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Stable determination of the initial data in an IBVP for the wave equation outside a non-trapping obstacle

Mourad Choulli

TL;DR

This work addresses recovering the initial data of a wave equation posed outside a non-trapping obstacle from localized boundary measurements. It combines exponential local energy decay for odd dimensions with a quantized unique continuation via a Laplace-transform framework to establish a $\text{double logarithmic stability inequality}$, linking initial data errors to boundary measurements. The results are extended to variable sound speed and linked to applications in thermoacoustic tomography, with discussion of when unique recovery is possible or limited by the data and model assumptions. The analysis provides a rigorous stability mechanism for interior data recovery in exterior-domain wave problems and informs practical inversion scenarios in imaging modalities that rely on boundary data.

Abstract

We establish a double logarithmic stability inequality for the problem of determining the initial data in an IBVP for the wave equation outside a non-trapping obstacle from two localized measurements.

Stable determination of the initial data in an IBVP for the wave equation outside a non-trapping obstacle

TL;DR

This work addresses recovering the initial data of a wave equation posed outside a non-trapping obstacle from localized boundary measurements. It combines exponential local energy decay for odd dimensions with a quantized unique continuation via a Laplace-transform framework to establish a , linking initial data errors to boundary measurements. The results are extended to variable sound speed and linked to applications in thermoacoustic tomography, with discussion of when unique recovery is possible or limited by the data and model assumptions. The analysis provides a rigorous stability mechanism for interior data recovery in exterior-domain wave problems and informs practical inversion scenarios in imaging modalities that rely on boundary data.

Abstract

We establish a double logarithmic stability inequality for the problem of determining the initial data in an IBVP for the wave equation outside a non-trapping obstacle from two localized measurements.
Paper Structure (3 sections, 3 theorems, 42 equations)

This paper contains 3 sections, 3 theorems, 42 equations.

Table of Contents

  1. Introduction
  2. Proof of Theorem \ref{['theorem1.0']}
  3. Comments

Key Result

Theorem 1.1

There exist $\varkappa=\varkappa (\Omega,Q) >0$ and $\delta=\delta (\Omega,Q) >0$ so that for any $(f,g)\in \mathbf{D}$ we have where $u=u(f,g)$.

Theorems & Definitions (3)

  • Theorem 1.1
  • Theorem 1.2
  • Corollary 3.1