Scattering Problems Research Articles

Inverse Scattering Problem for the Sturm-Liouville Equation with Infinite Range of Discontinuous Conditions

In this paper, we construct the new integral representation of the Jost solution of Sturm-Liouville equation with impuls in the semi axis $[0,+\infty )$ and we give this type of relation, examine the properties of the Kernel function and their partial derivatives with $x$ and $\ t$, constructed integral representation and obtain the partial differential equation provided by this Kernel function. Finally, in the paper we prove uniqueness of the determination of the potential by the scattering data.

Application of Skeletonization-Based Method in Solving Inverse Scattering Problems

In electromagnetic inverse scattering problems, Scattered field commonly needs to be measured by a large number of receiving antennas to provide enough scattered information for image reconstruction, which may increase the cost of the experimental system and require a long testing time. In this paper, a skeletonization-based method was proposed to reduce the number of actual receiving antennas involved in an inverse scattering system. The skeleton points were obtained by performing a strong-rank-revealing QR factorization of Green’s function matrix. By measuring the scattered field only at the skeleton points, the number of receiving antennas could be effectively reduced, while the scattered field data at other receiving points could be accurately restored from the skeleton points. The numerical results show that, compared with the frequency domain zero-padding (FDZP) method, the skeletonization-based method was more accurate for antennas distributed in an elliptical shape (such as thorax imaging). In addition, the inverse scattering method using the skeletonization-based method was able to reduce the number of measurements while maintaining an image quality comparable to that of the actual full measurement system. The proposed method can serve as a guidance for building an experimental system for inverse scattering problems, especially for cases when the antennas are elliptically distributed.

A Modified HODLR Solver Based on Higher Order Basis Functions for Solving Electromagnetic Scattering Problems

In this letter, we present a novel fast direct solver based on a modified hierarchically off-diagonal low-rank (M-HODLR) matrix format with higher order basis functions for analyzing electromagnetic scattering by electrically large objects. The M-HODLR matrix compresses the far-group submatrices by an extended admissibility condition, which enhances the efficiency and accuracy of the original HODLR. To further alleviate the burden of the fast direct solution of the conventional surface integral equation (SIE), the higher order hierarchical vector basis (HOHVB) functions defined over a curved surface are employed. To account for HOHVB, an efficient assembly scheme based on patch coupling is developed to accelerate the adaptive across approximation implementation in M-HODLR. The proposed method can significantly reduce the computational time and memory cost of a conventional SIE solver. Several numerical examples are presented to validate the accuracy and efficiency of the proposed method in solving electrically large objects. Owing to the characteristics of HOHVB, different patch sizes or basis functions with different orders can be chosen according to the local geometry of the object. Therefore, the proposed method is very appealing for large- and multiscale simulations.

Windowed Green Function MoM for Second-Kind Surface Integral Equation Formulations of Layered Media Electromagnetic Scattering Problems

This paper presents a second-kind surface integral equation method for the numerical solution of frequency-domain electromagnetic scattering problems by locally perturbed layered media in three spatial dimensions. Unlike standard approaches, the proposed methodology does not involve the use of layer Green functions. It instead leverages an indirect M\"uller formulation in terms of free-space Green functions that entails integration over the entire unbounded penetrable boundary. The integral equation domain is effectively reduced to a small-area surface by means of the windowed Green function method, which exhibits high-order convergence as the size of the truncated surface increases. The resulting (second-kind) windowed integral equation is then numerically solved by means of the standard Galerkin method of moments (MoM) using RWG basis functions. The methodology is validated by comparison with Mie-series and Sommerfeld-integral exact solutions as well as against a layer Green function-based MoM. Challenging examples including realistic structures relevant to the design of plasmonic solar cells and all-dielectric metasurfaces, demonstrate the applicability, efficiency, and accuracy of the proposed methodology.

Validation of HBF-Based MoM Approach to Solution of Radiation and Scattering Problems on Microwave Antennas With Linear Elements

A recently proposed method of moment (MoM)-based approach is extended to efficiently solve broadband radiation and scattering problems on wire and mixed geometries. This approach is based on expanding the total current on a conducting structure at any desired frequency in terms of entire domain hyper basis functions (HBFs), calculated at only one frequency. The developed approach is validated by comparing the characteristics of canonical microwave antennas with those obtained by direct MoM and characteristic mode analysis (CMA).

An Improved GMRES Method for Solving Electromagnetic Scattering Problems by MoM

Method of moments (MoM) is a classical numerical method for analyzing electromagnetic (EM) scattering from various objects. Generalized minimal residual (GMRES) is a kind of widely used solver for linear systems of equations generated by MoM, whose calculation complexity mainly includes two parts, matrix-vector multiplications and the orthogonalization procedure. In order to accelerate the solution of matrix equations arising in MoM, an efficient strategy for constructing nonorthonormal Krylov bases is proposed in this article, which can be introduced into standard GMRES and its variants (e.g., GMRES( <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"> <tex-math notation="LaTeX">$m$ </tex-math></inline-formula> ) and quasi-GMRES) and can be easily combined with the fast approaches for MOM (e.g., FMM and ACA). The principle of the method is described in detail, and the effectiveness is verified by numerical results.

Quantum-Mechanical Scattering Problem in Lobachevsky Space at Low Energies

Based on the use of the asymptotics for the wave function of a scattered particle in a Lobachevsky space in a form close to the asymptotics in flat space, general formulas for the theory of quantum mechanical scattering in this space are derived. This approach makes it possible to represent the basic formulas of the theory of scattering in the Lobachevsky space in the form that coincides with the corresponding expressions in three-dimensional Euclidean space. We o.er quantities (length of scattering, effective scattering radius), that are used in describing scattering at short-range potentials and are convenient as phenomenological parameters in describing nuclear interactions at low energies. Numerical estimates of these quantities and cross sections at low energies, that are characteristic of nuclear physics, are given.

MoM Solution of Broadband Radiation and Scattering Problems Based on Entire-Domain Hyperbasis Functions Approach

A new method of moments (MoM)-based approach is proposed to efficiently solve broadband radiation and scattering problems on triangulated, quadrilateral, and mixed mesh geometries. This approach is based on expanding the total current on a conducting body at any desired frequency in terms of entire-domain hyperbasis functions (HBFs), calculated at only one frequency. These HBFs must be found as eigenfunctions of the generalized eigenvalue equation (GEE), defined at that frequency. The developed approach is validated by comparing results with those obtained by direct MoM for different types of mesh. Finally, it is demonstrated for solving practical antenna problems.

Accelerated Adaptive Error Control and Refinement for SIE Scattering Problems

We present an application of adjoint-based adaptive error control and refinement for scattering problems solved using the method of moments (MoM) in the electric field integral equation (EFIE) and the coupled EFIE and magnetic field integral equation (MFIE) formulations. Specifically, we examine the Poggio–Miller–Chang–Harrington–Wu–Tsai (PMCHWT) formulation of the EFIE-MFIE. We first construct the adjoint problems of the EFIE and the EFIE-MFIE for estimating the error of radar cross section (RCS) quantities of interest (QoIs) directly. We then introduce an effective adaptive refinement algorithm based on an error prediction heuristic and a posteriori error estimation, which enables rapid and consistent convergence regardless of a chosen tolerance and the coarseness of the starting discretization. The approach, moreover, inherently promotes equilibration of the QoI error contributions and produces, therefore, consistently balanced meshes. Numerical examples with canonical scattering targets and adaptive <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"> <tex-math notation="LaTeX">$p$ </tex-math></inline-formula> -refinement confirm the strength of the proposed refinement method, demonstrating the ability to generate high-quality discretizations, both in terms of accuracy and efficiency, without expert user intervention.

A Hybrid Born Iterative Bayesian Inversion Method for Electromagnetic Imaging of Moderate-Contrast Scatterers With Piecewise Homogeneities

To tackle the problem of imaging moderate-contrast scatterers with piecewise homogeneities, a hybrid Born iterative (BI) Bayesian inversion method is proposed. First, we establish the nonlinear inverse scattering problem in the context of multiple measurement vector (MMV). Then, a computationally efficient BI formulation is used to account for the strong nonlinear relationship between the scattered field and the contrast. In each iteration, a regularization strategy that exploits the piecewise homogeneous <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"> <tex-math notation="LaTeX">$a$ </tex-math></inline-formula> priori information of the investigated domain is adopted to overcome the ill-posedness. Our proposed method can tackle the inverse scattering problem in its full nonlinearity and incorporate the available <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"> <tex-math notation="LaTeX">$a$ </tex-math></inline-formula> priori information to stabilize the inversion procedure. Several representative tests are carried out to evaluate the performance of the proposed method. It is demonstrated that the proposed method outperforms other existing state-of-the-art algorithms in terms of accuracy, convergence, robustness, and computational efficiency.

High Order Complex Contour Discretization Methods to Simulate Scattering Problems in Locally Perturbed Periodic Waveguides

High Order Complex Contour Discretization Methods to Simulate Scattering Problems in Locally Perturbed Periodic Waveguides

Structural performance of a submerged bottom-mounted compound porous cylinder on the water wave interaction in the presence of a porous sea-bed

Scattering problem of a submerged bottom-mounted compound porous cylinder located on a porous sea-bed is theoretically investigated under the assumption of linear potential flow theory. The compound cylinder is comprised of an impermeable inner cylinder and a porous outer cylinder. The boundary conditions on the porous boundaries follow Darcy's law by assuming fine pores in the porous structure. The whole fluid region is split into three bounded and unbounded sub-regions, within which the individual velocity potentials are found by using the eigenfunction expansion technique. Furthermore, utilization of the matching conditions along the boundaries of individual successive regions leads to a semi-analytical solution of the proposed problem. The impact of the non-dimensional porous-effect parameter of the cylindrical wall, the draft ratio, radius ratio, and the sea-bed porosity on wave loads and free-surface elevation are studied. In addition, the wave power dissipation for the system is calculated by integrating the power absorbed by the outer cylinder porous wall via direct method. Also, the far-field scattering coefficients are obtained with the help of asymptotic forms of Hankel functions in the plane wave representation form. Numerical results for the far-field scattering coefficient and power dissipation are investigated for various parameters. The theoretical model is verified by comparing it with the results of the conventional analytical work and experimental work. The results show that suitable consideration of porosity and structure parameters enhances the efficiency of the proposed compound cylinder in mitigating wave impact. Furthermore, the hydrodynamic wave load acting on the inner and outer cylinders can be reduced by the suitable positioning of the annular spacing of the system, which will provide explicit information for the purpose of engineering design in offshore and coastal regions.

Application of Mosaic-Skeleton Approximations of Matrices in the Physical Optics Method for Electromagnetic Scattering Problems

Application of Mosaic-Skeleton Approximations of Matrices in the Physical Optics Method for Electromagnetic Scattering Problems

Fast Analysis of Bistatic Scattering Problems for Three-Dimensional Objects Using Compressive Sensing and Characteristic Modes

In this letter, a combination of characteristic mode (CM) and compressive sensing (CS) is utilized to accelerate the solution of bistatic scattering problems of three-dimensional (3-D) objects. The surface induced currents discretized by Rao–Wilton–Glisson basis functions are successfully sparsely transformed by CM basis functions based on domain decomposition. Different from the dense matrix equation of traditional method of moments (MoM), a low-rank CS model with smaller size is constructed in the proposed method, wherein the induced currents can be reconstructed efficiently. Due to partially filling of the impedance matrix and efficient recovery algorithm, the time cost in the proposed method is reduced dramatically. Last, the results from several numerical simulations validate the efficiency and accuracy of the proposed method.

Fast High-Order Algorithms for Electromagnetic Scattering Problem from Finite Array of Cavities in TE Case with High Wave Numbers

In this paper, fast high-order finite difference algorithms for solving the electromagnetic scattering from the finite array of two-dimensional rectangular cavities are proposed in TE polarization. The scattering problem from the cavity array is described as coupled Helmholtz equations with transparent boundary conditions on open apertures. Second-order and fourth-order schemes for solving the coupled systems are developed in TE polarization respectively. A special technique is applied to construct a fourth-order scheme for transparent boundary conditions. Further, we propose fast algorithms which can simplify the larger global system to a small linear interface system on the apertures of cavities. Numerical experiments show the validity and efficiency of the proposed fast algorithms for solving the scattering problem with high wave numbers.

Learned Global Optimization for Inverse Scattering Problems: Matching Global Search With Computational Efficiency

The computationally-efficient solution of fully non-linear microwave inverse scattering problems (ISPs) is addressed. An innovative System-by-Design (SbD) based method is proposed to enable, for the first time to the best of the authors knowledge, an effective, robust, and time-efficient exploitation of an evolutionary algorithm (EA) to perform the global minimization of the data-mismatch cost function. According to the SbD paradigm as suitably applied to ISPs, the proposed approach founds on (i) a smart re-formulation of the ISP based on the definition of a minimum-dimensionality and representative set of degrees-of-freedom (DoFs) and on (ii) the artificial-intelligence (AI)-driven integration of a customized global search technique with a digital twin (DT) predictor based on the Gaussian Process (GP) theory. Representative numerical and experimental results are provided to assess the effectiveness and the efficiency of the proposed approach also in comparison with competitive state-of-the-art inversion techniques.

Enhanced Supervised Descent Learning Technique for Electromagnetic Inverse Scattering Problems by the Deep Convolutional Neural Networks

This work proposes a novel deep learning (DL) framework to solve the electromagnetic inverse scattering (EMIS) problems. The proposed framework integrates the complex-valued deep convolutional neural network (DConvNet) into the supervised descent method (SDM) to realize both off-line training and on-line “imaging” prediction for EMIS. The offline training consists of two parts: 1) DConvNet training: the training dataset is created, and the proposed DConvNet is trained to realize the EM forward process and 2) SDM training: the trained DConvNet is integrated into the SDM framework, and the average descent directions between the initial prediction and the true label of SDM iterative schemes are learned based on the same dataset in part 1). In the online step, the contrasts (permittivities) reconstruction of scatterers is realized by the SDM iteration process based on learned descent directions, while its forward process is achieved by the trained complex-valued DConvNet. Ultimately, this framework provides a new perspective to integrate the prior information into the EMIS solving process with the maintained accuracy. Unlike the conventional SDM, the novel proposed framework can significantly shorten the computation and realize the real-time imaging. Various numerical examples and discussions are provided to demonstrate the efficiency and accuracy of the proposed novel framework.

A Remark on the Inverse Scattering Problem for the Perturbed Hill Equation

A Remark on the Inverse Scattering Problem for the Perturbed Hill Equation

Cascaded Complex U-Net Model to Solve Inverse Scattering Problems With Phaseless-Data in the Complex Domain

The phaseless-data inverse scattering problems (PD-ISPs) are non-trivial due to their serious nonlinearity and ill-posed nature. The conventional methods for PD-ISPs can be categorized into single-step methods and two-step methods, where the first ones directly reconstruct the image and the second ones retrieve the phase and amplitude of scattered fields, and then reconstruct the image as the traditional FD-ISPs. However, the single-step methods correspond to solve higher nonlinear optimization problems and the two-step methods highly depend on the quality of the retrieved phase and amplitude information. To solve the above issues, we propose a two-step method with a cascaded complex U-net (CCU-net) model to solve the PD-ISPs in the complex domain. The CCU-net consists of two parts, that is, Phase Retrieval Net (PRNet) and Image Reconstruction Net (IRNet), where the PRNet recovers the phase and amplitude of the scattered field from the measured modulus of the total field and the IRNet takes the recovered scattered field as input to reconstruct the image. These two parts can be independently controlled and achieve joint optimization. Thanks to the strong nonlinear representation ability of complex neural networks, the physical relationship between the scattered fields and the images can be reserved and constructed. Several representative tests including both the synthetic and experimental examples verify that the proposed CCU-net in the complex domain has good robustness, generalization ability, and strong inversion capability when tackling high contrast scatterers and can also fulfill one-step implementation in real time.

Physics Embedded Deep Neural Network for Solving Full-Wave Inverse Scattering Problems

In this work, we design an iterative deep neural network to solve full-wave inverse scattering problems (ISPs) in the 2-D case. Forward modeling neural networks that predict the scattered field are embedded in an inversion neural network. In an iterative manner, the inversion network predicts the model update from the residual between the simulated data and the observed data. The proposed inversion network can achieve super-resolution reconstruction meanwhile keeping the simulated data of reconstructed models well consistent with the observed data. We validate this method with both synthetic and experimental data inversion. Results show that the inversion network can predict models with high accuracy, efficiency, and good generalization ability. By combining deep learning and physical simulation together, the proposed method provides a way for real-time imaging with high reliability and accuracy.