The ill-posedness and nonlinearity are two factors causing the phenomenon of multiple local minima and ravines of conventional least squares cost functionals for Coefficient Inverse Problems. Since any minimization method can stop at any point of a local minimum, then the problem of numerical solution of any Coefficient Inverse Problems becomes inherently unstable and so obtained solutions are unreliable.
To overcome this phenomenon, we have developed the so-called “convexification” method, It works for a broad variety of Coefficient Inverse Problems with non redundtant data. Coefficient Inverse Problems for Helmholtz equation, parabolic equation, elliptic equation of electrical impedance tomography and even for the eikonal equation of travel time tomography can be handled by a rather unified way. We will present both analytical results and numerical ones. Numerical results include both computationally simulated and experimental data collected by our research team. The convexification constructs globally strictly convex weighted Tikhonov-like functional. The key element of this functional is the presence in it of the so-called Carleman Weight Function, i.e. the function involved in the Carleman estimate for the corresponding PDE operator. The global convergence to the correct solution of the gradient projection method of the minimization of this functional is rigorously established and tested numerically on many problems.