Speaker：Ziqing Xie (Hunan Normal University)

Time：2023-8-15, 16:30

Location：Conference Room 105 at Experiment Building at Haiyun Campus

Abstract:

In this report we will systematically introduce the monotonic and non-monotonic local minima-max methods (LMM) and the algorithm framework for the saddle point calculation problem of semilinear partial differential equations with variational structure. Among them, the monotonic LMM is based on three standardized non-exact search strategies of Armijo, Goldstein and (strong) Wolf-Powell respectively, and the selection of the descending direction can be extended from the steepest descending direction to a more general descending direction, thus making it possible to accelerate the algorithm. Non-monotonic LMM is based on BB step size and Zhang-Hager non-monotonic search strategy. Numerical results show that this method can greatly improve the algorithm efficiency of LMM. Our work overcomes the inherent difficulties of nonlinear, non-convex, multiple solutions and instability in the saddle point calculation of semi-linear partial differential equations, and rigorously proves the feasibility and large-scale convergence of the method. As an application of LMM, which is used to calculate the saddle point solution of a class of semi-linear singularly perturbed Neumann problems, we found and accurately characterized the corresponding critical perturbation value.