A Lipschitz stable reconstruction formula for the inverse problem for the wave equation

Abstract:

We consider the problem to reconstruct a wave speed $c \in C^\infty (M)$ in a domain $M \subset \mathbb {R}^n$ from acoustic boundary measurements modelled by the hyperbolic Dirichlet-to-Neumann map $\Lambda$. We introduce a reconstruction formula for $c$ that is based on the Boundary Control method and incorporates features also from the complex geometric optics solutions approach. Moreover, we show that the reconstruction formula is locally Lipschitz stable for a low frequency component of $c^{-2}$ under the assumption that the Riemannian manifold $(M, c^{-2} dx^2)$ has a strictly convex function with no critical points. That is, we show that for all bounded $C^2$ neighborhoods $U$ of $c$, there is a $C^1$ neighborhood $V$ of $c$ and constants $C, R > 0$ such that \begin{align*} |\mathcal {F}\left (\widetilde c^{-2} - c^{-2}\right )(\xi )| \le C e^{2R|\xi |}\left \|\widetilde \Lambda - \Lambda \right \|_*, \quad \xi \in \mathbb {R}^n, \end{align*} for all $\widetilde c \in U \cap V$, where $\widetilde \Lambda$ is the Dirichlet-to-Neumann map corresponding to the wave speed $\widetilde c$ and $\left |\cdot \right |_*$ is a norm capturing certain regularity properties of the Dirichlet-to-Neumann maps.