Global Convexity in a Three-Dimensional Inverse Acoustic Problem

We consider an inverse scattering problem (ISP) for the acoustic equation % uₜₜ=c_²(x)Δ u,u|ₜ₌₀=0,uₜ|ₜ₌₀=δ (x),x∈ R³. The ISP consists of the determination of the speed of sound $c(x)$ inside a bounded domain Ω ⊂ R³ given $c(x)$ outside Ω and measurements of the amplitude $u(x,t)$ of the sound at the boundary % ∂ Ω ,\;u|_∂ Ω=φ (x,t). This problem is nonoverdetermined since only a single source location at \ 0\ is counted. Assuming regularity of the rays generated by $c(x)$ and using the Carleman's weight functions, we construct a cost functional J_λ. The main result is Theorem 3.1, which claims global strict convexity of J_λ 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_λ (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.