Projection Method for Saddle Points of Energy Functional in $H^{-1}$ Metric

Abstract

Saddle points play important roles as the transition states of activated process in gradient systems driven by energy functional. However, for the same energy functional, the saddle points, as well as other stationary points, are different in different metrics such as the $L^2$ metric and the $H^{-1}$ metric. The saddle point calculation in $H^{-1}$ metric is more challenging with much higher computational cost since it involves higher order derivative in space and the inner product calculation needs to solve another Possion equation to get the $\Delta ^{-1}$ operator. In this paper, we introduce the projection idea to the existing saddle point search methods, gentlest ascent dynamics (GAD) and iterative minimization formulation (IMF), to overcome this numerical challenge due to $H^{-1}$ metric. Our new method in the $L^2$ metric can locate the saddle point in $H^{-1}$ metric only by carefully incorporating a simple linear projection step. We show that our projection method maintains the same convergence speed of the original GAD and IMF, but the new algorithm is much faster than the direct method for $H^{-1}$ problems. The numerical results of saddle points in the one dimensional Ginzburg-Landau free energy and the two dimensional Landau-Brazovskii free energy in $H^{-1}$ metric are presented to demonstrate the efficiency of this new method.