverse medium problem for the generalized Helmholtz equation

A 3-D inverse medium problem in the frequency domain is considered. Another name for this problem is Coefficient Inverse Problem. The goal is to reconstruct spatially distributed dielectric constants from scattering data. Potential applications are in detection and identification of explosive-like targets. A single incident plane wave and multiple frequencies are used. A new numerical method is proposed. A theorem is proved, which claims that a small neigborhood of the exact solution of that problem is reached by this method without any advanced knowledge of that neighborhood. We call this property of that numerical method "global convergence". Results of numerical experiments for the case of the backscattering data are presented.

math.NA↗

arXiv2016

A globally convergent numerical method for a 1-d inverse medium problem with experimental data

In this paper, a reconstruction method for the spatially distributed dielectric constant of a medium from the back scattering wave field in the frequency domain is considered. Our approach is to propose a globally convergent algorithm, which does not require any knowledge of a small neighborhood of the solution of the inverse problem in advance. The Quasi-Reversibility Method (QRM) is used in the algorithm. The convergence of the QRM is proved via a Carleman estimate. The method is tested on both computationally simulated and experimental data.

math.AP↗

arXiv2015

Nanostructures imaging via numerical solution of a 3-d inverse scattering problem without the phase information

Inverse scattering problems without the phase information arise in imaging of nanostructures whose sizes are hundreds of nanometers as well as in imaging of biological cells. The governing equation is the 3-d generalized Helmholtz equation with the unknown coefficient, which represents the spatially distributed dielectric constant. It is assumed that only the modulus of the complex valued wave field is measured on a frequency range. The phase is not measured. Two rigorous numerical methods are tested and their performances are compared for realistic ranges of parameters.

math.AP↗