On an inverse problem of nonlinear imaging with fractional damping

Kaltenbacher, Barbara; Rundell, William

doi:10.1090/mcom/3683

On an inverse problem of nonlinear imaging with fractional damping
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Math. Comp. 91 (2022), 245-276 Request permission

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.

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