Computational Theory and Methods for Finding Multiple Critical Points

Let H be a Hilbert space and J ∈ C(H, ). Denote δJ its Frechet derivative and J ′ its gradient. The objective of this research is to develop computational theory and methods for finding multiple critical points, i.e., solutions to the Euler-Lagrange equation J (u) = 0. A critical point u is nondegenerate if J (u) is invertible. Otherwise u is degenerate. The first candidates for a critical point are the local extrema to which the classical critical point theory was devoted in calculus of variation. Most conventional numerical algorithms focus on finding such stable solutions. Critical points that are not local extrema are unstable and called saddle points. In physical systems, saddle points appear as unstable equilibria or transient excited states. (In)Stability is one of the main concerns in system design and control. However, in many applications, performance or maneuverability is more desirable. An unstable solution can be stable enough to accomplish a short term mission before being excited to decay to another desirable state [14,15,22] and it may have much higher performance or maneuverability index. Due to unstable nature, those multiple solutions are very elusive to be numerically captured. Little is known in the literature. Minimax method is one of the most popular approaches in critical point theory, But most minimax theorems, such as, the mountain pass, various linking and saddle point theorems, focus on the existence issue. They require one to solve a two-level global optimization problem and therefore are not for algorithm implementation. Inspired by the numerical work of Chio-McKenna and Ding-Costa-Chen in [9,12], the idea of Nehari and Ding-Ni to define a solution manifold and the Morse theory, we present a local minimax method (LMM). Consider a semilinear elliptic BVP as a model problem