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1. Imaging of bi-anisotropic periodic structures from electromagnetic near-field data

   https://doi.org/10.1515/jiip-2020-0114  

   Nguyen, Dinh-Liem; Truong, Trung (November 2020, Journal of Inverse and Ill-posed Problems)

   null (Ed.)

   Abstract This paper is concerned with the inverse scattering problem for the three-dimensional Maxwell equations in bi-anisotropic periodic structures.The inverse scattering problem aims to determine the shape of bi-anisotropic periodic scatterers from electromagnetic near-field data at a fixed frequency.The factorization method is studied as an analytical and numerical tool for solving the inverse problem.We provide a rigorous justification of the factorization method which results in the unique determination and a fast imaging algorithm for the periodic scatterer.Numerical examples for imaging three-dimensional periodic structures are presented to examine the efficiency of the method. 

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2. On fast reconstruction of periodic structures with partial scattering data

   https://doi.org/10.3934/era.2024303  

   Daugherty, John; Kaduk, Nate; Morgan, Elena; Nguyen, Dinh-Liem; Snidanko, Peyton; Truong, Trung (January 2024, Electronic Research Archive)

   This paper presents a numerical method for solving the inverse problem of reconstructing the shape of periodic structures from scattering data. This inverse problem is motivated by applications in the nondestructive evaluation of photonic crystals. The numerical method belongs to the class of sampling methods that aim to construct an imaging function for the shape of the periodic structure using scattering data. By extending the results of Nguyen, Stahl, and Truong [Inverse Problems, 39:065013, 2023], we studied a simple imaging function that uses half the data required by the numerical method in the cited paper. Additionally, this imaging function is fast, simple to implement, and very robust against noise in the data. Both isotropic and anisotropic cases were investigated, and numerical examples were presented to demonstrate the performance of the numerical method. 

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3. Fast numerical solutions to direct and inverse scattering for bi-anisotropic periodic Maxwell's equations

   Nguyen, Dinh-Liem; Truong, Trung (January 2023, AMS Contemporary Mathematics)

   Nguyen, Dinh-Liem; Nguyen, Loc; Nguyen, Thi-Phong (Ed.)

   This paper is concerned with the numerical solution to the direct and inverse electromagnetic scattering problem for bi-anisotropic periodic structures. The direct problem can be reformulated as an integro-di erential equation. We study the existence and uniqueness of solution to the latter equation and analyze a spectral Galerkin method to solve it. This spectral method is based on a periodization technique which allows us to avoid the evaluation of the quasiperiodic Green's tensor and to use the fast Fourier transform in the numerical implementation of the method. For the inverse problem, we study the orthogonality sampling method to reconstruct the periodic structures from scattering data generated by only two incident fields. The sampling method is fast, simple to implement, regularization free, and very robust against noise in the data. Numerical examples for both direct and inverse problems are presented to examine the efficiency of the numerical solvers. 

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4. Solving the inverse scattering problem via Carleman-based contraction mapping

   Nguyen, P M; Nguyen, L H; Vu, H T (June 2024, preprint arXiv:2404.04145)

   This paper addresses the inverse scattering problem in a domain $$\Omega$$. The input data, measured outside $$\Omega$$, involve the waves generated by the interaction of plane waves with various directions and unknown scatterers that are fully occluded inside $$\Omega$$. The output of this problem is the spatial dielectric constant of these scatterers. Our approach to solving this problem consists of two primary stages. Initially, we eliminate the unknown dielectric constant from the governing equation, resulting in a system of partial differential equations. Subsequently, we develop the Carleman contraction mapping method to effectively tackle this system. It is noteworthy to highlight the robustness of this method. It does not require a precise initial guess of the true solution, and its computational cost is relatively inexpensive. Some numerical examples are presented. 

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5. Inverse scattering without phase: Carleman convexification and phase retrieval via the Wentzel--Kramers--Brillouin approximation

   Le, T T; Nguyen, P M; Nguyen, L H (June 2025, Preprint arXiv:2506.21699)

   This paper addresses the challenging and interesting inverse problem of reconstructing the spatially varying dielectric constant of a medium from phaseless backscattering measurements generated by single-point illumination. The underlying mathematical model is governed by the three-dimensional Helmholtz equation, and the available data consist solely of the magnitude of the scattered wave field. To address the nonlinearity and servere ill-posedness of this phaseless inverse scattering problem, we introduce a robust, globally convergent numerical framework combining several key regularization strategies. Our method first employs a phase retrieval step based on the Wentzel--Kramers--Brillouin (WKB) ansatz, where the lost phase information is reconstructed by solving a nonlinear optimization problem. Subsequently, we implement a Fourier-based dimension reduction technique, transforming the original problem into a more stable system of elliptic equations with Cauchy boundary conditions. To solve this resulting system reliably, we apply the Carleman convexification approach, constructing a strictly convex weighted cost functional whose global minimizer provides an accurate approximation of the true solution. Numerical simulations using synthetic data with high noise levels demonstrate the effectiveness and robustness of the proposed method, confirming its capability to accurately recover both the geometric location and contrast of hidden scatterers. 

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