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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, 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. [ABSTRACT FROM AUTHOR]

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. [ABSTRACT FROM AUTHOR]

The quasi-potential is a key function in the Large Deviation Theory. It characterizes the difficulty of the escape from the neighborhood of an attractor of a stochastic non-gradient dynamical system due to the influence of small white noise. It also gives an estimate of the invariant probability distribution in the neighborhood of the attractor up to the exponential order. We present a new family of methods for computing the quasi-potential on a regular mesh named the ordered line integral methods (OLIMs). In comparison with the first proposed quasi-potential finder based on the ordered upwind method (OUM) (Cameron in Phys D Nonlinear Phenom 241:1532-1550, 2012), the new methods are 1.5-4 times faster, can produce error two to three orders of magnitude smaller, and may exhibit faster convergence. Similar to the OUM, OLIMs employ the dynamical programming principle. Contrary to it, they (1) have an optimized strategy for the use of computationally expensive triangle updates leading to a notable speed-up, and (2) directly solve local minimization problems using quadrature rules instead of solving the corresponding Hamilton-Jacobi-type equation by the first order finite difference upwind scheme. The OLIM with the right-hand quadrature rule is equivalent to OUM. The use of higher order quadrature rules in local minimization problems dramatically boosts up the accuracy of OLIMs. We offer a detailed discussion on the origin of numerical errors in OLIMs and propose rules-of-thumb for the choice of the important parameter, the update factor, in the OUM and OLIMs. Our results are supported by extensive numerical tests on two challenging 2D examples. [ABSTRACT FROM AUTHOR]

We model an isothermal aggregation process of particles/atoms interacting according to the Lennard-Jones pair potential by mapping the energy landscapes of each cluster size N onto stochastic networks, computing transition probabilities from the network for an N-particle cluster to the one for $$N+1$$ , and connecting these networks into a single joint network. The attachment rate is a control parameter. The resulting network representing the aggregation of up to 14 particles contains 6427 vertices. It is not only time-irreversible but also reducible. To analyze its transient dynamics, we introduce the sequence of the expected initial and pre-attachment distributions and compute them for a wide range of attachment rates and three values of temperature. As a result, we find the configurations most likely to be observed in the process of aggregation for each cluster size. We examine the attachment process and conduct a structural analysis of the sets of local energy minima for every cluster size. We show that both processes taking place in the network, attachment and relaxation, lead to the dominance of icosahedral packing in small (up to 14 atom) clusters. [ABSTRACT FROM AUTHOR]

Large continuous-time Markov chains with exponentially small transition rates arise in modeling complex systems in physics, chemistry, and biology. We propose a constructive graph-algorithmic approach to determine the sequence of critical timescales at which the qualitative behavior of a given Markov chain changes, and give an effective description of the dynamics on each of them. This approach is valid for both time-reversible and time-irreversible Markov processes, with or without symmetry. Central to this approach are two graph algorithms, Algorithm 1 and Algorithm 2, for obtaining the sequences of the critical timescales and the hierarchies of Typical Transition Graphs or T-graphs indicating the most likely transitions in the system without and with symmetry, respectively. The sequence of critical timescales includes the subsequence of the reciprocals of the real parts of eigenvalues. Under a certain assumption, we prove sharp asymptotic estimates for eigenvalues (including pre-factors) and show how one can extract them from the output of Algorithm 1. We discuss the relationship between Algorithms 1 and 2 and explain how one needs to interpret the output of Algorithm 1 if it is applied in the case with symmetry instead of Algorithm 2. Finally, we analyze an example motivated by R. D. Astumian's model of the dynamics of kinesin, a molecular motor, by means of Algorithm 2. [ABSTRACT FROM AUTHOR]

A set of analytical and computational tools based on transition path theory (TPT) is proposed to analyze flows in complex networks. Specifically, TPT is used to study the statistical properties of the reactive trajectories by which transitions occur between specific groups of nodes on the network. Sampling tools are built upon the outputs of TPT that allow to generate these reactive trajectories directly, or even transition paths that travel from one group of nodes to the other without making any detour and carry the same probability current as the reactive trajectories. These objects permit to characterize the mechanism of the transitions, for example by quantifying the width of the tubes by which these transitions occur, the location and distribution of their dynamical bottlenecks, etc. These tools are applied to a network modeling the dynamics of the Lennard-Jones cluster with 38 atoms ( $$\mathrm{LJ}_{38}$$ ) and used to understand the mechanism by which this cluster rearranges itself between its two most likely states at various temperatures. [ABSTRACT FROM AUTHOR]

This paper is devoted to the theoretical analysis of the zero-temperature string method, a scheme for identifying minimum energy paths (MEPs) on a given energy landscape. By definition, MEPs are curves connecting critical points on the energy landscape which are everywhere tangent to the gradient of the potential except possibly at critical points. In practice, MEPs are mountain pass curves that play a special role, e.g., in the context of rare reactive events that occur when one considers a steepest descent dynamics on the potential perturbed by a small random noise. The string method aims to identify MEPs by moving each point of the curve by steepest descent on the energy landscape. Here we address the question of whether such a curve evolution necessarily converges to an MEP. Surprisingly, the answer is no, for an interesting reason: MEPs may not be isolated, in the sense that there may be families of them that can be continuously deformed into one another. This degeneracy is related to the presence of critical points of Morse index 2 or higher along the MEP. In this paper, we elucidate this issue and completely characterize the limit set of a curve evolving by the string method. We establish rigorously that the limit set of such a curve is again a curve when the MEPs are isolated. We also show under the same hypothesis that the string evolution converges to an MEP. However, we identify and classify situations where the limit set is not a curve and may contain higher dimensional parts. We present a collection of examples where the limit set of a path contains a 2D region, a 2D surface, or a region of an arbitrary dimension up to the dimension of the space. In some of our examples the evolving path wanders around without converging to its limit set. In other examples it fills a region, converging to its limit set, which is not an MEP. [ABSTRACT FROM AUTHOR]