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Increasing stability for inverse source problem with limited-aperture far field data at multi-frequencies

Ibtissem Ben Aïcha, Guanghui Hu, Suliang Si

Abstract

We study the increasing stability of an inverse source problem for the Helmholtz equation from limited-aperture far field data at multiple wave numbers. The measurement data are givenby the far field patterns $u^\infity(\hat{x},k)$ for all observation directions in some neighborhood of a fixed direction $\hat{x}$ and for all wave numbers k belonging to a finite interval $(0,K)$. In this paper, we discuss the increasing stability with respect to the width of the wavenumber interval $K>1$. In three dimensions we establish stability estimates of the $L^2$-norm and $H^{-1}$-norm of the source function from the far field data. The ill-posedness of the inverse source problem turns out to be of Hölder type while increasing the wavenumber band K. We also discuss an analytic continuation argument of the far-field data with respect to the wavenumbers at a fixed direction.

Increasing stability for inverse source problem with limited-aperture far field data at multi-frequencies

Abstract

We study the increasing stability of an inverse source problem for the Helmholtz equation from limited-aperture far field data at multiple wave numbers. The measurement data are givenby the far field patterns for all observation directions in some neighborhood of a fixed direction and for all wave numbers k belonging to a finite interval . In this paper, we discuss the increasing stability with respect to the width of the wavenumber interval . In three dimensions we establish stability estimates of the -norm and -norm of the source function from the far field data. The ill-posedness of the inverse source problem turns out to be of Hölder type while increasing the wavenumber band K. We also discuss an analytic continuation argument of the far-field data with respect to the wavenumbers at a fixed direction.
Paper Structure (7 sections, 12 theorems, 96 equations)

This paper contains 7 sections, 12 theorems, 96 equations.

Table of Contents

  1. Introduction
  2. Main results
  3. Stability estimate of the $L^2$-norm: proof of Theorem \ref{['THM1']}
  4. Preliminaries
  5. Proof of Theorem \ref{['THM1']}
  6. Stability estimate of the $H^{-1}$-norm: proof of Theorem \ref{['THM2']}
  7. Analytic continuation with respect to wavenumbers and observation angles: proof of Theorems \ref{['THM3']} and \ref{['THM4']}

Key Result

Theorem 2.1

Let $f\in \mathcal{C}_{M,2n+1}$ and let $u\in H^{2n+3}_{loc}(\mathbb{R}^{3})$ be the unique solution to the equation (1). Then there exists $\alpha>0$ such that where $C>0$ is a constant independent of $K>0$.

Theorems & Definitions (17)

  • Theorem 2.1
  • Theorem 2.2
  • Theorem 2.3
  • Theorem 2.4
  • Lemma 3.1
  • Lemma 3.2
  • proof
  • Lemma 3.3
  • proof
  • Lemma 3.4
  • ...and 7 more