On an inverse problem of nonlinear imaging with fractional damping

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 nonlocal 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 used to recover it and from this basis develop and analyse Newton-type schemes for its effective recovery.