## On an inverse problem of nonlinear imaging with fractional damping

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Barbara Kaltenbacher and William Rundell
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## Abstract:

This paper considers the attenuated Westervelt equation in pressure formulation. The attenuation is by various models proposed in the literature and characterised by the inclusion of non-local operators that give power law damping as opposed to the exponential of classical models. The goal is the inverse problem of recovering a spatially dependent coefficient in the equation, the parameter of nonlinearity $\kappa (x)$, in what becomes a nonlinear hyperbolic equation with non-local terms. The overposed measured data is a time trace taken on a subset of the domain or its boundary. We shall show injectivity of the linearised map from $\kappa$ to the overposed data and from this basis develop and analyse Newton-type schemes for its effective recovery.## References

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## Additional Information

**Barbara Kaltenbacher**- Affiliation: Department of Mathematics, Alpen-Adria-Universität Klagenfurt, Universitatsstrasse 65-67, 9020 Klagenfurt Austria
- MR Author ID: 616341
- ORCID: 0000-0002-3295-6977
- Email: barbara.kaltenbacher@aau.at
**William Rundell**- Affiliation: Department of Mathematics, Texas A&M University, College Station, Texas 77843
- MR Author ID: 213241
- Email: rundell@math.tamu.edu
- Received by editor(s): February 12, 2021
- Received by editor(s) in revised form: May 19, 2021
- Published electronically: September 20, 2021
- Additional Notes: The work of the first author was supported by the Austrian Science Fund fwf under the grants P30054 and doc78. The work of the second author was supported in part by the National Science Foundation through awards dms-1620138 and dms-2111020
- © Copyright 2021 American Mathematical Society
- Journal: Math. Comp.
**91**(2022), 245-276 - MSC (2020): Primary 35R30, 35R11, 35L70, 78A46
- DOI: https://doi.org/10.1090/mcom/3683
- MathSciNet review: 4350539