  Saddle points play important roles as the transition states of activated
process in gradient system 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
only by carefully incorporates 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
$H^{-1}$ problem. 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.
