Source and metric estimation in the eikonal equation using optimization on a manifold

We address the estimation of the source(s) location in the eikonal equation on a Riemann surface, as well as the determination of the metric when it depends on a few parameters. The available observations are the arrival times or are obtained indirectly from the arrival times by an observation operator, this frame is intended to describe electro-cardiographic imaging. The sensitivity of the arrival times is computed from \begin{document}$ {{{\rm{Log}}}}_x $\end{document} the log map wrt to the source \begin{document}$ x $\end{document} on the surface. The \begin{document}$ {{{\rm{Log}}}}_x $\end{document} map is approximated by solving an elliptic vectorial equation, using the Vector Heat Method. The \begin{document}$ L^2 $\end{document}-error function between the model predictions and the observations is minimized using Gauss-Newton optimization on the Riemann surface. This allows to obtain fast convergence. We present numerical results, where coefficients describing the metric are also recovered like anisotropy and global orientation.