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The temporal domain derivative in inverse acoustic obstacle scattering

by M. Knöller and J. Nick

(Report number 2024-17)

Abstract
This work describes and analyzes the domain derivative for a time-dependent acoustic scattering problem. We study the nonlinear operator that maps a sound-soft scattering object to the solution of the time-dependent wave equation evaluated at a finite number of points away from the obstacle. The Fréchet derivative of this operator with respect to variations of the scatterer coincides with point evaluations of the temporal domain deriva- tive. The latter is the solution to another time-dependent scattering problem, for which a well-posedness result is shown under sufficient temporal regularity of the incoming wave. Applying convolution quadrature to this scattering problem gives a stable and provably con- vergent semi-discretization in time, provided that the incoming wave is sufficient regular. Using the discrete domain derivative in a Gauss–Newton method, we describe an efficient algorithm to reconstruct the boundary of an unknown scattering object from time domain measurements in a few points away from the boundary. Numerical examples for the acoustic wave equation in two dimensions demonstrate the performance of the method.

Keywords: inverse scattering, wave equation, temporal domain derivative, convolution quadrature

@Techreport{KN24_1099,
  author = {M. Kn\"oller and J. Nick},
  title = {The temporal domain derivative in inverse acoustic obstacle scattering},
  institution = {Seminar for Applied Mathematics, ETH Z{\"u}rich},
  number = {2024-17},
  address = {Switzerland},
  url = {https://www.sam.math.ethz.ch/sam_reports/reports_final/reports2024/2024-17.pdf },
  year = {2024}
}

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