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Dynamic Inverse Problems for the Acoustic Wave Equation

In: Time-dependent Problems in Imaging and Parameter Identification

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  • Thies Gerken

    (University of Bremen)

Abstract

We consider the identification of a time- and space dependent wave speed and mass density based on the knowledge of the wave field. The wave propagation is modeled by the acoustic wave equation. By making use of an abstract framework for parameter reconstruction in hyperbolic partial differential equations, we are able to obtain a well-defined forward operator. Furthermore, we prove the Fréchet-differentiability of this forward operator and the local ill-posedness of the inverse problems. In order to facilitate the application of regularization schemes, we also calculate the necessary adjoint operators. The theoretical considerations are complemented by a numerical demonstration of the inversion using the regularization method cg-reginn in two- and three-dimensional settings. There we present the numerically obtained convergence rates and show that even in this time-dependent setting one can obtain good reconstructions with reasonable computational effort.

Suggested Citation

  • Thies Gerken, 2021. "Dynamic Inverse Problems for the Acoustic Wave Equation," Springer Books, in: Barbara Kaltenbacher & Thomas Schuster & Anne Wald (ed.), Time-dependent Problems in Imaging and Parameter Identification, pages 25-49, Springer.
    • Handle: RePEc:spr:sprchp:978-3-030-57784-1_2
    DOI: 10.1007/978-3-030-57784-1_2
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