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

The authors consider nonoverdetermined 3-D inverse scattering problems based on the telegraph equation u(u) = Delta u + a(x)u(t) + b(x)u + delta(x, t), u/(t<0) = 0, x is an element of R(3). The goal is to recover the media properties represented by the coefficients a(x) or b(x) from, for example, backscattering data. 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 finitely many Fourier harmonics of the solution u(x, t) (in the time variable t only). This seems to be acceptable for practical computations. The main mathematical tool is the construction of uniformly convex cost functionals on compact convex subsets of the solutions. This assures a global convergence of the minimization algorithm, which can be applied in the case of large media inhomogeneities. Since the technique is based on the so-called Carleman's weight functions, the approach is called Carleman's weight method.