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

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 $κ$ 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 $Σ$ representing the receiving transducer array. After an analysis of the map from $κ$ to the overposed data, we show injectivity of its linearisation and use this as motivation for several iterative schemes to recover $κ$. Numerical simulations will also be shown to illustrate the efficiency of the methods.

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