On an inverse problem of nonlinear imaging with fractional damping

TL;DR

This paper investigates an inverse problem for a nonlinear, fractional damping wave equation, aiming to recover a spatially varying nonlinearity parameter using overposed data and Newton-type algorithms.

Contribution

It introduces a novel inverse problem involving nonlocal fractional damping models and develops Newton-type schemes for recovering the nonlinear coefficient.

Findings

Proved injectivity of the linearized inverse map.

Developed Newton-type algorithms for coefficient recovery.

Analyzed the convergence of the proposed schemes.

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.