arXiv subjects

Loc Hoang Nguyen

Publications and source records attributed to Loc Hoang Nguyen.

14 recordsLinked to original sources

arXiv2023

Numerical verification of the convexification method for a frequency-dependent inverse scattering problem with experimental data

The reconstruction of physical properties of a medium from boundary measurements, known as inverse scattering problems, presents significant challenges. The present study aims to validate a newly developed convexification method for a 3D coefficient inverse problem in the case of buried unknown objects in a sandbox, using experimental data collected by a microwave scattering facility at The University of North Carolina at Charlotte. Our study considers the formulation of a coupled quasilinear elliptic system based on multiple frequencies. The system can be solved by minimizing a weighted Tikhonov-like functional, which forms our convexification method. Theoretical results related to the convexification are also revisited in this work.

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arXiv2022

Numerical reconstruction for 3D nonlinear SAR imaging via a version of the convexification method

This work extends the applicability of our recent convexification-based algorithm for constructing images of the dielectric constant of buried or occluded target. We are orientated towards the detection of explosive-like targets such as antipersonnel land mines and improvised explosive devices in the non-invasive inspections of buildings. In our previous work, the method is posed in the perspective that we use multiple source locations running along a line of source to get a 2D image of the dielectric function. Mathematically, we solve a 1D coefficient inverse problem for a hyperbolic equation for each source location. Different from any conventional Born approximation-based technique for synthetic-aperture radar, this method does not need any linearization. In this paper, we attempt to verify the method using several 3D numerical tests with simulated data. We revisit the global convergence of the gradient descent method of our computational approach.

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arXiv2022

The Carleman contraction mapping method for quasilinear elliptic equations with over-determined boundary data

We propose a globally convergent numerical method to compute solutions to a general class of quasi-linear PDEs with both Neumann and Dirichlet boundary conditions. Combining the quasi-reversibility method and a suitable Carleman weight function, we define a map of which fixed point is the solution to the PDE under consideration. To find this fixed point, we define a recursive sequence with an arbitrary initial term using the same manner as in the proof of the contraction principle. Applying a Carleman estimate, we show that the sequence above converges to the desired solution. On the other hand, we also show that our method delivers reliable solutions even when the given data are noisy. Numerical examples are presented.

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arXiv2020

Convexification for a 3D inverse scattering problem with the moving point source

For the first time, we develop in this paper the globally convergent convexification numerical method for a Coefficient Inverse Problem for the 3D Helmholtz equation for the case when the backscattering data are generated by a point source running along an interval of a straight line and the wavenumber is fixed. Thus, by varying the wavenumber, one can reconstruct the dielectric constant depending not only on spatial variables but the wavenumber (i.e. frequency) as well. Our approach relies on a new derivation of a boundary value problem for a system of coupled quasilinear elliptic partial differential equations. This is done via an application of a special truncated Fourier-like method. First, we prove the Lipschitz stability estimate for this problem via a Carleman estimate. Next, using the Carleman Weight Function generated by that estimate, we construct a globally strictly convex cost functional and prove the global convergence to the exact solution of the gradient projection method. Finally, our theoretical finding is verified via several numerical tests with computationally simulated data. These tests demonstrate that we can accurately recover all three important components of targets of interest: locations, shapes and dielectric constants. In particular, large target/background contrasts in dielectric constants (up to 10:1) can be accurately calculated.

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arXiv2019

A new algorithm to determine the creation or depletion term of parabolic equations from boundary measurements

We propose a robust numerical method to find the coefficient of the creation or depletion term of parabolic equations from the measurement of the lateral Cauchy information of their solutions. Most papers in the field study this nonlinear and severely ill-posed problem using optimal control. The main drawback of this widely used approach is the need of some advanced knowledge of the true solution. In this paper, we propose a new method that opens a door to solve nonlinear inverse problems for parabolic equations without any initial guess of the true coefficient. This claim is confirmed numerically. The key point of the method is to derive a system of nonlinear elliptic equations for the Fourier coefficients of the solution to the governing equation with respect to a special basis of L^2. We then solve this system by a predictor-corrector process, in which our computation to obtain the first and second predictors is effective. The desired solution to the inverse problem under consideration follows.

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arXiv2019

A numerical method for an inverse source problem for parabolic equations and its application to a coefficient inverse problem

Two main aims of this paper are to develop a numerical method to solve an inverse source problem for parabolic equations and apply it to solve a nonlinear coefficient inverse problem. The inverse source problem in this paper is the problem to reconstruct a source term from external observations. Our method to solve this inverse source problem consists of two stages. We first establish an equation of the derivative of the solution to the parabolic equation with respect to the time variable. Then, in the second stage, we solve this equation by the quasi-reversibility method. The inverse source problem considered in this paper is the linearization of a nonlinear coefficient inverse problem. Hence, iteratively solving the inverse source problem provides the numerical solution to that coefficient inverse problem. Numerical results for the inverse source problem under consideration and the corresponding nonlinear coefficient inverse problem are presented.

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arXiv2018

An inverse source problem for hyperbolic equations and the Lipschitz-like convergence of the quasi-reversibility method

We propose in this paper a new numerical method to solve an inverse source problem for general hyperbolic equations. This is the problem of reconstructing sources from the lateral Cauchy data of the wave field on the boundary of a domain. In order to achieve the goal, we derive an equation involving a Volterra integral, whose solution directly provides the desired solution of the inverse source problem. Due to the presence of such a Volterra integral, this equation is not in a standard form of partial differential equations. We employ the quasi-reversibility method to find its regularized solution. Using Carleman estimates, we show that the obtained regularized solution converges to the exact solution with the Lipschitz-like convergence rate as the measurement noise tends to 0. This is one of the novelties of this paper since currently, convergence results for the quasi-reversibility method are only valid for purely differential equations. Numerical tests demonstrate a good reconstruction accuracy.

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arXiv2015

Cloaking using complementary media for the Helmholtz equation and a three spheres inequality for second order elliptic equations

Cloaking using complementary media was suggested by Lai et al. This was proved in the quasistatic regime by H. M. Nguyen. One of the difficulties in the study of this problem is the appearance of the localized resonance, i.e., the fields blow up in some regions and remain bounded in some others as the loss goes to 0. To this end, the author introduced the technique of removing localized singularity and used a standard three spheres inequality. This method also works for the Helmholtz equation. However, it requires small size of the cloaked region for large frequency due to the use of the (standard) three spheres inequality. In this paper, we give a proof of cloaking using complementary media in the finite frequency regime without imposing any condition on the cloaked region; hence the cloak works for all frequency. To successfully apply the approach of H.M. Nguyen, we establish a new three spheres inequality. A modification of the cloaking setting to obtain illusion optics is also discussed.

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arXiv2014

Reconstruction and stability in acousto-optic imaging for absorption maps with bounded variation

The aim of this paper is to propose for the first time a reconstruction scheme and a stability result for recovering from acoustic-optic data absorption distributions with bounded variation. The paper extends earlier results on smooth absorption distributions. It opens a door for a mathematical and numerical framework for imaging, from internal data, parameter distributions with high contrast in biological tissues.

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arXiv2014

Admittivity imaging from multi-frequency micro-electrical impedance tomography

The aim of this paper is to propose an optimal control optimization algorithm for reconstructing admittivity distributions (i.e., both conductivity and permittivity) from multi-frequency micro-electrical impedance tomography. A convergent and stable optimization scheme is shown to be obtainable from multi-frequency data. The results of this paper have potential applicability in cancer imaging, cell culturing and differentiation, food sciences, and biotechnology.

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arXiv2013

Localized and complete resonance in plasmonic structures

This paper studies a possible connection between the way the time averaged electromagnetic power dissipated into heat blows up and the anomalous localized resonance in plasmonic structures. We show that there is a setting in which the localized resonance takes place whenever the resonance does and moreover, the power is always bounded and might go to $0$. We also provide another setting in which the resonance is complete and the power goes to infinity whenever resonance occurs; as a consequence of this fact there is no localized resonance. This work is motivated from recent works on cloaking via anomalous localized resonance.

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arXiv2012

Nonlinear Elliptic Dirichlet and No-Flux Boundary Value Problems

This paper is devoted to establishing results for semilinear elliptic boundary value problems where the solvability of problems subject to {\it No Flux} boundary conditions follows from the solvability of related {\it Dirichlet} boundary value problems. Throughout it is assumed that the nonlinear perturbation terms are gradient dependent. An extension of \textit{No-Flux} problems is discussed, as well.

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arXiv2012

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 establish a new equation in the sense of distributions for the optical absorption coefficient. 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 and finally construct the true coefficient by providing a Landweber type iteration and proving 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 [Ammari, Garnier, Nguyen and Seppecher, 2012] for piecewise smooth optical absorption distributions.

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arXiv2011

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

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 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

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