论文标题
鞍点功能的鞍点的投影方法在$ h^{ - 1} $ metric中
Projection Method for Saddle Points of Energy Functional in $H^{-1}$ Metric
论文作者
论文摘要
鞍点在能量功能驱动的梯度系统中激活过程的过渡状态起着重要作用。但是，对于相同的能量功能，在不同的指标中，鞍点以及其他固定点也有所不同，例如$ l^2 $ metric和$ h^{ - 1} $ metric。 $ h^{ - 1} $公制中的鞍点计算在更高的计算成本方面更具挑战性，因为它涉及空间中的高阶导数，并且内部产品计算需要求解另一个可能的方程，以获取$δ^{ - 1} $操作员。在本文中，我们将投影概念介绍给现有的鞍点搜索方法，最温和的上升动力学（GAD）和迭代最小化公式（IMF），以克服由于$ h^{ - 1} $ Metric而引起的这一数字挑战。我们在$ l^2 $公制中的新方法仅通过仔细地包含一个简单的线性投影步骤。我们表明，我们的投影方法保持原始GAD和IMF的收敛速度相同，但是新算法比$ h^{ - 1} $问题的直接方法要快得多。提出了$ H^{ - 1} $公制中的骑马点和二维的Landau-Brazovskii自由能的数值结果，以证明这种新方法的效率。
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 $Δ^{-1}$ 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.
