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An inverse random source problem for the fractional Helmholtz equation

Peijun Li, Zhenqian Li

Abstract

This paper investigates an inverse random source problem for the stochastic fractional Helmholtz equation. The source is modeled as a centered, complex-valued, microlocally isotropic generalized Gaussian random field whose covariance and relation operators are described by classical pseudo-differential operators. For sufficiently large wavenumbers, we first establish the well-posedness of the direct problem in the distributional sense by analyzing the corresponding Lippmann--Schwinger integral equation. For the inverse problem, we show that the principal symbols of both the covariance and relation operators can be uniquely determined, with probability one, from the far-field patterns generated by a single realization of the random source. The approach employs a combination of the Born linearization, asymptotic expansions of the fractional Helmholtz Green kernel at high wavenumbers, and microlocal analysis of associated Fourier integral operators.

An inverse random source problem for the fractional Helmholtz equation

Abstract

This paper investigates an inverse random source problem for the stochastic fractional Helmholtz equation. The source is modeled as a centered, complex-valued, microlocally isotropic generalized Gaussian random field whose covariance and relation operators are described by classical pseudo-differential operators. For sufficiently large wavenumbers, we first establish the well-posedness of the direct problem in the distributional sense by analyzing the corresponding Lippmann--Schwinger integral equation. For the inverse problem, we show that the principal symbols of both the covariance and relation operators can be uniquely determined, with probability one, from the far-field patterns generated by a single realization of the random source. The approach employs a combination of the Born linearization, asymptotic expansions of the fractional Helmholtz Green kernel at high wavenumbers, and microlocal analysis of associated Fourier integral operators.
Paper Structure (13 sections, 14 theorems, 185 equations)

This paper contains 13 sections, 14 theorems, 185 equations.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Sobolev spaces
  4. Complex-valued GMIG random fields
  5. The direct problem
  6. The inverse problem
  7. The Born series
  8. Estimates of $F_{0}$
  9. Estimates of $F_{1}$
  10. Estimates of $F_{ 2 }$
  11. Uniqueness
  12. Conclusion
  13. The Green's function

Key Result

Lemma 2.1

Let $h$ be a GMIG field of order $- m$ supported in an open set $D \subset \mathbb{R}^{ d }$, with $m<d$. Then, for any $\epsilon > 0$ and any $p \in ( 1, \infty )$, it holds almost surely that $h \in W^{ - \frac{ d - m }{ 2 } - \epsilon, p } ( D )$.

Theorems & Definitions (28)

  • Lemma 2.1
  • Remark 2.3
  • Theorem 3.1
  • proof
  • Theorem 3.2
  • proof
  • Theorem 3.3
  • proof
  • Theorem 3.4
  • proof
  • ...and 18 more