turkmath.org

Coefficient Inverse Problems (CIPs) are both ill-posed and highly nonlinear. These two factors cause the non-convexity of conventional least squares cost functionals, which are constructed for numerical solutions of CIPs. The speaker with coauthors has developed a new approach to numerical solutions of CIPs, called convexification. The convexification constructs globally strictly convex cost functionals for a broad class of CIPs. This functional is defined on a bounded convex set of an arbitrary but fixed diameter in an appropriate Hilbert space, which we call Carleman Weighted Hilbert Space. The weight is the Carleman Weight Function, which is used in the Carleman estimate for a corresponding PDE operator. Uniqueness and existence of the minimizer of such a functional on that set is established. Convergence of minimizers to the true solution of the CIP is proven, provided that the noise in the data tends to zero. Many numerical examples, including ones with experimentally collected data, confirm the theory. Some of these results will be presented in my talk. Main contributers are (in the alphabetial order): Vo Khoa, Thuy Le and Loc Nguyen.