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Many processes in nature such as conformal changes in biomolecules and clusters of interacting particles, genetic switches, noisy mechanical or electromechanical oscillators, and many others are modeled using stochastic differential equations with small white noise. The study of rare transitions between metastable states in such systems is of great interest and importance, but direct simulations are difficult due to long waiting times. Transition path theory is a mathematical framework for the quantitative description of rare events. Its direct implementation the key component of which is the solution of the committor problem, a boundary value problem for the backward Kolmogorov equation, is often challenging due to high dimensionality or other numerical issues. This work exploits the key fact that the optimal controller constructed from the committor leads to generation of transition trajectories exclusively.

An experimental arrangement and a set of experiments are developed to generate empirical evidence of the effect of noise on a rotating, macro-scale cantilever structure. The experiment is a controlled representation of a rotating machinery blade. Due to the nature of the nonlinear restoring forces acting on the cantilever structure, the structure's response includes regions of multi-stability and hysteresis. Here, a large number of trials are used to show that random perturbations can be used to create a transition between a high amplitude response and a low amplitude response of the cantilever. The observed transition behavior occurs from a high amplitude response to a low amplitude response, but not vice versa. Stochastic modeling of the system, Monte Carlo simulations, and calculations of the stochastic system's quasipotential are used to explain the nearly one-directional transition behavior. These noise-influenced transitions can also occur in other physical systems., Comment: 25 pages, 22 figures

We present a new algorithm, the efficient jet marching method (EJM), for computing the quasipotential and its gradient for two-dimensional SDEs. The quasipotential is a potential-like function for nongradient SDEs that gives asymptotic estimates for the invariant probability measure, expected escape times from basins of attractors, and maximum likelihood escape paths. The quasipotential is a solution to an optimal control problem with an anisotropic cost function which can be solved for numerically via Dijkstra-like label-setting methods. Previous Dijkstra-like quasipotential solvers have displayed in general 1st order accuracy in the mesh spacing. However, by utilizing higher order interpolations of the quasipotential as well as more accurate approximations of the minimum action paths (MAPs), EJM achieves second-order accuracy for the quasipotential and nearly second-order for its gradient. Moreover, by using targeted search neighborhoods for the fastest characteristics following the ideas of Mirebeau, EJM also enjoys a reduction in computation time. This highly accurate solver enables us to compute the prefactor for the WKB approximation for the invariant probability measure and the Bouchet-Reygner sharp estimate for the expected escape time for the Maier-Stein SDE. Our codes are available on GitHub., Comment: 21 figures, 3 tables

We present a family of fast and accurate Dijkstra-like solvers for the eikonal equation and factored eikonal equation which compute solutions on a regular grid by solving local variational minimization problems. Our methods converge linearly but compute significantly more accurate solutions than competing first order methods. In 3D, we present two different families of algorithms which significantly reduce the number of FLOPs needed to obtain an accurate solution to the eikonal equation. One method employs a fast search using local characteristic directions to prune unnecessary updates, and the other uses the theory of constrained optimization to achieve the same end. The proposed solvers are more efficient than the standard fast marching method in terms of the relationship between error and CPU time. We also modify our method for use with the additively factored eikonal equation, which can be solved locally around point sources to maintain linear convergence. We conduct extensive numerical simulations and provide theoretical justification for our approach. A library that implements the proposed solvers is available on GitHub., Comment: 43 pages, 21 figures. Submitted to Springer Journal of Scientific Computing

The concept of metastability has caused a lot of interest in recent years. The spectral decomposition of the generator matrix of a stochastic network exposes all of the transition processes in the system. The assumption of the existence of a low lying group of eigenvalues separated by a spectral gap, leading to factorization of the dynamics, has become a popular theme. We consider stochastic networks representing potential energy landscapes where the states and the edges correspond to local minima and transition states respectively, and the pairwise transition rates are given by the Arrhenuis formula. Using the minimal spanning tree, we construct the asymptotics for eigenvalues and eigenvectors of the generator matrix starting from the low lying group. This construction gives rise to an efficient algorithm for computing the asymptotic spectrum suitable for large and complex networks. We apply it to Wales's Lennard-Jones-38 network with 71887 states and 119853 edges where the underlying potential energy landscape has a double-funnel structure. Our results demonstrate that the concept of metastability should be applied with care to this system. In particular, for the full network, there is no significant spectral gap separating the eigenvalue corresponding to the exit from the wider and shallower icosahedral funnel at any reasonable temperature range., Comment: Submitted to Journal Networks and Heterogeneous Media on Feb. 25, 2014, 36 pages, 14 figures