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Shape and parameter identification by the linear sampling method for a restricted Fourier integral operator

Lorenzo Audibert, Shixu Meng

Abstract

In this paper we provide a new linear sampling method based on the same data but a different definition of the data operator for two inverse problems: the multi-frequency inverse source problem for a fixed observation direction and the Born inverse scattering problems. We show that the associated regularized linear sampling indicator converges to the average of the unknown in a small neighborhood as the regularization parameter approaches to zero. We develop both a shape identification theory and a parameter identification theory which are stimulated, analyzed, and implemented with the help of the prolate spheroidal wave functions and their generalizations. We further propose a prolate-based implementation of the linear sampling method and provide numerical experiments to demonstrate how this linear sampling method is capable of reconstructing both the shape and the parameter.

Shape and parameter identification by the linear sampling method for a restricted Fourier integral operator

Abstract

In this paper we provide a new linear sampling method based on the same data but a different definition of the data operator for two inverse problems: the multi-frequency inverse source problem for a fixed observation direction and the Born inverse scattering problems. We show that the associated regularized linear sampling indicator converges to the average of the unknown in a small neighborhood as the regularization parameter approaches to zero. We develop both a shape identification theory and a parameter identification theory which are stimulated, analyzed, and implemented with the help of the prolate spheroidal wave functions and their generalizations. We further propose a prolate-based implementation of the linear sampling method and provide numerical experiments to demonstrate how this linear sampling method is capable of reconstructing both the shape and the parameter.
Paper Structure (15 sections, 11 theorems, 105 equations, 6 figures)

This paper contains 15 sections, 11 theorems, 105 equations, 6 figures.

Table of Contents

  1. Introduction
  2. The Mathematical Model for Inverse Source and Born Inverse Scattering Problems
  3. Multi-frequency inverse source problem for a fixed observation direction
  4. Born inverse scattering problem
  5. A model that summarizes the inverse source and Born inverse scattering problem
  6. PSWFs and their generalization
  7. Data operator, factorization, and range identity
  8. Linear sampling and factorization methods for shape identification
  9. Linear sampling and factorization methods for parameter identification
  10. An explicit example and numerical preliminaries
  11. An explicit example
  12. Legendre polynomials and Legendre-Gauss-Lobatto quadrature
  13. Computation of PSWFs system
  14. A prolate-based formulation of the linear sampling method
  15. Numerical experiments for parameter and shape identification

Key Result

Theorem 1

Let the data operator $\mathcal{N}: L^2(B) \to L^2(B)$ be given by Section operator N def. Then it holds that where $\mathcal{S}_\Omega$, $\mathcal{S}^*_\Omega$, and $\mathcal{T}_\Omega$ are given by Section operator S_omega def, Section operator S*_omega def, and Section operator T_omega def, respectively.

Figures (6)

  • Figure 1: $\|A^{\mathbbm{J}}_{i,j}\|$ on the left and $\|A^{DFT}_{i,j}\|$ on the right in both case $1\leq i,j \leq 400$
  • Figure 2: Plot of $I(z)$, $q_{avg}$, and $q$ for four different types of parameters with noiseless data and $\epsilon=5\times 10^{-2}$. Left column: $\dim(\mathbb{J})=37$. Right column: $\dim(\mathbb{J})=54$. Row $j$ corresponds to type $j$, j=1,2,3,4.
  • Figure 3: Plot of $I(z)$, $q_{avg}$, and $q$ for four different types of parameters with noisy level $5\%$. $\epsilon=5\times 10^{-2}$ and $c=20$.
  • Figure 4: Plot of $I(z)$, $q_{avg}$, and $q$ for noiseless data (left column) and noisy data with noise level $\delta = 5\%$ (right column). $\epsilon=1\times 10^{-1}$. From top to bottom $c=3,5,7,10$.
  • Figure 5: Plot of $I(z)$, $q+q_{inf}$ on the left and $I(z)-q_{inf}(z)$, $I(z)-I_{inf}(z)$, $q$ on the right and noiseless data with $\epsilon=5\times 10^{-1}$ and $c=40$.
  • ...and 1 more figures

Theorems & Definitions (28)

  • Remark 1
  • Remark 2
  • Theorem 1
  • proof
  • Proposition 1
  • proof
  • Proposition 2
  • proof
  • Lemma 1
  • proof
  • ...and 18 more