Finding Multiple Saddle Points for G-differential Functionals and Defocused Nonlinear Problems
Year: 2017
Type: dissertation
Abstract: We study computational theory and numerical methods for finding multiple unstable\nsolutions (saddle points) for two types of nonlinear variational functionals. The first type\nconsists of Gateaux differentiable (G-differentiable) M-type (focused) problems. Motivated\nby quasilinear elliptic problems from physical applications, where energy functionals\nare at most lower semi-continuous with blow-up singularities in the whole space and\nG-differntiable in a subspace, and mathematical results and numerical methods for C1 or\nnonsmooth/Lipschitz saddle points existing in the literature are not applicable, we establish\na new mathematical frame-work for a local minimax method and its numerical implementation\nfor finding multiple G-saddle points with a new strong-weak topology approach.\nNumerical implementation in a weak form of the algorithm is presented. Numerical examples\nare carried out to illustrate the method. The second type consists of C^1 W-type\n(defocused) problems. In many applications, finding saddles for W-type functionals is desirable,\nbut no mathematically validated numerical method for finding multiple solutions\nexists in literature so far. In this dissertation, a new mathematical numerical method called\na local minmaxmin method (LMMM) is proposed and numerical examples are carried out\nto illustrate the efficiency of this method. We also establish computational theory and\npresent the convergence results of LMMM under much weaker conditions. Furthermore,\nwe study this algorithm in depth for a typical W-type problem and analyze the instability\nperformances of saddles by LMMM as well.
Author: Meiqin Li
Language: English
Subfield: Computational Theory and Mathematics
Field: Computer Science
Domain: Physical Sciences
Open Access status: green
OpenAlex ID: W2998949554