Determining the nonlinearity in an acoustic wave equation

TL;DR

This paper investigates an inverse problem for a nonlinear acoustic wave equation with an unknown nonlinearity function, proposing methods to recover it from data and demonstrating their effectiveness through analysis and simulations.

Contribution

It introduces a generalized nonlinear model for ultrasound propagation, analyzes its well-posedness, and develops iterative schemes for recovering the unknown nonlinearity function from measurements.

Findings

Injectivity of the linearized forward map established

Iterative schemes successfully recover the nonlinearity function

Numerical simulations demonstrate method efficiency

Abstract

We consider an undetermined coefficient inverse problem for a nonlinear partial differential equation describing high intensity ultrasound propagation as widely used in medical imaging and therapy. The usual nonlinear term in the standard model using the Westervelt equation in pressure formulation is of the form $p p_t$. However, this should be considered as a low order approximation to a more complex physical model where higher order terms will be required. Here we assume a more general case where the form taken is $f(p)\,p_t$ and $f$ is unknown and must be recovered from data measurements. Corresponding to the typical measurement setup, the overposed data consists of time trace observations of the acoustic pressure at a single point or on a one dimensional set $\Sigma$ representing the receiving transducer array at a fixed time. Additionally to an analysis of well-posedness of the resulting {\sc pde}, we show injectivity of the linearized forward map from $f$ to the overposed data and use this as motivation for several iterative schemes to recover $f$. Numerical simulations will also be shown to illustrate the efficiency of the methods.