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Determining evolutionary equations by a single passive boundary observation

Lu Chen, Yan Jiang, Hongyu Liu, Catharine W. K. Lo, Longyue Tao

Abstract

This paper addresses the longstanding inverse problem of simultaneously recovering both causal sources and medium parameters in evolutionary PDEs from a single passive boundary observation. We develop a mathematical framework focusing on second-order hyperbolic systems, where the measurement consists of the Cauchy pair of the wave field restricted to an open subset of the boundary over all positive time. Under a structural invariance condition on the source-to-speed ratio, we prove that the initial source \(f\) and wave speed \(c\) of the wave equation are uniquely determined by such boundary data. The proof combines intricate integral identities, Fourier and harmonic analysis, and subtle high-frequency asymptotics, avoiding artificial decoupling assumptions and accommodating physically realistic scenarios. As key extensions, we demonstrate the recovery of multiple unknowns in more general hyperbolic systems and establish analogous unique determination results for parabolic and Schrödinger-type equations, showcasing the versatility of the methodology. These results resolve a major open problem in coupled-physics imaging and provide a rigorous mathematical foundation for similar inverse problems arising in more sophisticated evolutionary settings.

Determining evolutionary equations by a single passive boundary observation

Abstract

This paper addresses the longstanding inverse problem of simultaneously recovering both causal sources and medium parameters in evolutionary PDEs from a single passive boundary observation. We develop a mathematical framework focusing on second-order hyperbolic systems, where the measurement consists of the Cauchy pair of the wave field restricted to an open subset of the boundary over all positive time. Under a structural invariance condition on the source-to-speed ratio, we prove that the initial source and wave speed of the wave equation are uniquely determined by such boundary data. The proof combines intricate integral identities, Fourier and harmonic analysis, and subtle high-frequency asymptotics, avoiding artificial decoupling assumptions and accommodating physically realistic scenarios. As key extensions, we demonstrate the recovery of multiple unknowns in more general hyperbolic systems and establish analogous unique determination results for parabolic and Schrödinger-type equations, showcasing the versatility of the methodology. These results resolve a major open problem in coupled-physics imaging and provide a rigorous mathematical foundation for similar inverse problems arising in more sophisticated evolutionary settings.
Paper Structure (24 sections, 10 theorems, 158 equations)

This paper contains 24 sections, 10 theorems, 158 equations.

Table of Contents

  1. Introduction
  2. TAT/PAT formulation and main results
  3. First uniqueness result: determining the ratio $f/c^2$ independent of $c$
  4. Second uniqueness result: simultaneously determining $f$ and $c$
  5. Discussion and relation to existing literature
  6. Organization of the paper
  7. Mathematical setting and hyperbolic forward problems
  8. Well-posedness of the forward hyperbolic problem
  9. Temporal Fourier transform and non-trapping condition
  10. Proof of Theorem \ref{['thm0']}
  11. First integral identity
  12. Construction of CGO test waves
  13. Proof of Theorem \ref{['thm0']}
  14. High‐frequency analysis and simultaneous recovery
  15. Frequency-domain reduction
  16. ...and 9 more sections

Key Result

Theorem 1.2

Assume that $c,\tilde{c}\in L^\infty(\mathbb{R}^3)$ and $f,\tilde{f}\in H^{2}(\mathbb{R}^3)$ satisfy eq:fhsupp, eq:unifbdd and eq:esssupp. Let $(f,c)$ and $(\tilde{f},\tilde{c})$ be two configurations fulfilling the admissibility condition ass:x3_invariance_cf, and the non-trapping condition in Def Then

Theorems & Definitions (21)

  • Remark 1.1
  • Theorem 1.2
  • Theorem 1.3
  • Remark 1.4
  • Lemma 2.1
  • Definition 2.2
  • Definition 2.3
  • Lemma 3.1
  • proof
  • Lemma 3.2
  • ...and 11 more