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Exploring low-rank structure for an inverse scattering problem with far-field data

Yuyuan Zhou, Lorenzo Audibert, Shixu Meng, Bo Zhang

TL;DR

The paper tackles the challenging ill-posed inverse scattering problem by exploiting a low-rank structure built from disk PSWFs, enabling stable recovery of the contrast $q$ from far-field data. The method maps measurements to the unit disk, projects the data and unknown onto a disk-PSWF basis with a spectral cutoff, and obtains a stable reconstruction via a direct, explicit inversion $q_{m,n,l}=u_{m,n,l}/\alpha_{m,n}$. Theoretical support includes explicit stability bounds for Sobolev spaces and Lipschitz stability within the finite low-rank subspace, while extensive numerical experiments demonstrate improved resolution, robustness to noise and modeling errors, and increasing stability with frequency. The approach offers a computationally efficient alternative to iterative methods and FFT-based techniques, with demonstrated applicability to both Born and full nonlinear data and potential for extrapolating sparse measurements.

Abstract

In this work, we introduce a novel low-rank structure tailored for solving the inverse scattering problem. The particular low-rank structure is given by the generalized prolate spheroidal wave functions, computed stably and accurately via a Sturm-Liouville problem. We first process the far-field data to obtain a post-processed data set within a disk domain. Subsequently, the post-processed data are projected onto a low-rank space given by the low-rank structure. The unknown is approximately solved in this low-rank space, by dropping higher-order terms. The low-rank structure leads to an explicit stability estimate for unknown functions belonging to standard Sobolev spaces, and a Lipschitz stability estimate for unknowns belonging to a finite dimensional low-rank space. Various numerical experiments are conducted to validate its performance, encompassing assessments of resolution capability, robustness against randomly added noise and modeling errors, and demonstration of increasing stability.

Exploring low-rank structure for an inverse scattering problem with far-field data

TL;DR

The paper tackles the challenging ill-posed inverse scattering problem by exploiting a low-rank structure built from disk PSWFs, enabling stable recovery of the contrast from far-field data. The method maps measurements to the unit disk, projects the data and unknown onto a disk-PSWF basis with a spectral cutoff, and obtains a stable reconstruction via a direct, explicit inversion . Theoretical support includes explicit stability bounds for Sobolev spaces and Lipschitz stability within the finite low-rank subspace, while extensive numerical experiments demonstrate improved resolution, robustness to noise and modeling errors, and increasing stability with frequency. The approach offers a computationally efficient alternative to iterative methods and FFT-based techniques, with demonstrated applicability to both Born and full nonlinear data and potential for extrapolating sparse measurements.

Abstract

In this work, we introduce a novel low-rank structure tailored for solving the inverse scattering problem. The particular low-rank structure is given by the generalized prolate spheroidal wave functions, computed stably and accurately via a Sturm-Liouville problem. We first process the far-field data to obtain a post-processed data set within a disk domain. Subsequently, the post-processed data are projected onto a low-rank space given by the low-rank structure. The unknown is approximately solved in this low-rank space, by dropping higher-order terms. The low-rank structure leads to an explicit stability estimate for unknown functions belonging to standard Sobolev spaces, and a Lipschitz stability estimate for unknowns belonging to a finite dimensional low-rank space. Various numerical experiments are conducted to validate its performance, encompassing assessments of resolution capability, robustness against randomly added noise and modeling errors, and demonstration of increasing stability.
Paper Structure (24 sections, 4 theorems, 62 equations, 15 figures, 4 algorithms)

This paper contains 24 sections, 4 theorems, 62 equations, 15 figures, 4 algorithms.

Table of Contents

  1. Introduction
  2. Mathematical model of inverse scattering
  3. Generalization of prolate spheroidal wave functions
  4. Disk PSWFs
  5. Evaluation of disk PSWFs and prolate eigenvalues
  6. Jacobi polynomials
  7. Appoximation of disk PSWFs system
  8. Quadrature
  9. Proposed method based on the low-rank structure
  10. Data processing
  11. Low-rank reconstruction
  12. Numerical experiments
  13. Born model
  14. Data generation
  15. Noiseless and noisy post-processed data
  16. ...and 9 more sections

Key Result

Lemma 1

\newlabellemma: SL0 For any $c>0$, $\{\psi_{m,n,l}(x;c)\}^{l\in\mathbb{I}(m)}_{m,n\in \mathbb{N}}$ forms a complete and orthonormal system of $L^2(B(0,1))$, i.e., for $\forall~m,~n,~m',~n'\in\mathbb{N},~l\in\mathbb{I}(m),~l'\in\mathbb{I}(m')$ it holds that where $\delta$ denotes the Kronecker delta. The corresponding Sturm-Liouville eigenvalues $\{\chi_{m,n}\}_{m,n\in \mathbb{N}}$ in sturm-liouv

Figures (15)

  • Figure 1: (a) exact quadrature nodes. (b) $N_1 \times N_2 = 100\times 100$ grid nodes before dimentionality reduction. (c) approximate or mock-quadrature nodes.
  • Figure 1: Reconstruction of a disk PSWF. Left: $\psi_{3,2,2}(x;30)$, right: reconstruction.
  • Figure 2: Distributions of the absolute value of eigenvalues $\alpha_{m,n}(c)$ for $m=0,~5,~10,~15,~20,~25,~30,~35$. (a) $c=30$, (b) $c=90$.
  • Figure 2: Reconstruction of three different types of unknowns. First row: exact. Second row: Noiseless data. Third row: $20\%$ noisy data.
  • Figure 3: Reconstruction of the rectangle with the following dimensions of the low-rank space: $N_{\rm PSWFs}=1,~3,~5,~71,~73,~75,~134,~144,~192$. $k=15$ and $\delta=20\%$.
  • ...and 10 more figures

Theorems & Definitions (5)

  • Lemma 1
  • Lemma 2
  • Lemma 1
  • Theorem 2
  • Proof 1