Scattering Problems Research Articles

Fast Algorithms for Solving the Inverse Scattering Problem for the Zakharov–Shabat System of Equations and Their Applications

Fast Algorithms for Solving the Inverse Scattering Problem for the Zakharov–Shabat System of Equations and Their Applications

A Quantized Tensor Train Method for High-Frequency Scattering Problems Involving Heterogeneous Dielectric Layers

We present a new numerical scheme to solve efficiently scattering problems involving an elongated and flat heterogeneous dielectric material assumed to be invariant along a direction of space. The technique consists of compressing the integral operators of an integro-differential formulation with a so-called quantized tensor train (QTT) algorithm whose use is rather original in this context. We show that it allows to compute and store operators with a notably small memory footprint while having at the same time a fast matrix-vector product (MVP) leading to a competitive method compared with the more classical <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"> <tex-math notation="LaTeX">$\mathcal {H}$ </tex-math></inline-formula> -matrix approach.

Enhanced Machine Learning Approach for Accurate and Fast Resolution of Inverse Scattering Problem in Breast Cancer Detection

An improved machine learning approach is presented in this paper to guarantee the fast convergence of the Born Iterative Method, even in the presence of strong scatterers, by assuming a single operating frequency and a reduced number of antennas in the scattering setup. The initial estimation of the dielectric profile, provided by the Born Iterative Method, was processed by a specific convolutional neural network to improve the reconstruction using a fast machine learning approach. To ensure rapid convergence, a proper choice of the initial guess in terms of the minimum permittivity value over the entire domain was also made. Numerical validations on realistic breast phantoms were illustrated, demonstrating an average error of 2.4% and an accuracy greater than 96% for all considered tests, even when considering a single operating frequency and a reduced amount of training data.

Solving Electromagnetic Scattering Problems With Tens of Billions of Unknowns Using GPU Accelerated Massively Parallel MLFMA

In this article, a massively parallel approach of the multilevel fast multipole algorithm (PMLFMA) on graphics processing unit (GPU) heterogeneous platform, noted as GPU-PMLFMA, is presented for solving extremely large electromagnetic scattering problems involving tens of billions of unknowns, In this approach, the flexible and efficient ternary partitioning scheme is employed at first to partition the MLFMA octree among message-passing interface (MPI) processes. Then, the computationally intensive parts of the PMLFMA on each MPI process, matrix filling, aggregation and disaggregation, and so on are accelerated by using the GPU. Different parallelization strategies in coincidence with the ternary parallel MLFMA approach are designed for GPU to ensure high computational throughput. Special memory usage strategy is designed to improve computational efficiency and benefit data reusing. The CPU/GPU asynchronous computing pattern is designed with the OpenMP and compute unified device architecture (CUDA), respectively, for accelerating the CPU and GPU execution parts and computation time overlapped. GPU architecture-based optimization strategies are implemented to further improve the computational efficiency. Numerical results demonstrate that the proposed GPU-PMLFMA can achieve over three times speedup, compared with the eight-threaded conventional PMLFMA. Solutions of scattering by electrically large and complicated objects with about 24 000 wavelengths and over 41.8 billion unknowns are presented.

A Modified Major Current Coefficients Method for Inverse Scattering Problems Using Multiple-Frequency Data

In this letter, a modified major current coefficients method (M-MCC), which combines the major current coefficients method (MCC), the hierarchical scheme, generalized Tikhonov regularization, and frequency rejection mechanism is proposed to solve inverse scattering problems using multiple-frequency data. The M-MCC utilizes the reconstruction results obtained at lower frequencies as prior information to improve the higher frequency result. The hierarchical scheme utilizes the reconstruction results of the lower frequency to refine the area of objects by using a threshold to judge if the area includes the objects. The inversion at higher frequencies will only be done for the refined area instead of the domain of interests. The generalized Tikhonov regularization is used at higher frequencies, which makes use of the reconstruction results at lower frequencies. In addition, the frequency rejection mechanism also utilizes the result of the lower frequency to exclude the frequencies that have invalid data. The proposed method outperforms the MCC method, which is verified through simulations.

Z-Transform-Based FDTD Implementations of Biaxial Anisotropy for Radar Target Scattering Problems

In this article, an efficient Z-transform-based finite-difference time-domain (Z-FDTD) is developed to implement and analyze electromagnetic scatterings in the 3D biaxial anisotropy. In terms of the Z-transform technique, we first discuss the conversion relationship between time- or frequency-domain derivative operators and the corresponding Z-domain operator, then build up the Z-transform-based iteration from the electric flux D converted to the electric field E based on dielectric tensor ε (and from the magnetic flux B converted to the magnetic field H in line with permeability tensor μ) by combining the constitutive formulations about the biaxial anisotropy. As a result, the iterative process about the Z-FDTD implementation can be smoothly carried out by means of combining with the Maxwell’s equations. To our knowledge, it is inevitably necessary for the absorbing boundary condition (ABC) to be considered in the electromagnetic scattering; hence, we utilize the unsplit-field complex-frequency-shifted perfectly matched layer (CFS-PML) to truncate the Z-FDTD’s physical region, and then capture time- and frequency-domain radiation with the electric dipole. In the 3D simulations, we select two different biaxial anisotropic models to validate the proposed formulations by using the popular commercial software COMSOL. Moreover, it is certain that those results are effective and available for electromagnetic scattering problems under the oblique incidence executed by the Z-FDTD method.

Theoretical Study on Non-Improvement of the Multi-Frequency Direct Sampling Method in Inverse Scattering Problems

Generally, it has been confirmed that applying multiple frequencies guarantees a successful imaging result for various non-iterative imaging algorithms in inverse scattering problems. However, the application of multiple frequencies does not yield good results for direct sampling methods (DSMs), which has been confirmed through simulation but not theoretically. This study proves this premise theoretically by showing that the indicator function with multi-frequency can be expressed by the Bessel and Struve functions and the propagation direction of the incident field. This is based on the fact that the indicator function with single frequency can be expressed by the exponential and Bessel function of order zero of the first kind. Various simulation outcomes are shown to support the theoretical result.

Resolution Enhancement for Mixed Boundary Conditions in Inverse Scattering Problems

Resolution Enhancement for Mixed Boundary Conditions in Inverse Scattering Problems

A Conditioned Probabilistic Method for the Solution of the Inverse Acoustic Scattering Problem

In the present work, a novel stochastic method has been developed and investigated in order to face the time-reduced inverse scattering problem, governed by the Helmholtz equation, outside connected or disconnected obstacles supporting boundary conditions of Dirichlet type. On the basis of the stochastic analysis, a series of efficient and alternative stochastic representations of the scattering field have been constructed. These novel representations constitute conceptually the probabilistic analogue of the well known deterministic integral representations involving the famous Green’s functions, and so merit special importance. Their advantage lies in their intrinsic probabilistic nature, allowing to solve the direct and inverse scattering problem in the realm of local methods, which are strongly preferable in comparison with the traditional global ones. The aforementioned locality reflects the ability to handle the scattering field only in small bounded portions of the scattering medium by monitoring suitable stochastic processes, confined in narrow sub-regions where data are available. Especially in the realm of the inverse scattering problem, two different schemes are proposed facing reconstruction from the far field and near field data, respectively. The crucial characteristic of the inversion is that the reconstruction is fulfilled through stochastic experiments, taking place in the interior of conical regions whose base belong to the data region, while their vertices detect appropriately the supporting surfaces of the sought scatterers.

High order local farfield expansions absorbing boundary conditions for multiple scattering

In this paper, we derive high order and local absorbing boundary conditions (ABC) for two- and three-dimensional multiple acoustic scattering problems. For this purpose, we employ Karp and Wilcox farfield expansions, which are exact representations of purely outgoing waves in two- and three-dimensions, respectively. We seek solutions of multiple scattering problems for scatterers of arbitrary shape in both homogeneous and locally inhomogeneous media by assuming that each scatterer and its surrounding inhomogeneity (if any) can be well-separated by artificial boundaries from the other scatterers. As a consequence, the computational domain is reduced to the union of relatively small disjoint subdomains bounded in the interior by a scatterer boundary and in the exterior by a circular (2D) or spherical (3D) artificial boundary. The novel ABC is defined at these artificial boundaries by considering continuities of the scattered field and its derivatives at the artificial boundaries and by representing the scattered wave as a superposition of outgoing waves from each subdomain. Finally, we numerically solve the new bounded boundary value problem for numerous examples including complexly shaped obstacles, such as prototype submarines and aircrafts. Accurate results are obtained at relatively low computational cost for problems in both homogeneous and inhomogeneous media.

Exponential Convergence of Perfectly Matched Layers for Scattering Problems with Periodic Surfaces

The main task of this paper is to prove that the perfectly matched layers (PML) method converges exponentially with respect to the PML parameter for scattering problems with periodic surfaces. In [S. N. Chandler-Wilde and P. Monk, Appl. Numer. Math., 59 (2009), pp. 2131--2154], a linear convergence is proved for the PML method for scattering problems with rough surfaces. At the end of that paper, three important questions are asked, and the third question is whether exponential convergence holds locally. In our paper, we answer this open question for a special case, when the rough surface is actually periodic. Due to technical reasons, we have to exclude all the wavenumbers which are half integers. The main idea of the proof is to apply the Floquet--Bloch transform to rewrite the problem as an equivalent family of quasi-periodic problems, and then study the analytic extension of the quasi-periodic problems with respect to the Floquet--Bloch parameters. Then the Cauchy integral formula is applied to avoid linear convergent points. Finally, the exponential convergence is proved from the inverse Floquet--Bloch transform. Numerical results are also presented at the end of this paper.

Oblique Wave Scattering Problems Involving Vertical Porous Membranes

Oblique Wave Scattering Problems Involving Vertical Porous Membranes

Target signatures for thin surfaces

We investigate an inverse scattering problem for a thin inhomogeneous scatterer in , m = 2, 3, which we model as an m − 1 dimensional open surface. The scatterer is referred to as a screen. The goal is to design target signatures that are computable from scattering data in order to detect changes in the material properties of the screen. This target signature is characterized by a mixed Steklov eigenvalue problem for a domain whose boundary contains the screen. We show that the corresponding eigenvalues can be determined from appropriately modified scattering data by using the generalized linear sampling method. A weaker justification is provided for the classical linear sampling method. Numerical experiments are presented to support our theoretical results.

Analytical Solutions to Definite Integrals for Combinations of Legendre, Bessel and Trigonometric Functions Encountered in Propagation and Scattering Problems in Spherical Coordinates

Mie theory is a rigorous solution to scattering problems in spherical coordinate system. The first step in applying Mie theory is expansion of some arbitrary incident field in terms of spherical harmonics fields in terms of spherical which in turn requires evaluation of certain definite integrals whose integrands are products of Bessel functions, associated Legendre functions and periodic functions. Here we present analytical results for two specific integrals that are encountered in expansion of arbitrary fields in terms of summation of spherical waves. The analytical results are in terms of finite summations which include Lommel functions. A concise analytical expression is also derived for the special case of Lommel functions that arise, rendering expensive numerical integration or other iterative techniques unnecessary.

SOM-Net: Unrolling the Subspace-Based Optimization for Solving Full-Wave Inverse Scattering Problems

In this paper, an unrolling algorithm of the iterative subspace-based optimization method (SOM) is proposed for solving full-wave inverse scattering problems (ISPs). The unrolling network, named SOM-Net, inherently embeds the Lippmann- Schwinger physical model into the design of network structures. The SOM-Net takes the deterministic induced current and the raw permittivity image obtained from back-propagation (BP) as the input. It then updates the induced current and the permittivity successively in sub-network blocks of the SOM- Net by imitating iterations of the SOM. The final output of the SOM-Net is the full predicted induced current, from which the scattered field and the permittivity image can also be deduced analytically. The parameters of the SOM-Net are optimized in a supervised manner with the total loss to simultaneously ensure the consistency of the induced current, the scattered field, and the permittivity in the governing equations. Numerical tests on both synthetic and experimental data verify the superior performance of the proposed SOM-Net over typical ones. The results on challenging examples like scatterers with tough profiles or high permittivity demonstrate the good generalization ability of the SOM-Net. With the use of deep unrolling technology, this work builds a bridge between traditional iterative methods and deep learning methods for solving ISPs.

The Half-Space Matching Method for Elastodynamic Scattering Problems in Unbounded Domains

The Half-Space Matching Method for Elastodynamic Scattering Problems in Unbounded Domains

Time Domain Linear Sampling Method for Inverse Scattering Problems with Cracks

Time Domain Linear Sampling Method for Inverse Scattering Problems with Cracks

Unrolled Optimization With Deep Learning-Based Priors for Phaseless Inverse Scattering Problems

Inverse scattering problems, such as those in electromagnetic imaging using phaseless data (PD-ISPs), involve imaging objects using phaseless measurements of wave scattering. Such inverse problems can be highly non-linear and ill-posed under extremely strong scattering conditions such as when the objects have very high permittivity or are large in size. In this work, we propose an end-to-end reconstruction framework using unrolled optimization with deep priors to solve PD-ISPs under very strong scattering conditions. We incorporate an approximate linear physics-based model into our optimization framework along with a deep learning-based prior and solve the resulting problem using an iterative algorithm which is unfolded into a deep network. This network not only learns data-driven regularization, but also overcomes the shortcomings of approximate linear models and learns non-linear features. More important, unlike existing PD-ISP methods, the proposed framework learns optimum values of all tunable parameters (including multiple regularization parameters) as a part of the framework. Results from simulations and experiments are shown for the use case of indoor imaging using 2.4 GHz phaseless Wi-Fi measurements, where the objects exhibit extremely strong scattering and low-absorption. Results show that the proposed framework outperforms existing model-driven and data-driven techniques by a significant margin and provides up to 20 times higher validity range.

A Wavelet-Based Compressive Deep Learning Scheme for Inverse Scattering Problems

Recently, physics-assisted deep learning schemes (DLSs) have demonstrated state-of-the-art performance for solving inverse scattering problems (ISPs). However, most learning approaches typically require a high computational overhead and a big memory footprint, which prohibits further applications. In this work, a wavelet-based compressive scheme (WCS) is proposed in solving ISPs, where the multi-subspace information is explored by wavelet bases and branched between each encoder and decoder path. It is shown that the proposed WCS can be simply adapted to commonly used DLSs, such as the back-propagation scheme (BPS) and the dominant current scheme (DCS), to reduce the computational and storage load. Specifically, benefiting from compressive and multi-resolution properties of wavelet and with the help of the factorized convolution method, more than 99.7% trainable weights are reduced in both illustrated BP-WCS and DC-WCS, whereas the performance deterioration is limited around 1% in terms of traditional BPS and DCS. Extensive numerical and experimental tests are conducted for quantitative validations. Comparisons are also made among UNet, a well-known compressive method (Mobile-UNet), and the proposed method. It is expected that the suggested compression technique would find its applications on deep learning-based electromagnetic inverse problems under source-limited scenarios.

An Approximation of 2-D Inverse Scattering Problems From a Convex Optimization Perspective

We present a two-step strategy to solve an inverse scattering problem in 2-D geometry. The first step approximates the inverse scattering as a convex optimization problem and provides an estimation of the total field inside the domain under investigation without <italic xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">a priori</i> knowledge or tuning parameters. In the second step, the previously estimated total field is used to reconstruct the unknown contrast permittivity, which is represented by a superposition of level-1 Haar wavelet transform basis functions. Subject to <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"> <tex-math notation="LaTeX">${\ell _{1}}$ </tex-math></inline-formula> -norm constraints of the wavelet coefficients, a least absolute shrinkage and selection operator (LASSO) problem that searches for the global minimum of the <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"> <tex-math notation="LaTeX">${\ell _{2}}$ </tex-math></inline-formula> -norm residual is exploited by accounting for the sparsity of the wavelet-based permittivity representation. Numerical results are presented to assess the effectiveness of the proposed formulation against objects with relatively small electric size. Finally, the approach is validated against experimental data.