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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. 

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. 

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. 

Abstract This paper is concerned with imaging of 3D scattering objects with experimental data from the Fresnel database. The first goal of the paper is to investigate a modified version of the orthogonality sampling method (OSM) by Harris and Nguyen [2020 SIAM J. Sci. Comput. 42 B72–737] for the imaging problem. The advantage of the modified OSM over its original version lies in its applicability to more types of polarization vectors associated with the electromagnetic scattering data. We analyze the modified OSM using the factorization analysis for the far field operator and the Funk–Hecke formula. The second goal is to verify the performance of the modified OSM, the OSM, and the classical factorization method for the 3D Fresnel database. The modified OSM we propose is able to invert the sparse and limited-aperture real data in a fast, simple, and efficient way. It is also shown in the real data verification that the modified OSM performs better than its original version and the factorization method. 

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. 

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.