Global Convexity in a Three-Dimensional Inverse Acoustic Problem

Abstract

We consider an inverse scattering problem (ISP) for the acoustic equation $% u_{tt}=c_{}^2(x)\Delta u,u|_{t=0}=0,u_t|_{t=0}=\delta (x),x\in {\Bbb R}^3.$ The ISP consists of the determination of the speed of sound $c(x)$ inside a bounded domain $\Omega \subset {\Bbb R}^3$ given $c(x)$ outside $\Omega $ and measurements of the amplitude $u(x,t)$ of the sound at the boundary $% \partial \Omega ,\;u|_{_{\partial \Omega }}=\varphi (x,t).$ This problem is nonoverdetermined since only a single source location at $\left\{ 0\right\}$ is counted. Assuming regularity of the rays generated by $c(x)$ and using the Carleman's weight functions, we construct a cost functional $J_\lambda $. The main result is Theorem 3.1, which claims global strict convexity of $J_\lambda $ on "reasonable" compact sets of solutions. Therefore, global convergence on such a set of a number of standard minimization algorithms to the unique global minimum of $J_\lambda $ (i.e., solution of the ISP) is guaranteed. This in turn shows a possibility of constructions of numerical methods for this ISP which would not be affected by the problem of local minima.