GLOBAL CONVEXITY IN A 3-DIMENSIONAL INVERSE ACOUSTIC PROBLEM

We consider an inverse scattering problem (ISP) for the acoustic equat ion u(tt) = c(2)(x)Delta u, u/(t=0) = 0, ut/(t=0) = delta(x),x epsilon R-3. The ISP consists of the determination of the speed of sound c(x) inside a bounded domain Omega subset of R-3 given c(x) outside Omega and measurements of the amplitude u(x, t) of the sound at the boundary partial derivative Omega, u/(partial derivative Omega) = phi(x, t). T his problem is nonoverdetermined since only a single source location a t {0} is counted. Assuming regularity of the rays generated by c(x) an d using the Carleman's weight functions, we construct a cost functiona l J(lambda). The main result is Theorem 3.1, which claims global stric t convexity of J(lambda) on ''reasonable'' compact sets of solutions. Therefore, global convergence on such a set of a number of standard mi nimization 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.