Multilevel and adaptive methods for some nonlinear optimization problems

by Emelianenko, Maria.

In this thesis, we propose new multilevel and adaptive methods for solving nonlinear non-convex optimization problems without relying on the linearization. We focus on two particular applications, that come from the fields of quantization and materials science. For the first problem, a multilevel quantization scheme is developed, that possesses a uniform convergence independent of the problem size. This is the first multilevel quantization scheme in the literature with a rigorous proof of uniform convergence with respect to the grid size and the number of grid levels for nonconstant densities. The proposed scheme can be generalized to higher dimensions, and both scalar and vector versions demonstrate significant speedup comparing to the traditional Lloyd method. We also provide some new characterizations for the convergence of the Lloyd iteration and other possible acceleration techniques including Newton-like methods. For the second optimization problem, this thesis presents a novel algorithm aimed at automating phase diagram construction in complex multicomponent systems. The new method utilizes the geometric properties of the energy surfaces together with adaptivity and effective sampling techniques to improve on the starting points for the minimization. It is shown that the new approach overcomes the drawbacks of the previously known algorithms, at the same time giving comparable accuracy to the solution. iii