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.

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.

Decay of Turbulent Axisymmetrical Free Flows With Rotation

1967 ◽  
Vol 34 (4) ◽  
pp. 806-812 ◽  
Author(s):  
A. Chervinsky ◽  
D. Lorenz
Keyword(s):  
Boundary Layer ◽  
Differential Equations ◽  
Boundary Conditions ◽  
Algebraic Equations ◽  
Free Jets ◽  
Swirling Jets ◽  
First Order ◽  
Free Jet ◽  
Order Algebraic ◽  
First Order Differential Equations

Previous studies on turbulent swirling jets are extended to cover general axisymmetrical turbulent free flows with rotation, with particular considerations of free jets and wakes. The equations governing the flow are integrated subject to the pertaining boundary conditions making use of the usual boundary-layer approximations. The axial and tangential components of velocity are assumed to retain similar forms of radial distributions and a set of two ordinary first-order differential equations is derived, the solution of which describes the axial decay of the maximal axial and tangential velocities. Integration of the differential equations in the particular case of uniform external velocity subject to conditions at the orifice results in a set of two second-order algebraic equations which are readily solved. The theoretical solutions derived for a free jet are compared with available experimental results.

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.

Inverse Scattering Solutions by a Sinc Basis, Multiple Source, Moment Method -- Part III: Fast Algorithms

1984 ◽  
Vol 6 (1) ◽  
pp. 103-116
Author(s):  
S. A. Johnson ◽  
Y. Zhou ◽  
M. K. Tracy ◽  
M. J. Berggren ◽  
F. Stenger
Keyword(s):  
Wave Equation ◽  
Inverse Scattering ◽  
Scattering Problem ◽  
Inverse Scattering Problem ◽  
Algebraic Equations ◽  
The Past ◽  
Nonlinear Algebraic Equations ◽  
Detector Geometry ◽  
Scattering Potential ◽  
Helmholtz Wave Equation

Solving the inverse scattering problem for the Helmholtz wave equation without employing the Born or Rytov approximations is a challenging problem, but some slow iterative methods have been proposed. One such method suggested by us is based on solving systems of nonlinear algebraic equations that are derived by applying the method of moments to a sinc basis function expansion of the fields and scattering potential. In the past, we have solved these equations for a 2-D object of n by n pixels in a time proportional to n5. In the present paper, we demonstrate a new method based on FFT convolution and the concept of backprojection which solves these equations in time proportional to n3 • log(n). Several numerical examples are given for images up to 7 by 7 pixels in size. Analogous algorithms to solve the Riccati wave equation in n3 • log(n) time are also suggested, but not verified. A method is suggested for interpolating measurements from one detector geometry to a new perturbed detector geometry whose measurement points fall on a FFT accessible, rectangular grid and thereby render many detector geometrics compatible for use by our fast methods.

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.

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.

Sign in / Sign up

Export Citation Format

Share Document