Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

A global approach for the inverse scattering problem using a Carleman contraction map

Phuong M. Nguyen, Loc H. Nguyen, Huong T. T. Vu

Abstract

This paper addresses the inverse scattering problem in the domain Omega. The input data, measured outside Omega, involve the waves generated by the interaction of plane waves with various directions and unknown scatterers fully occluded inside Omega. The output of this problem is the spatially dielectric constant of these scatterers. Our approach to solving this problem consists of two primary stages. Initially, we eliminate the unknown dielectric constant from the governing equation, resulting in a system of partial differential equations. Subsequently, we develop the Carleman contraction mapping method to effectively tackle this system. It is noteworthy to highlight this method's robustness. It does not request a precise initial guess of the true solution, and its computational cost is not expensive. Some numerical examples are presented.

A global approach for the inverse scattering problem using a Carleman contraction map

Abstract

This paper addresses the inverse scattering problem in the domain Omega. The input data, measured outside Omega, involve the waves generated by the interaction of plane waves with various directions and unknown scatterers fully occluded inside Omega. The output of this problem is the spatially dielectric constant of these scatterers. Our approach to solving this problem consists of two primary stages. Initially, we eliminate the unknown dielectric constant from the governing equation, resulting in a system of partial differential equations. Subsequently, we develop the Carleman contraction mapping method to effectively tackle this system. It is noteworthy to highlight this method's robustness. It does not request a precise initial guess of the true solution, and its computational cost is not expensive. Some numerical examples are presented.
Paper Structure (8 sections, 2 theorems, 65 equations, 6 figures, 1 algorithm)

This paper contains 8 sections, 2 theorems, 65 equations, 6 figures, 1 algorithm.

Table of Contents

  1. Introduction
  2. An approximate model
  3. The Carleman contraction mapping method for \ref{['2.13']}
  4. Numerical study
  5. Data generation
  6. The implementations
  7. Numerical examples
  8. Further discussions and concluding remarks

Key Result

Lemma 3.1

Fix a point ${\bf x}_0 \in \mathbb{R}^d \setminus \Omega$. Define $r({\bf x}) = |{\bf x} - {\bf x}_0|$ for all ${\bf x} \in \Omega.$ There exist positive constants $\beta_0$ depending only on ${\bf x}_0$, $\Omega$, and $d$ such that for all function $v \in C^2(\overline \Omega)$ satisfying the following estimate holds true for all $\beta > \beta_0$ and $\lambda \geq \lambda_0$. Here, $\lambda_0

Figures (6)

  • Figure 1: A visualization of the inverse scattering process: An incident wave hits a scatterer in an inaccessible domain $\Omega$. The interaction between the wave and the scatterer produces a scattered wave, which contains important information about the scatterer (shape, location, and some physical properties). The primary aim of the inverse scattering problem is to use the measurements of the scattering wave outside $\Omega$ to find this information.
  • Figure 2: The graphs of the functions $e(N)$ are computed by the data, each containing 10% noise, generated for all four numerical tests in Subsection \ref{['sec_num_example']}. These graphs suggest us to choose $N = 42$.
  • Figure 3: The actual and computed spatial dielectric constants $c$ along with the $L^2$ relative difference of the computed $\mathbf{v}$ are presented. It is seen that the computed function $c$ meets the acceptable criteria. Additionally, Figure (c) provides numerical evidence supporting the rapid convergence of Algorithm \ref{['alg']}. Notably, it can be observed that only 3 iterations are adequate to achieve an accurate reconstruction. The data in this test is corrupted with 10% of noise.
  • Figure 4: The actual and computed spatial dielectric constants $c$ along with the $L^2$ relative difference of the computed $\mathbf{v}$ are presented. It is seen that the computed function $c$ meets the acceptable criteria. Additionally, Figure (c) provides numerical evidence supporting the rapid convergence of Algorithm \ref{['alg']}. Notably, it can be observed that only 3 iterations are adequate to achieve an accurate reconstruction. The data in this test is corrupted with 10% of noise.
  • Figure 5: The actual and computed spatial dielectric constants $c$ along with the $L^2$ relative difference of the computed $\mathbf{v}$ are presented. It is seen that the computed function $c$ meets the acceptable criteria. Additionally, Figure (c) provides numerical evidence supporting the rapid convergence of Algorithm \ref{['alg']}. Notably, it can be observed that only 3 iterations are adequate to achieve an accurate reconstruction. The data in this test is corrupted with 10% of noise.
  • ...and 1 more figures

Theorems & Definitions (7)

  • Remark 2.1
  • Remark 2.2
  • Lemma 3.1: Carleman estimate
  • Theorem 3.1
  • Remark 3.1
  • proof : Proof of Theorem \ref{['thm']}
  • Remark 3.2