A Local Minimax Method Using Virtual Geometric Objects: Part I—For Finding Saddles

Abstract

By a dynamics of points on virtual geometric objects such as curves, surfaces, etc., with a flexible endpoint, this paper is to develop a new local minimax method for finding the first few unconstrained saddles of a functional, so that different types of saddle point problems in infinite-dimensional spaces can be solved. Since the mathematical framework of the method is general, it covers several existing algorithms in the literature. Algorithm justifications including a strong energy dissipation law and convergence are established. The new algorithm is implemented and tested on several benchmark examples commonly used in the literature to show its stability and efficiency, and then applied to numerically compute saddles of a semilinear elliptic PDE for both (focusing) M-type and (defocusing) W-type cases, where it is shown that those virtual geometric objects can be easily defined without knowing their explicit expressions or representative (interpolation) points and the method can be easily extended to find k-saddles or modified for other purposes, e.g., to compute constrained k-saddles.