Imaging nonlinearity coefficient and sound speed with the JMGT equation in frequency domain
Barbara Kaltenbacher
TL;DR
This work develops a frequency-domain framework for imaging spatially varying sound speed $c(x)$ and nonlinearity $\eta(x)$ in the Jordan-Moore-Gibson-Thompson (JMGT) equation using data from two sources. It leverages a multiharmonic expansion to capture higher harmonics from nonlinearity and analyzes a linearized forward map, proving continuous invertibility and local stability. The authors establish local uniqueness and stability for the nonlinear inverse problem via an inverse-function approach and show that increasing the relaxation time $\tau$ improves stability, enabling a quasi-reversibility interpretation that regularizes toward the Westervelt model as $\tau\to0$. They further develop a tau-regularization scheme, proving convergence of tau-regularized reconstructions for noisy data under suitable choices of $\tau(\delta)$ and data smoothing, with careful handling of norms and trace operators. The results provide a two-source identification pathway that yields both linear and nonlinear imaging parameters and offer practical regularization insights for quasi-reversibility in nonlinear acoustics.
Abstract
In this paper we prove uniqueness and stability of reconstruction of two coefficients (sound speed and nonlinearity parameter) in the Jordan-Moore-Gibson-Thompson JMGT equation of nonlinear acoustics, relying on observations resulting from only two sources. A key tool for this purpose is a multiharmonic expansion of the PDE solution, which reflects the physical phenomenon of higher harmonics appearing due to nonlinearity and allows us to work in frequency domain. Based on this result, we derive a regularization property of reconstruction with JMGT as the relexation time tends to zero (in the spirit of a quasi reversibility method) for reconstruction from the classical Westervelt equation.