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

Abstract

<p style='text-indent:20px;'>We consider an undetermined coefficient inverse problem for a nonlinear 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 <inline-formula><tex-math id="M1">\begin{document}$ B/A $\end{document}</tex-math></inline-formula> in the literature which is part of a space dependent coefficient <inline-formula><tex-math id="M2">\begin{document}$ \kappa $\end{document}</tex-math></inline-formula> in the Westervelt equation governing nonlinear acoustics. Corresponding to the typical measurement setup, the overposed data consists of time trace measurements on some zero or one dimensional set <inline-formula><tex-math id="M3">\begin{document}$ \Sigma $\end{document}</tex-math></inline-formula> representing the receiving transducer array. After an analysis of the map from <inline-formula><tex-math id="M4">\begin{document}$ \kappa $\end{document}</tex-math></inline-formula> to the overposed data, we show injectivity of its linearisation and use this as motivation for several iterative schemes to recover <inline-formula><tex-math id="M5">\begin{document}$ \kappa $\end{document}</tex-math></inline-formula>. Numerical simulations will also be shown to illustrate the efficiency of the methods.</p>