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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. 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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3. Controlling three-dimensional optical fields via inverse Mie scattering

   https://doi.org/10.1126/sciadv.aax4769  

   Zhan, Alan ; Gibson, Ricky ; Whitehead, James ; Smith, Evan ; Hendrickson, Joshua R. ; Majumdar, Arka ( October 2019 , Science Advances)

   Controlling the propagation of optical fields in three dimensions using arrays of discrete dielectric scatterers is an active area of research. These arrays can create optical elements with functionalities unrealizable in conventional optics. Here, we present an inverse design method based on the inverse Mie scattering problem for producing three-dimensional optical field patterns. Using this method, we demonstrate a device that focuses 1.55-μm light into a depth-variant discrete helical pattern. The reported device is fabricated using two-photon lithography and has a footprint of 144 μm by 144 μm, the largest of any inverse-designed photonic structure to date. This inverse design method constitutes an important step toward designer free-space optics, where unique optical elements are produced for user-specified functionalities. 

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4. Sampling type method combined with deep learning for inverse scattering with one incident wave

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

   We consider the inverse problem of determining the geometry of penetrable objects from scattering data generated by one incident wave at a fixed frequency. We first study an orthogonality sampling type method which is fast, simple to implement, and robust against noise in the data. This sampling method has a new imaging functional that is applicable to data measured in near field or far field regions. The resolution analysis of the imaging functional is analyzed where the explicit decay rate of the functional is established. A connection with the orthogonality sampling method by Potthast is also studied. The sampling method is then combined with a deep neural network to solve the inverse scattering problem. This combined method can be understood as a network using the image computed by the sampling method for the first layer and followed by the U-net architecture for the rest of the layers. The fast computation and the knowledge from the results of the sampling method help speed up the training of the network. The combination leads to a significant improvement in the reconstruction results initially obtained by the sampling method. The combined method is also able to invert some limited aperture experimental data without any additional transfer training. 

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5. Transformer Meets Boundary Value Inverse Problem

   Guo, Ruchi ; Cao, Shuhao ; Chen, Long ( May 2023 , International Conference on Learning Representations)

   A Transformer-based deep direct sampling method is proposed for electrical impedance tomography, a well-known severely ill-posed nonlinear boundary value inverse problem. A real-time reconstruction is achieved by evaluating the learned inverse operator between carefully designed data and the reconstructed images. An effort is made to give a specific example to a fundamental question: whether and how one can benefit from the theoretical structure of a mathematical problem to develop task-oriented and structure-conforming deep neural networks? Specifically, inspired by direct sampling methods for inverse problems, the 1D boundary data in different frequencies are preprocessed by a partial differential equation-based feature map to yield 2D harmonic extensions as different input channels. Then, by introducing learnable non-local kernels, the direct sampling is recast to a modified attention mechanism. The new method achieves superior accuracy over its predecessors and contemporary operator learners and shows robustness to noises in benchmarks. This research shall strengthen the insights that, despite being invented for natural language processing tasks, the attention mechanism offers great flexibility to be modified in conformity with the a priori mathematical knowledge, which ultimately leads to the design of more physics-compatible neural architectures. 

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