Bauver Martha, Forgoston Eric, Billings Lora
Department of Mathematical Sciences, Montclair State University, 1 Normal Avenue, Montclair, New Jersey 07043, USA.
Chaos. 2016 Aug;26(8):083101. doi: 10.1063/1.4958926.
In stochastic systems, one is often interested in finding the optimal path that maximizes the probability of escape from a metastable state or of switching between metastable states. Even for simple systems, it may be impossible to find an analytic form of the optimal path, and in high-dimensional systems, this is almost always the case. In this article, we formulate a constructive methodology that is used to compute the optimal path numerically. The method utilizes finite-time Lyapunov exponents, statistical selection criteria, and a Newton-based iterative minimizing scheme. The method is applied to four examples. The first example is a two-dimensional system that describes a single population with internal noise. This model has an analytical solution for the optimal path. The numerical solution found using our computational method agrees well with the analytical result. The second example is a more complicated four-dimensional system where our numerical method must be used to find the optimal path. The third example, although a seemingly simple two-dimensional system, demonstrates the success of our method in finding the optimal path where other numerical methods are known to fail. In the fourth example, the optimal path lies in six-dimensional space and demonstrates the power of our method in computing paths in higher-dimensional spaces.
在随机系统中，人们常常希望找到最优路径，该路径能使从亚稳态逃逸或在亚稳态之间切换的概率最大化。即使对于简单系统，也可能无法找到最优路径的解析形式，而在高维系统中，几乎总是如此。在本文中，我们制定了一种建设性方法，用于数值计算最优路径。该方法利用有限时间李雅普诺夫指数、统计选择标准和基于牛顿法的迭代最小化方案。该方法应用于四个例子。第一个例子是一个二维系统，描述了具有内部噪声的单一种群。该模型具有最优路径的解析解。使用我们的计算方法找到的数值解与解析结果吻合良好。第二个例子是一个更复杂的四维系统，必须使用我们的数值方法来找到最优路径。第三个例子虽然看似是一个简单的二维系统，但展示了我们的方法在找到其他数值方法已知会失败的最优路径方面的成功。在第四个例子中，最优路径位于六维空间，展示了我们的方法在计算高维空间路径方面的能力。
