Doctoral defense of Bingbing Ji

Date: October 12, 2017

Time: 3:00PM - 4:00PM

Location: BLOC 612

Speaker: Bingbing Ji

Description: Title: A Local Minimax Method Using the Generalized Nehari Manifold for Finding Differential Saddles Abstract: A new local minimax method (LMM) for finding the first few unconstrained saddles of a functional is developed, so that different types of saddle point problems in infinite-dimensional spaces can be solved. This method is based on a dynamics of points on virtual geometric objects such as curves, surfaces, etc. and it covers several existing algorithms in the literature as its algorithm framework is general. Algorithm stability and convergence are mathematically verified. The new algorithm is tested on several benchmark problems commonly used in the literature to show its stability and efficiency, then it is applied to numerically compute saddles of a semilinear elliptic PDE for both (focusing) M-type and (defocusing) W-type cases. It is shown that those virtual geometric objects can be easily defined without knowing their explicit expressions and extended to find k-saddles so there is a great flexibility to choose preferred geometric objects for some purposes, such as convergence acceleration. Inspired by this feature, since all the non-trivial critical points stay on the Nehari manifold, it is used to accelerate the convergence and a comparison of computation speed between using the Nehari manifold and quadratic geometric objects on the same semilinear elliptic PDEs is given, then a mixed M and W type case is solved by LMM with the Nehari manifold. To solve the indefinite M-type problems, the generalized Nehari manifold is introduced in detail and a more general dynamic system of points on it is given. The corresponding local minimax method is justified by establishing a strong energy dissipation law and showing the convergence of the algorithm. The new algorithm with the generalized Nehari manifold is then applied to solve some indefinite M-type cases. A numerical investigation of bifurcation for indefinite problems will be given in order to provide numerical evidence for PDE analysts for future stud