Convexification Method in Inverse Problems

Coefficient Inverse Problems are nonlinear. Therefore, least squares cost functionals for them are non convex. This means that they have local minima and ravines. Hence, optimization numerical methods for these problems are not reliable since it is unclear which exactly point of a minimum do they produce and the minimizer can be located far from the solution. Starting from 1995 and 1997, the speaker has developed a numerical procedure, which is free from that drawback. This is the so-called convexification method. It works for a broad class of Coefficient Inverse Problems and nonlinear ill-posed problems for Partial Differential Equations. The convexification method is based on Carleman estimates. This method converges to the true solution globally. In other words, a proper initial guess is not required.