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Increasing resolution and instability for linear inverse scattering problems

Pu-Zhao Kow, Mikko Salo, Sen Zou

TL;DR

This work analyzes how increasing frequency improves the resolvable content in linear inverse scattering problems under a fixed frequency. It establishes a clear division between a stable region, where forward-operator singular values remain bounded as frequency grows, and an unstable region, where they decay exponentially, thereby formalizing an increasing-resolution phenomenon. The authors derive rigorous singular-value asymptotics for the Herglotz operator and its linearized scattering counterpart using structural tools (Agmon– Hörmander estimates, Courant min–max, and the coarea formula) and provide optimality results, together with stability-into-resolution trade-offs via an instability framework. Numerical experiments support the theoretical bounds. Overall, the paper reframes increasing stability as increasing resolution and quantifies the limits of stable recovery in high-frequency linear inverse scattering.

Abstract

In this work we study the increasing resolution of linear inverse scattering problems at a large fixed frequency. We consider the problem of recovering the density of a Herglotz wave function, and the linearized inverse scattering problem for a potential. It is shown that the number of features that can be stably recovered (stable region) becomes larger as the frequency increases, whereas one has strong instability for the rest of the features (unstable region). To show this rigorously, we prove that the singular values of the forward operator stay roughly constant in the stable region and decay exponentially in the unstable region. The arguments are based on structural properties of the problems and they involve the Courant min-max principle for singular values, quantitative Agmon-Hörmander estimates, and a Schwartz kernel computation based on the coarea formula.

Increasing resolution and instability for linear inverse scattering problems

TL;DR

This work analyzes how increasing frequency improves the resolvable content in linear inverse scattering problems under a fixed frequency. It establishes a clear division between a stable region, where forward-operator singular values remain bounded as frequency grows, and an unstable region, where they decay exponentially, thereby formalizing an increasing-resolution phenomenon. The authors derive rigorous singular-value asymptotics for the Herglotz operator and its linearized scattering counterpart using structural tools (Agmon– Hörmander estimates, Courant min–max, and the coarea formula) and provide optimality results, together with stability-into-resolution trade-offs via an instability framework. Numerical experiments support the theoretical bounds. Overall, the paper reframes increasing stability as increasing resolution and quantifies the limits of stable recovery in high-frequency linear inverse scattering.

Abstract

In this work we study the increasing resolution of linear inverse scattering problems at a large fixed frequency. We consider the problem of recovering the density of a Herglotz wave function, and the linearized inverse scattering problem for a potential. It is shown that the number of features that can be stably recovered (stable region) becomes larger as the frequency increases, whereas one has strong instability for the rest of the features (unstable region). To show this rigorously, we prove that the singular values of the forward operator stay roughly constant in the stable region and decay exponentially in the unstable region. The arguments are based on structural properties of the problems and they involve the Courant min-max principle for singular values, quantitative Agmon-Hörmander estimates, and a Schwartz kernel computation based on the coarea formula.
Paper Structure (13 sections, 15 theorems, 171 equations, 3 figures)

This paper contains 13 sections, 15 theorems, 171 equations, 3 figures.

Table of Contents

  1. Introduction
  2. Unique continuation for Herglotz wave functions
  3. Linearized inverse scattering
  4. Methods
  5. Organization
  6. Singular values of Herglotz operator
  7. The stable region
  8. The unstable region
  9. Singular values of linearized scattering matrix
  10. From singular values to instability estimates
  11. Numerical simulations for singular values
  12. Unique continuation for Herglotz waves
  13. Linearized inverse scattering problem

Key Result

Theorem 1.1

Let $n\ge 2$ and $\kappa \ge 1$. The singular values $\sigma_{j}(A_{\kappa})$ of eq:linear-injective-compact-map1 satisfy where the constant $c>0$ and the implied constants are independent of $\kappa$ and $j$.

Figures (3)

  • Figure A.1: The singular values of $A_{\kappa}$ (first row) and $Q_{\kappa}$ (second row) in descending order (counting multiplicity). In the first column, the $y$-axis is of $\log$-scale, while in the second column, both $x$ and $y$-axis are of $\log$-scale.
  • Figure A.2: Plot of singular values $\hat{\sigma}_{j}$ when $n=2$ (first row) and $n=3$ (second row). The singular values for normalized far-field operator $F_{\kappa}$ are given in the first column while operators $\tilde{F}_{\kappa}$ are given in the second column. In each figure, both $x$ and $y$-axis are of $\log$-scale.
  • Figure A.3: Plot of $\hat{\sigma}_{1}$ versus wave number $\kappa$

Theorems & Definitions (17)

  • Theorem 1.1
  • Theorem 1.2
  • Theorem 1.3: See also \ref{['thm_scattering_refined']}
  • Remark 1.4
  • Theorem 1.5
  • Remark 1.6
  • Lemma 2.1
  • Lemma 2.2
  • Lemma 2.3
  • Lemma 3.1
  • ...and 7 more