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.

鞍点在能量泛函驱动的梯度系统中作为激活过程的过渡态发挥着重要作用。然而，对于相同的能量泛函，鞍点以及其他静止点在不同的度量中是不同的，例如\(L^2\)度量和\(H^{-1}\)度量。\(H^{-1}\)度量中的鞍点计算更具挑战性，计算成本更高，因为它涉及空间中的高阶导数，并且内积计算需要求解另一个 Possion 方程以获得\(\Delta ^{-1}\)操作员。在本文中，我们将投影思想引入到现有的鞍点搜索方法、最温和上升动力学 (GAD) 和迭代最小化公式 (IMF) 中，以克服由\(H^{-1}\)度量引起的数值挑战。我们在\(L^2\)度量中的新方法可以通过仔细合并一个简单的线性投影步骤来定位\(H^{-1}\)度量中的鞍点。我们表明，我们的投影方法保持与原始 GAD 和 IMF 相同的收敛速度，但新算法比解决\(H^{-1}\)问题的直接方法要快得多。一维Ginzburg-Landau自由能和二维Landau-Brazovskii自由能中鞍点的数值结果\(H^{-1}\)度量被提出来证明这种新方法的效率。