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Convergence analysis of contrast source inversion type methods for acoustic inverse medium scattering problems

Qiao Hu, Bo Zhang, Haiwen Zhang

TL;DR

This work tackles the fixed-frequency acoustic inverse medium scattering problem by examining the CSI-type methods (CSI and SOM) and introducing two iteratively regularized variants (IRCSI and IRSOM) with an $\ell_1$ proximal term. The authors prove global convergence of the IRCSI/IRSOM schemes to an $\varepsilon$-stationary point under mild assumptions, marking the first such result for nonlinear inverse scattering at fixed frequency. Numerical experiments show that IRCSI/IRSOM achieve better stability and convergence than the original methods, especially under noisy data, while maintaining comparable computational cost. The study provides both theoretical guarantees and practical guidance for robust inverse scattering reconstructions.

Abstract

The contrast source inversion (CSI) method and the subspace-based optimization method (SOM) are first proposed in 1997 and 2009, respectively, and subsequently modified. The two methods and their variants share several properties and thus are called the CSI-type methods. The CSI-type methods are efficient and popular methods for solving inverse medium scattering problems, but their rigorous convergence remains an open problem. In this paper, we propose two iteratively regularized CSI-type (IRCSI-type) methods with a novel $\ell_1$ proximal term as the iteratively regularized term: the iteratively regularized CSI (IRCSI) method and the iteratively regularized SOM (IRSOM) method, which have a similar computation complexity to the original CSI and SOM methods, respectively, and prove their global convergence under natural and weak conditions on the original objective function. To the best of our knowledge, this is the first convergence result for iterative methods of solving nonlinear inverse scattering problems with a fixed frequency. The convergence and performance of the two IRCSI-type algorithms are illustrated by numerical experiments.

Convergence analysis of contrast source inversion type methods for acoustic inverse medium scattering problems

TL;DR

This work tackles the fixed-frequency acoustic inverse medium scattering problem by examining the CSI-type methods (CSI and SOM) and introducing two iteratively regularized variants (IRCSI and IRSOM) with an proximal term. The authors prove global convergence of the IRCSI/IRSOM schemes to an -stationary point under mild assumptions, marking the first such result for nonlinear inverse scattering at fixed frequency. Numerical experiments show that IRCSI/IRSOM achieve better stability and convergence than the original methods, especially under noisy data, while maintaining comparable computational cost. The study provides both theoretical guarantees and practical guidance for robust inverse scattering reconstructions.

Abstract

The contrast source inversion (CSI) method and the subspace-based optimization method (SOM) are first proposed in 1997 and 2009, respectively, and subsequently modified. The two methods and their variants share several properties and thus are called the CSI-type methods. The CSI-type methods are efficient and popular methods for solving inverse medium scattering problems, but their rigorous convergence remains an open problem. In this paper, we propose two iteratively regularized CSI-type (IRCSI-type) methods with a novel proximal term as the iteratively regularized term: the iteratively regularized CSI (IRCSI) method and the iteratively regularized SOM (IRSOM) method, which have a similar computation complexity to the original CSI and SOM methods, respectively, and prove their global convergence under natural and weak conditions on the original objective function. To the best of our knowledge, this is the first convergence result for iterative methods of solving nonlinear inverse scattering problems with a fixed frequency. The convergence and performance of the two IRCSI-type algorithms are illustrated by numerical experiments.

Paper Structure

This paper contains 7 sections, 3 theorems, 98 equations, 8 figures, 2 algorithms.

Table of Contents

  1. Introduction
  2. The direct and inverse medium scattering problems
  3. The original CSI and SOM methods for inverse medium scattering problems
  4. Iteratively regularized CSI and SOM algorithms
  5. Convergence analysis of the IRCSI and IRSOM algorithms
  6. Numerical experiments
  7. Conclusion

Key Result

Theorem 5.4

Suppose Assumption ass1 holds and let $\{z^k=(x^k, y^k)\}_{k\in{\mathbb N}}$ be the sequence generated by the IRCSI-type method e5-1--e5-3 with $\gamma,\beta>0$. Then the following statements hold: $(i)$ The sequence $\{z^k\}_{k\in{\mathbb N}}$ is convergent. $(ii)$ Let $L\geq\max_{1\leq j\leq J} \s

Figures (8)

  • Figure 1: The ground truth $m(x)=e^{-1/(1-|x|^2)}$ and the reconstruction results by the IRCSI and IRSOM algorithms with different values of $\beta$ at the $30000$-th iteration from the noisy measurement data with $5\%$ noise. Top row from left to right: the ground truth and the reconstructions by the IRCSI algorithm with $\beta=0,10^{-6},10^{-5},10^{-4},10^{-3}$. Bottom row from left to right: the reconstructions by the IRSOM algorithm with $\beta=0,10^{-5},10^{-4},10^{-3},10^{-2}$.
  • Figure 2: The ground truth $m(x)=e^{-1/(1-|x|^2)}$ and the reconstruction results by the IRCSI and IRSOM algorithms with different values of $\beta$ at the $5000$-th iteration from the noiseless measurement data. Top row from left to right: the ground truth and the reconstructions by the IRCSI algorithm with $\beta=0,10^{-6},10^{-5},10^{-4},10^{-3}$. Bottom row from left to right: the reconstructions by the IRSOM algorithm with $\beta=0,10^{-5},10^{-4},10^{-3},10^{-2}$.
  • Figure 3: The relative error between the ground truth and the reconstruction results by the IRCSI algorithm with $\beta=0,10^{-6},10^{-5},10^{-4},10^{-3}$ (left figure) and the IRSOM algorithm with $\beta=0,10^{-5},10^{-4},10^{-3},10^{-2}$ (right figure) against the iteration step, where the ground truth contrast is $m(x)=e^{-1/(1-|x|^2)}$ and the relative noise of the measurement data is $5\%$.
  • Figure 4: The relative error between the ground truth and the reconstruction results by the IRCSI algorithm with $\beta=0,10^{-6},10^{-5},10^{-4},10^{-3}$ (left figure) and the IRSOM algorithm with $\beta=0,10^{-5},10^{-4},10^{-3},10^{-2}$ (right figure) against the iteration step, where the ground truth contrast is $m(x)=e^{-1/(1-|x|^2)}$ and the measurement data has no noise.
  • Figure 5: The ground truth of the handwritten digit $0$ image and the reconstructions by the IRCSI and IRSOM algorithms with different values of $\beta$ at the $50000$-th iteration from the noisy data with $5\%$ noise. Top row from left to right: the ground truth and the reconstructions by the IRCSI algorithm with $\beta=0,10^{-6},10^{-5},10^{-4},10^{-3}$. Bottom row from left to right: the reconstructions by the IRSOM algorithm with $\beta=0,10^{-5},10^{-4},10^{-3},10^{-2}$.
  • ...and 3 more figures

Theorems & Definitions (7)

  • Remark 2.1
  • Definition 5.1
  • Remark 5.3
  • Theorem 5.4
  • Proof 1
  • Corollary 5.5
  • Corollary 5.6