Global convexity in a single-source 3-D inverse scattering problem

The authors consider nonoverdetermined 3-D inverse scatteringproblems based on the telegraph equation The goal is to recover the media properties represented by thecoefficients a(x) or b(x) from, for example, backscatteringdata. Such a problem models imaging in certain biological tissues,in murky water, and in some geophysical and atmospheric phenomena.The main restriction is in the consideration of only finitelymany Fourier harmonics of the solution u(x, t) (in the timevariable t only). This seems to be acceptable for practicalcomputations. The main mathematical tool is the constructionof uniformly convex cost functionals on compact convex subsetsof the solutions. This assures a global convergence of the minimizationalgorithm, which can be applied in the case of large media inhomogeneities.Since the technique is based on the so-called Carleman's weightfunctions, the approach is called Carleman's weight method.