On the identification of the nonlinearity parameter in the Westervelt equation from boundary measurements

TL;DR

This paper addresses the inverse problem of determining the nonlinearity parameter in the Westervelt equation from boundary measurements, proposing iterative methods and demonstrating their effectiveness through numerical simulations.

Contribution

It introduces new iterative schemes for recovering the nonlinearity coefficient in the Westervelt equation based on boundary data analysis.

Findings

Injectivity of the linearized map from coefficient to data

Development of efficient iterative reconstruction algorithms

Numerical simulations confirm the effectiveness of the proposed methods

Abstract

We consider an undetermined coefficient inverse problem for a non-\\linear partial differential equation occurring in high intensity ultrasound propagation as used in acoustic tomography. In particular, we investigate the recovery of the nonlinearity coefficient commonly labeled as $B/A$ in the literature which is part of a space dependent coefficient $\kappa$ in the Westervelt equation governing nonlinear acoustics. Corresponding to the typical measurement setup, the overposed data consists of time trace measurements on some zero or one dimensional set $\Sigma$ representing the receiving transducer array. After an analysis of the map from $\kappa$ to the overposed data, we show injectivity of its linearisation and use this as motivation for several iterative schemes to recover $\kappa$. Numerical simulations will also be shown to illustrate the efficiency of the methods.