American Institute of Mathematical Sciences

doi: 10.3934/ipi.2021020

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

 1 Department of Mathematics, Alpen-Adria-Universität Klagenfurt, 9020 Klagenfurt, Austria 2 Department of Mathematics, Texas A&M University, Texas 77843, USA

* Corresponding author: Barbara Kaltenbacher

Received  August 2020 Revised  November 2020 Published  February 2021

Fund Project: Supported by the Austrian Science Fund fwf under grant P30054 and the National Science Foundation through award dms-1620138

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 $B/A$ 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.

Citation: Barbara Kaltenbacher, William Rundell. On the identification of the nonlinearity parameter in the Westervelt equation from boundary measurements. Inverse Problems & Imaging, doi: 10.3934/ipi.2021020
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The surface Σ
Reconstructions of a smooth $\kappa(x)$ from time trace data at $\,x = 1\,$ under 0.1% (left) and 1% (right) noise using Newton's method
Reconstructions of piecewise linear $\kappa(x)$ from time trace data at $\,x = 1$ under 0.1% (left) and 1% (right) noise using Newton's method
Reconstructions of a piecewise constant $\kappa(x)$ from time trace data at $x = 1$ under $0.1\%$ noise using Newton iteration
Comparison of Newton (in red) and Halley (in blue) final reconstructions under $0.1\%$ noise
Comparison of Newton (in red) and Halley (in blue) final reconstructions and norm differences of the $n^{\rm th}$ iterate $\kappa_n$ and the actual $\kappa$. Noise level was $1\%$
Reconstructions of a piecewise linear $\kappa(x)$ from time trace data at $x = 1$ under $1\%$ noise using Landweber iteration
The leftmost figure shows reconstructions of $\kappa(x)$ under $0.1\%$ noise using Landweber iteration. The rightmost figure shows the decay of the norm $\kappa_n(x)-\kappa_{\rm act}(x)$
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