Algorithms for solving scattering problems for the Manakov model of nonlinear Schrödinger equations

2020 ◽  
Vol 0 (0) ◽  
Author(s):  
Leonid L. Frumin
Keyword(s):  
Scattering Problem ◽  
Inverse Scattering Problem ◽  
Numerical Algorithms ◽  
Scattering Problems ◽  
Block Matrices ◽  
Nonlinear Schrödinger ◽  
Direct Scattering ◽  
Vector Case ◽  
Manakov Model ◽  
Vector Variant

AbstractWe introduce numerical algorithms for solving the inverse and direct scattering problems for the Manakov model of vector nonlinear Schrödinger equation. We have found an algebraic group of 4-block matrices with off-diagonal blocks consisting of special vector-like matrices for generalizing the scalar problem’s efficient numerical algorithms to the vector case. The inversion of block matrices of the discretized system of Gelfand–Levitan–Marchenko integral equations solves the inverse scattering problem using the vector variant the Toeplitz Inner Bordering algorithm of Levinson’s type. The reversal of steps of the inverse problem algorithm gives the solution of the direct scattering problem. Numerical tests confirm the proposed vector algorithms’ efficiency and stability. We also present an example of the algorithms’ application to simulate the Manakov vector solitons’ collision.

The inverse elastic scattering by an impenetrable obstacle and a crack

2020 ◽  
Author(s):  
Jianli Xiang ◽  
Guozheng Yan
Keyword(s):  
Mixed Type ◽  
Scattering Problem ◽  
Inverse Scattering Problem ◽  
Field Operator ◽  
Integral Equation Method ◽  
Factorization Method ◽  
Boundary Integral ◽  
Far Field ◽  
Direct Scattering ◽  
Time Harmonic

Abstract This paper is concerned with the inverse scattering problem of time-harmonic elastic waves by a mixed-type scatterer, which is given as the union of an impenetrable obstacle and a crack. We develop the modified factorization method to determine the shape of the mixed-type scatterer from the far field data. However, the factorization of the far field operator $F$ is related to the boundary integral matrix operator $A$, which is obtained in the study of the direct scattering problem. So, in the first part, we show the well posedness of the direct scattering problem by the boundary integral equation method. Some numerical examples are presented at the end of the paper to demonstrate the feasibility and effectiveness of the inverse algorithm.

Uniqueness and stability of the recovery of an absorbing obstacle from a knowledge of its scattering resonances

2000 ◽  
Vol 130 (1) ◽  
pp. 63-80
Author(s):  
C. Labreuche
Keyword(s):  
A Priori ◽  
Scattering Problem ◽  
Inverse Scattering Problem ◽  
Large Error ◽  
Small Error ◽  
Compact Set ◽  
Far Field ◽  
Scattering Problems ◽  
Resonant Frequencies ◽  
Ill Posed

In a previous paper, I investigated the use (for the inverse scattering problem) of the resonant frequencies and the associated eigen far-fields. I showed that the shape of a sound soft obstacle is uniquely determined by a knowledge of one resonant frequency and one associated eigen far-field. Inverse obstacle scattering problems are ill-posed in the sense that a small error in the measurement may imply a large error in the reconstruction. This is contrary to the idea of continuity. I proved that, by adding some a priori information, the reconstruction becomes continuous. More precisely, continuity holds if we assume that the obstacle lies a fixed and known compact set.The goal of this paper is to extend these results to the case of absorbing obstacles.

scholarly journals Interior Elastic Scattering by a Non-Penetrable Partially Coated Obstacle and Its Shape Recovering

Mathematics ◽  
2021 ◽  
Vol 9 (19) ◽  
pp. 2485
Author(s):  
Angeliki Kaiafa ◽  
Vassilios Sevroglou
Keyword(s):  
Mixed Boundary ◽  
Variational Equation ◽  
Scattering Problem ◽  
Inverse Scattering Problem ◽  
Point Of View ◽  
Inversion Algorithm ◽  
Direct Scattering ◽  
Time Harmonic ◽  
Navier Equation ◽  
Robin Boundary

In this paper, the interior elastic direct and inverse scattering problem of time-harmonic waves for a non-penetrable partially coated obstacle placed in a homogeneous and isotropic medium is studied. The scattering problem is formulated via the Navier equation, considering incident circular waves due to point-source fields, where the corresponding scattered data are measured on a closed curve inside the obstacle. Our model, from the mathematical point of view, is described by a mixed boundary value problem in which the scattered field satisfies mixed Dirichlet-Robin boundary conditions on the Lipschitz boundary of the obstacle. Using a variational equation method in an appropriate Sobolev space setting, uniqueness and existence results as well as stability ones are established. The corresponding inverse problem is also studied, and using some specific auxiliary integral operators an appropriate modified factorisation method is given. In addition, an inversion algorithm for shape recovering of the partially coated obstacle is presented and proved. Last but not least, useful remarks and conclusions concerning the direct scattering problem and its linchpin with the corresponding inverse one are given.

scholarly journals Two-Step Extended Sampling Method for the Inverse Acoustic Source Problem

2020 ◽  
Vol 2020 ◽  
pp. 1-8
Author(s):  
Jiyu Sun ◽  
Yuhui Han
Keyword(s):  
Inverse Scattering ◽  
Sampling Method ◽  
Scattering Problem ◽  
Inverse Scattering Problem ◽  
Indicator Function ◽  
Scattering Data ◽  
Acoustic Source ◽  
Scattering Problems ◽  
Linear Sampling Method ◽  
Ill Posed

Recently, a new method, called the extended sampling method (ESM), was proposed for the inverse scattering problems. Similar to the classical linear sampling method (LSM), the ESM is simple to implement and fast. Compared to the LSM which uses full-aperture scattering data, the ESM only uses the scattering data of one incident wave. In this paper, we generalize the ESM for the inverse acoustic source problems. We show that the indicator function of ESM, which is defined using the approximated solutions of some linear ill-posed integral equations, is small when the support of the source is contained in the sampling disc and is large when the source is outside. This behavior is similar to the ESM for the inverse scattering problem. Numerical examples are presented to show the effectiveness of the method.

scholarly journals Improved TV-CS Approaches for Inverse Scattering Problem

2015 ◽  
Vol 2015 ◽  
pp. 1-9 ◽  
Author(s):  
M. T. Bevacqua ◽  
L. Di Donato
Keyword(s):  
Compressive Sensing ◽  
Total Variation ◽  
Inverse Scattering ◽  
Scattering Problem ◽  
Inverse Scattering Problem ◽  
Scattering Problems ◽  
Piecewise Constant ◽  
Numerical Examples ◽  
Inverse Scattering Problems

Total Variation and Compressive Sensing (TV-CS) techniques represent a very attractive approach to inverse scattering problems. In fact, if the unknown is piecewise constant and so has a sparse gradient, TV-CS approaches allow us to achieve optimal reconstructions, reducing considerably the number of measurements and enforcing the sparsity on the gradient of the sought unknowns. In this paper, we introduce two different techniques based on TV-CS that exploit in a different manner the concept of gradient in order to improve the solution of the inverse scattering problems obtained by TV-CS approach. Numerical examples are addressed to show the effectiveness of the method.

DISCUSSION OF QUANTUM INVERSE SCATTERING PROBLEMS FOR COUPLED CHANNELS AT FIXED ENERGY

2011 ◽  
Vol 20 (08) ◽  
pp. 1765-1773 ◽  
Author(s):  
WERNER SCHEID ◽  
BARNABAS APAGYI
Keyword(s):  
Angular Momentum ◽  
Inverse Scattering ◽  
Nuclear Physics ◽  
Total Angular Momentum ◽  
Scattering Problem ◽  
Inverse Scattering Problem ◽  
Practical Method ◽  
Scattering Problems ◽  
Coupled Channels ◽  
S Matrix

In nuclear physics, the inverse scattering problem for coupled channels at fixed energies searches for the coupling potentials by using the S matrix as information. On the basis of the Newton–Sabatier method we investigate the special case that the coupling is independent of the total angular momentum. We discuss transparent potentials and consider a principal, but not practical method for the solution of coupling potentials dependent on total angular momentum.

scholarly journals An Efficient Technique for Solving Inhomogeneous Electromagnetic Inverse Scattering Problems

2020 ◽  
Vol 20 (1) ◽  
pp. 64-72 ◽  
Author(s):  
Mohamed Elkattan
Keyword(s):  
Inverse Scattering ◽  
Scattering Problem ◽  
Three Dimensional ◽  
Inverse Scattering Problem ◽  
Scattered Data ◽  
Global Minimization ◽  
Scattering Problems ◽  
Inverse Scattering Problems ◽  
Cooling Schedules ◽  
Electromagnetic Inverse Scattering

The electromagnetic inverse scattering approach seeks to obtain the electric characteristics of a scatterer using information about the source and the scattered data. The inverse scattering problem usually suffers from limited knowledge about the scatterer used, which makes its solution more challenging than the forward problem. This paper presents an inversion approach to estimating the unknown electric properties of a two- and three-dimensional inhomogeneous scatterer. The presented approach considers the inverse scattering problem as a global minimization problem with a meshless forward formulation for the computation of the scattered electromagnetic field. Various simulated annealing cooling schedules are applied and assessed to solve the problem, and the results of several case studies are presented for both two- and three-dimensional electromagnetic inverse scattering problems.

Numerical solution of the system of Marchenko integral equations

2021 ◽  
Vol 65 (3) ◽  
pp. 159-165
Keyword(s):  
Differential Equations ◽  
Integral Equations ◽  
Inverse Scattering ◽  
Scattering Problem ◽  
Inverse Scattering Problem ◽  
Algebraic Equations ◽  
System Of Differential Equations ◽  
Scattering Problems ◽  
First Order ◽  
First Order Differential Equations

In this paper, inverse scattering problems for a system of differential equations of the first order are considered. The Marchenko approach is used to solve the inverse scattering problem. The system of Marchenko integral equations is reduced to a linear system of algebraic equations such that the solution of the resulting system yields to the unknown coefficients of the system of first-order differential equations. Illustrative examples are provided to demonstrate the preciseness and effectiveness of the proposed technique. The results are compared with the exact solution by using computer simulations.

Far field patterns and inverse scattering problems for imperfectly conducting obstacles

1989 ◽  
Vol 106 (3) ◽  
pp. 553-569 ◽  
Author(s):  
T. S. Angell ◽  
David Colton ◽  
Rainer Kress
Keyword(s):  
Boundary Condition ◽  
Inverse Scattering ◽  
Scattering Problem ◽  
Plane Waves ◽  
Inverse Scattering Problem ◽  
Impedance Boundary Condition ◽  
Far Field ◽  
Scattering Problems ◽  
Impedance Boundary ◽  
Field Patterns

AbstractWe first examine the class of far field patterns for the scalar Helmholtz equation in ℝ2 corresponding to incident time harmonic plane waves subject to an impedance boundary condition where the impedance is piecewise constant with respect to the incident direction and continuous with respect to x ε ∂ D where ∂ D is the scattering obstacle. We then examine the class of far field patterns for Maxwell's equations in subject to an impedance boundary condition with constant impedance. The results obtained are used to derive optimization algorithms for solving the inverse scattering problem.

scholarly journals Photonic-dispersion neural networks for inverse scattering problems

2021 ◽  
Vol 10 (1) ◽  
Author(s):  
Tongyu Li ◽  
Ang Chen ◽  
Lingjie Fan ◽  
Minjia Zheng ◽  
Jiajun Wang ◽  
...  
Keyword(s):  
Neural Networks ◽  
Inverse Scattering ◽  
Scattering Problem ◽  
Inverse Scattering Problem ◽  
Practical Significance ◽  
Single Shot ◽  
Inverse Mapping ◽  
Scattering Problems ◽  
Inverse Scattering Problems ◽  
Forward Mapping

AbstractInferring the properties of a scattering objective by analyzing the optical far-field responses within the framework of inverse problems is of great practical significance. However, it still faces major challenges when the parameter range is growing and involves inevitable experimental noises. Here, we propose a solving strategy containing robust neural-networks-based algorithms and informative photonic dispersions to overcome such challenges for a sort of inverse scattering problem—reconstructing grating profiles. Using two typical neural networks, forward-mapping type and inverse-mapping type, we reconstruct grating profiles whose geometric features span hundreds of nanometers with nanometric sensitivity and several seconds of time consumption. A forward-mapping neural network with a parameters-to-point architecture especially stands out in generating analytical photonic dispersions accurately, featured by sharp Fano-shaped spectra. Meanwhile, to implement the strategy experimentally, a Fourier-optics-based angle-resolved imaging spectroscopy with an all-fixed light path is developed to measure the dispersions by a single shot, acquiring adequate information. Our forward-mapping algorithm can enable real-time comparisons between robust predictions and experimental data with actual noises, showing an excellent linear correlation (R2 > 0.982) with the measurements of atomic force microscopy. Our work provides a new strategy for reconstructing grating profiles in inverse scattering problems.

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