More Like this

1. 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. 

   more » « less
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. 

   more » « less
3. Direct sampling methods for isotropic and anisotropic scatterers with point source measurements

   https://doi.org/10.3934/ipi.2022015  

   Harris, Isaac; Nguyen, Dinh-Liem; Nguyen, Thi-Phong (January 2022, Inverse Problems and Imaging)

   In this paper, we consider the inverse scattering problem for recovering either an isotropic or anisotropic scatterer from the measured scattered field initiated by a point source. We propose two new imaging functionals for solving the inverse problem. The first one employs a 'far-field' transform to the data which we then use to derive and provide an explicit decay rate for the imaging functional. In order to analyze the behavior of this imaging functional we use the factorization of the near field operator as well as the Funk-Hecke integral identity. For the second imaging functional the Cauchy data is used to define the functional and its behavior is analyzed using the Green's identities. Numerical experiments are given in two dimensions for both isotropic and anisotropic scatterers. 

   more » « less
4. 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. 

   more » « less
5. Fast imaging of local perturbations in a unknown bi-periodic layered medium

   https://doi.org/10.1016/j.jcp.2024.112773  

   Cakoni, Fioralba; Haddar, Houssem; Nguyen, Thi-Phong (March 2024, Journal of Computational Physics)

   We discuss a novel approach for imaging local faults inside an infinite bi-periodic layered medium in ℝ3 using acoustic measurements of scattered fields at the bottom or the top of the layer. The faulted area is represented by compactly supported perturbations with erroneous material properties. Our method reconstructs the support of perturbations without knowing or reconstructing the constitutive material parameters of healthy or faulty bi-period layer; only the size of the period is needed. This approach falls under the class of non-iterative imaging methods, known as the generalized linear sampling method with differential measurements, first introduced in [2] and adapted to periodic layers in [25]. The advantage of applying differential measurements to our inverse problem is that instead of comparing the measured data against measurements due to healthy structures, one makes use of periodicity of the layer where the data operator restricted to single Floquet-Bloch modes plays the role of the one corresponding to healthy material. This leads to a computationally efficient and mathematically rigorous reconstruction algorithm. We present numerical experiments that confirm the viability of the approach for various configurations of defects 

   more » « less