论文标题
瑞兹（Ritz
Ritz method for transition paths and quasipotentials of rare diffusive events
论文作者
论文摘要
弗里德林 - 韦泽尔 - 格雷厄姆的大偏差理论给出了弱扩散过程轨迹的轨迹的概率。从弗雷德林 - 韦兹尔动作函数的最小值中获得了两个状态（激体顿）和过程跃迁密度对数（准电位）中最可能的路径。在这里，我们提出了一种RITZ方法，该方法在从Chebyshev多项式的全局基础的路径空间中搜索最小值。从而将动作降低到基础系数的多元功能，该功能可以通过非线性优化找到其最小值。为了最小化，无论路径持续时间如何，当应用于重新构度的“壳上”动作时，此过程最有效，这是通过利用Noether对称性来获得的，并且是标量工作的概括[Olender and Elber，1997]，用于渐变的动力学和几何动力学[Heyman and vanden-eijndenden，2008]。我们的方法为链链方法提供了一种替代方法，用于复杂能量景观的最小能量路径和鞍座，以及用于循环场固定式准分子的汉密尔顿 - 雅各比方法。我们证明了三个基准问题的光谱收敛，涉及Muller-Brown电位，Maier-Stein Force Field和Egger天气模型。
The probability of trajectories of weakly diffusive processes to remain in the tubular neighbourhood of a smooth path is given by the Freidlin-Wentzell-Graham theory of large deviations. The most probable path between two states (the instanton) and the leading term in the logarithm of the process transition density (the quasipotential) are obtained from the minimum of the Freidlin-Wentzell action functional. Here we present a Ritz method that searches for the minimum in a space of paths constructed from a global basis of Chebyshev polynomials. The action is reduced, thereby, to a multivariate function of the basis coefficients, whose minimum can be found by nonlinear optimization. For minimisation regardless of path duration, this procedure is most effective when applied to a reparametrisation-invariant "on-shell" action, which is obtained by exploiting a Noether symmetry and is a generalisation of the scalar work [Olender and Elber, 1997] for gradient dynamics and the geometric action [Heyman and Vanden-Eijnden, 2008] for non-gradient dynamics. Our approach provides an alternative to chain-of-states methods for minimum energy paths and saddlepoints of complex energy landscapes and to Hamilton-Jacobi methods for the stationary quasipotential of circulatory fields. We demonstrate spectral convergence for three benchmark problems involving the Muller-Brown potential, the Maier-Stein force field and the Egger weather model.
