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1. 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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2. 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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3. A new sampling indicator function for stable imaging of periodic scattering media

   https://doi.org/10.1088/1361-6420/acce5f  

   Nguyen, Dinh-Liem; Stahl, Kale; Truong, Trung (May 2023, Inverse Problems)

   Abstract This paper is concerned with the inverse problem of determining the shape of penetrable periodic scatterers from scattered field data. We propose a sampling method with a novel indicator function for solving this inverse problem. This indicator function is very simple to implement and robust against noise in the data. The resolution and stability analysis of the indicator function is analyzed. Our numerical study shows that the proposed sampling method is more stable than the factorization method and more efficient than the direct or orthogonality sampling method in reconstructing periodic scatterers. 

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4. Fast Simulations with High Accuracy for Periodic Quantum Nanostructures and Photonic Crystals Enabled by a Physics-Informed Projection Learning Methodology

   Veresko, Martin; Liu, Yu; Hou, Daqing; Cheng, Ming-Cheng (July 2024, XXXV IUPAP. Comp. Physics (CCP2024))

   Fast solution of wave propagation in periodic structures usually relies on simplified approaches, such as analytical methods, transmission line models, scattering matrix approaches, plane wave methods, etc. For complex multi-dimensional problems, computationally intensive direct numerical simulation (DNS) is always needed. This study demonstrates a fast and accurate simulation methodology enabled by a physics-based learning methodology, derived from proper orthogonal decomposition (POD) and Galerkin projection, for periodic quantum nanostructure and photonic crystals. POD is a projection-based method that generates optimal basis functions (or POD modes) via solution data collected from DNSs. This process trains the POD modes to adapt parametric variations of the system and offers the best least squares (LS) fit to the solution using the smallest number of modes. This is very different from other projection approaches, e.g., Fourier, Legendre, Bessel, Airy functions, etc., that adopt assumed basis functions selected for the problem based on the solution form. After generating the optimal POD modes, Galerkin projection of the wave equation onto each of the POD modes is performed to close the model and incorporate physical principles guided by the wave equation. Such a rigorous approach offers efficient simulations with high accuracy and exhibits the extrapolation ability in cases reasonably beyond the training bounds. The POD-Galerkin methodology is applied in this study to predict band structures and wave solutions for 2D periodic quantum-dot and photonic-lattice structures. The plane-wave approach is also included in a periodic quantum-dot structure to illustrate the superior performance of the POD-Galerkin methodology. The POD-Galerkin approach offers a 2-order computing speedup for both nanostructure and optical superlattices, compared to DNS, when solving both the wave solution and band structure. If the band structure is the only concern, a 4-order improvement in computational efficiency can be achieved. Fig. 1(a) shows the optical superlattice in a demonstration, where a unit cell includes 22 discs with diagonally symmetrical refractive indices and the background index n = 1. The POD modes for this case are trained by TE mode electric field data collected from DNSs with variation of diagonally symmetrical refractive indices. The LS error of the predicted electric field wave solution from the POD-Galerkin approach, shown in Fig. 1(b) compared to DNS, is below 1% with just 8 POD modes that offer a more than 4-order reduction in the degrees of freedom, compared to DNS. In addition, an extremely accurate prediction of band structure is illustrated in Fig. 1(c) with a maximum error below 0.1% in the entire Brillouin zone. 

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5. Convexification numerical algorithm for a 2D inverse scattering problem with backscatter data

   https://doi.org/10.1080/17415977.2021.1943384  

   Truong, Trung; Nguyen, Dinh-Liem; Klibanov, Michael V. (January 2021, Inverse Problems in Science and Engineering)

   null (Ed.)

   This paper is concerned with the inverse scattering problem which aims to determine the spatially distributed dielectric constant coefficient of the 2D Helmholtz equation from multifrequency backscatter data associated with a single direction of the incident plane wave. We propose a globally convergent convexification numerical algorithm to solve this nonlinear and ill-posed inverse problem. The key advantage of our method over conventional optimization approaches is that it does not require a good first guess about the solution. First, we eliminate the coefficient from the Helmholtz equation using a change of variables. Next, using a truncated expansion with respect to a special Fourier basis, we approximately reformulate the inverse problem as a system of quasilinear elliptic PDEs, which can be numerically solved by a weighted quasi-reversibility approach. The cost functional for the weighted quasi-reversibility method is constructed as a Tikhonov-like functional that involves a Carleman Weight Function. Our numerical study shows that, using a version of the gradient descent method, one can find the minimizer of this Tikhonov-like functional without any advanced a priori knowledge about it. 

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