Scattering Problems Research Articles

Parallel Mosaic-Skeleton Algorithm for the Numerical Solution of a Three-Dimensional Scalar Scattering Problem in Integral Form

A three-dimensional scalar stationary scattering problem is considered. It is formulated in the form of a weakly singular Fredholm boundary integral equation of the first kind with a single unknown function. The equation is approximated by a system of linear algebraic equations, which is then solved numerically by an iterative method. The mosaic-skeleton method is used at the stage of the approximate solution of this system in order to reduce the computational complexity of the approach.

Efficient Solution of Electromagnetic Scattering Problems Using Multilevel Adaptive Cross Approximation and LU Factorization

The multilevel adaptive cross approximation (MLACA), previously described in the literature, extends the single-level ACA with a recursive multilevel algorithm that significantly improves compression of off-diagonal matrix blocks resulting from electromagnetic integral equations (IE) discretized via the method of moments (MoM). In this article, the MLACA approach is extended and applied to a direct solution of the MoM matrix system via LU factorization. It will be shown through numerical experiments that the off-diagonal LU blocks are also compressible using MLACA, yielding a compression rate superior to the single-level ACA and a memory complexity of O(N <sup xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">4/3</sup> log N). In addition, the MLACA LU block updates are performed in rank-reduced form, yielding a very efficient software implementation via a Level 3 BLAS optimized for the CPU or GPU.

A Practical Method for Solving the Inverse Quantum Scattering Problem on a Half Line

A method for solving inverse quantum scattering problems on a half line is proposed. It is based on the application of the transmutation operators and recent results on series expansion of the integral transmutation kernels. From the corresponding Gel’fand-Levitan equation a system of linear algebraic equations is derived for the coefficients of the Fourier-Legendre series expansion of the output (transmutation operator) kernel. It is shown that the knowledge of the very first coefficient is sufficient for recovering the potential and hence for solving the inverse problem. A numerical illustration is presented.

Transfer-matrix formulation of the scattering of electromagnetic waves and broadband invisibility in three dimensions

We develop a transfer-matrix formulation of the scattering of electromagnetic waves by a general isotropic medium which makes use of a notion of electromagnetic transfer matrix that does not involve slicing of the scattering medium or discretization of some of the position- or momentum-space variables. This is a linear operator that we can express as a matrix with operator entries and identify with the S-matrix of an effective nonunitary quantum system. We use this observation to establish the composition property of , obtain an exact solution of the scattering problem for a non-magnetic point scatterer that avoids the divergences of the Green’s function approaches, and prove a general invisibility theorem. The latter allows for an explicit characterization of a class of isotropic media displaying perfect broadband invisibility for electromagnetic waves of arbitrary polarization provided that their wavenumber k does not exceed a preassigned critical value , i.e. behaves exactly like vacuum for . Generalizing this phenomenon, we introduce and study -equivalent media that, by definition, have identical scattering features for .

Fast BEM Solution for 2-D Scattering Problems Using Quantized Tensor-Train Format

It is common to accelerate the boundary element method (BEM) by compression techniques [fast multipole method (FMM), H-matrix/adaptive cross approximation (ACA)] that enable a more accurate solution or a solution in higher frequency. In this article, we present a compression method based on a transformation of the linear system into the tensor-train format by the quantization technique. The method is applied to a scattering problem on a canonical object with a regular mesh and improves the performance obtained from existing methods.

X-ray scatter removal for artifact free CT imaging

In this paper we present a method to compensate for scattered radiation artifacts. Using a newly developed workflow to determine the scattered radiation, scatter artifacts can be dramatically reduced to a level, where further evaluation of the data becomes possible with state-of-the-art algorithms. The method intrinsically corrects for all kinds of scatter problems that may occur in a CT system including scattering inside the object. We will demonstrate the approach for the inspection of large aluminum castings. We will show that using this method an image quality can be achieved, which is otherwise only available using fan-beam tomography with line detectors. In comparison to line-detector based fan-beam CT, scans can be achieved in significantly reduced measurement times in the order of few minutes instead of many hours.

A Multiresolution Contraction Integral Equation Method for Solving Highly Nonlinear Inverse Scattering Problems

The solution of highly nonlinear inverse scattering problems is addressed. An innovative multiresolution microwave imaging technique, which leverages on the profitable integration of the contraction integral equation for inversion (CIE-I) and the iterative multiscaling approach (IMSA), is proposed. Thanks to the joint exploitation of the features of the CIE-I formulation and the IMSA, the arising IMSA-CIE-I inversion technique provides faithful reconstructions of complex-shaped strong scatterers with enhanced resolution and computational efficiency with respect to the state-of-the-art competitive alternatives. Numerical and experimental test cases verify the effectiveness and limitations of the proposed method.

KANTBP 4M Program for Solving the Scattering Problem for a System of Ordinary Second-Order Differential Equations

We report an upgrade of the program KANTBP 4M implemented in the computer algebra system MAPLE for solving, with a given accuracy, the multichannel scattering problem, which is reduced to a boundary-value problem for a system of ordinary differential equations of the second order with continuous or piecewise continuous real or complex-valued coeffcients. The solution over a finite interval is subject to mixed homogeneous boundary conditions: Dirichlet and/or Neumann, and/or of the third kind. The discretization of the boundary problem is implemented by means of the finite element method with the Lagrange or Hermite interpolation polynomials. The effciency of the proposed algorithm is demonstrated by solving a multichannel scattering problem with coupling of channels in both the reaction region and the asymptotic one.

A Higher Order Perturbation Approach for Electromagnetic Scattering Problems on Random Domains

We consider time-harmonic electromagnetic scattering problems on perfectly conducting scatterers with uncertain shape. Thus, the scattered field will also be uncertain. Based on the knowledge of the two-point correlation of the domain boundary variations around a reference domain, we derive a perturbation analysis for the mean of the scattered field. Therefore, we compute the second shape derivative of the scattering problem for a single perturbation. Taking the mean, this leads to an at least third order accurate approximation with respect to the perturbation amplitude of the domain variations. To compute the required second order correction term, a tensor product equation on the domain boundary has to be solved. We discuss its discretization and efficient solution using boundary integral equations. Numerical experiments in three dimensions are presented.

Matrix-Partitioned DDM for the Accurate Analysis of Challenging Scattering Problems

A matrix-partitioned domain decomposition method based on integral equation using the out-of-core iterative solver is presented for accurately analyzing challenging electromagnetic scattering problems with limited memory. The proposed method is based on the domain decomposition strategy, which decomposes the original large complex matrix of the electrically large problem into several sub-matrices of the electrically small sub-problems. Then, the out-of-core solver is used to solve the partitioned matrix equation panel by panel. In the process of constructing sub-problems, the proposed method does not introduce any additional unknowns. Thus, it can significantly reduce memory consumption, expanding the scale of the problem that can be solved. Numerical examples demonstrate that the method is very accurate even for the EM scattering targets the RCS of which are below -40 dBsm. And it can completely eliminate the pseudo edge effect which often occurs in the implementation of the domain decomposition method. In addition, modeling and partitioning the subdomains of the proposed method is easy and flexible.

Nonlinear Iterative Hard Thresholding for Inverse Scattering

We consider the inverse scattering problem for sparse scatterers. An image reconstruction algorithm is proposed that is based on a nonlinear generalization of iterative hard thresholding. The convergence and error of the method was analyzed by means of coherence estimates and compared to numerical simulations.

Two-Dimensional Problem of Electromagnetic Scattering by Mikaelian Lens, Two Methods of Solution, and Some Features of Focusing

Two-Dimensional Problem of Electromagnetic Scattering by Mikaelian Lens, Two Methods of Solution, and Some Features of Focusing

Single- and Multi-Frequency Direct Sampling Methods in a Limited-Aperture Inverse Scattering Problem

Although the direct sampling method (DSM) has demonstrated its feasibility and robustness for imaging of small inhomogeneities, mathematical analyses of DSM have been conducted only on the full-aperture inverse scattering problem. Numerous studies have shown that DSM can also be applied in the limited-aperture inverse scattering problem, but most of its applications are still heuristic. This study considers an application, mathematical analysis, and improvement of DSM with a single-incident field only in the limited-aperture inverse scattering problem. First, we introduce a traditional indicator function of DSM at a single frequency, establish its mathematical structure, and examine its inherent limitation. To demonstrate the theoretical result, various results of numerical simulations with synthetic and experimental data are presented. Next, we consider the multi-frequency indicator function of DSM with a single-incident direction to improve imaging performance. For this, we design a multi-frequency indicator function of MDSM, analyze its mathematical structure, and theoretically explain the improvement of the imaging of single inhomogeneity and the limitation on the identification of multiple inhomogeneities. Various numerical simulations with synthetic and experimental data are presented to validate our results.

Perfectly Matched Layer Method for Acoustic Scattering Problem by a Locally Perturbed Line with Impedance Boundary Condition

Perfectly Matched Layer Method for Acoustic Scattering Problem by a Locally Perturbed Line with Impedance Boundary Condition

Solving Phaseless Highly Nonlinear Inverse Scattering Problems With Contraction Integral Equation for Inversion

At high frequencies, phase measurement is difficult to meet the accuracy required for imaging, and it is more susceptible to noise pollution. Consequently, it is important to develop inversion models and algorithms for electromagnetic inverse scattering problems with phaseless data (PD-ISPs). Compared to the ISPs with full-data (FD-ISPs), the PD-ISPs are more nonlinear due to the lack of the phase information. Aiming at solving highly nonlinear PD-ISPs, in this article we present a contraction integral equation for inversion (CIE-I) regularized with Fourier subspace via contrast source inversion method, denoted as PD-CSI-CIE-I. Under the model of CIE-I, the nonlinearity of PD-ISPs can be effectively alleviated. In order to stabilize CIE-I, a nested optimization scheme with multi-round procedure is adopted by choosing the proper number of low-frequency components from unknown induced current spanned in the Fourier subspace. By the Fourier bases expansion of induced current, the computational cost can be further reduced. Numerical and experimental examples validate the efficiency of the proposed inversion method. Futher, it is shown that the proposed PD-CSI-CIE-I has a stronger inversion capability than the one with the Lippmann-Schwinger integral equation (LSIE) when tackling high contrast scatterers with phaseless data.

Diffraction Approach in the Scattering Problem for Three Charged Quantum Particles

Diffraction Approach in the Scattering Problem for Three Charged Quantum Particles

A CIP-FEM for High-Frequency Scattering Problem with the Truncated DtN Boundary Condition

A CIP-FEM for High-Frequency Scattering Problem with the Truncated DtN Boundary Condition

Fast Analysis of Local Current Distribution for Electromagnetic Scattering Problems of Electrically Large Objects

To improve computational efficiency of obtaining local induced current distribution under multiple incidents angles when electromagnetic scattering problems of electrically large objects are considered, a novel method is proposed to form small-scale impedance matrix and reduce number of repeated calculation for different incident angles, which is implemented by following three steps. Firstly, an undetermined matrix equation including multiple incident angles is constructed by interrelating part field nodes and all source nodes on object surface. Secondly, the undetermined matrix equation is transformed into a form that conforms to the compressive sensing (CS) framework, which is solved by reconstructed algorithm to obtain approximate induced currents of all source nodes under a wide incident angle range. Finally, exact wide-angle currents of a given node are obtained by calculating total magnetic field, and neighborhood wide-angle currents of the given node can be expanded by choosing appropriate technology in different boundary conditions. Numerical results of electrically large objects shown the efficiency of the proposed method.

Application of Subspace-Based Distorted-Born Iteration Method in Imaging Biaxial Anisotropic Scatterer

Various natural and artificial materials are anisotropic. The inverse scattering problem of anisotropic scatterers is widely involved in oil detection, nondestructive evaluation of composite materials and microscopic imaging of biological tissue. In this contribution, the two-dimensional inverse scattering problem of biaxial anisotropic scatterers illuminated by the TE-polarized incident wave is investigated. Since the biaxial anisotropic scatterer has different permittivity components along different transverse directions, the problem faced with is more complex than in the scalar TM-polarized case. The subspace-based distorted-Born iteration method (S-DBIM) is employed. Only one regularization term is involved in the inversion, which is proven to be quite robust against noise and flexible to be chosen. Both synthetic and experimental results are given to prove the validity of the proposed method. The results illustrate that as the interaction between the incident electric field and the scatterer induces a directional scattered field, the images constructed appear clear into the strongly scattered directions, but blurred into weakly ones. Overall, S-DBIM is shown to yield super-resolved images for the biaxial anisotropic scatterers, while being quite robust with respect to noise.

Convexification for a Three-Dimensional Inverse Scattering Problem with the Moving Point Source

For the first time, we develop in this paper the globally convergent convexification numerical method for a coefficient inverse problem for the three-dimensional Helmholtz equation for the case whe...