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Linearized boundary control method for density reconstruction in acoustic wave equations.

Lauri Oksanen1, Tianyu Yang2, Yang Yang2

  • 1Department of Mathematics and Statistics, University of Helsinki, Helsinki, Finland.

Inverse Problems
|December 16, 2024
PubMed
Summary
This summary is machine-generated.

This study introduces a boundary control method to reconstruct density variations in acoustic wave equations. The new approach offers stable algorithms for identifying unknown density perturbations from boundary measurements.

Keywords:
Neumann-to-Dirichlet mapacoustic wave equationboundary control methodincreasing stabilityinverse boundary value problem

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Area of Science:

  • Inverse problems
  • Acoustic wave equation
  • Mathematical modeling

Background:

  • Inverse boundary value problems are crucial for determining material properties from limited measurements.
  • Reconstructing density perturbations in acoustic wave equations is challenging due to wave propagation complexities.

Purpose of the Study:

  • To develop a linearized boundary control method for solving the inverse problem of determining density in the acoustic wave equation.
  • To reconstruct an unknown density perturbation within a known background density using the linearized Neumann-to-Dirichlet map.

Main Methods:

  • Linearized boundary control method.
  • Utilizing a linearized Blagoves̆c̆enskiĭ's identity with a free parameter.
  • Deriving reconstructive algorithms with stability estimates.

Main Results:

  • Two stable reconstruction algorithms are derived for constant background densities.
  • An increasing stability estimate is established for non-constant background densities.
  • Numerical experiments validate the feasibility and performance of the proposed algorithms.

Conclusions:

  • The developed linearized boundary control method provides a robust framework for density reconstruction in acoustic wave equations.
  • The method demonstrates stability and feasibility for both constant and non-constant background densities.
  • This work contributes to advancements in inverse problem methodologies for wave propagation analysis.