Xiang Zhou

Adaptive minimum action method for the study of rare events

An adaptive minimum action method is proposed for computing the most probable transition paths between stable equilibria in metastable systems that do not necessarily have an underlying energy function, by minimizing the action functional associated with such transition paths. This new algorithm uses the moving mesh strategy to adaptively adjust the grid points over the time interval of transition. Numerical examples are presented to demonstrate the efficiency of the adaptive minimum action method.

The gentlest ascent dynamics

Dynamical systems that describe the escape from the basins of attraction of stable invariant sets are presented and analysed. It is shown that the stable fixed points of such dynamical systems are the index-1 saddle points. Generalizations to high index saddle points are discussed. Both gradient and non-gradient systems are considered. Preliminary results on the nature of the dynamical behaviour are presented.

Escaping from an attractor: Importance sampling and rest points I

We discuss importance sampling schemes for the estimation of finite time exit probabilities of small noise diffusions that involve escape from an equilibrium. A factor that complicates the analysis is that rest points are included in the domain of interest. We build importance sampling schemes with provably good performance both pre-asymptotically, that is, for fixed size of the noise, and asymptotically, that is, as the size of the noise goes to zero, and that do not degrade as the time horizon gets large. Simulation studies demonstrate the theoretical results.

Study of the noise-induced transition and the exploration of the phase space for the Kuramoto?Sivashinsky equation using the minimum action method

Noise-induced transition in the solutions of the Kuramoto–Sivashinsky (K–S) equation is investigated using the minimum action method derived from the large deviation theory. This is then used as a starting point for exploring the configuration space of the K–S equation. The particular example considered here is the transition between a stable fixed point and a stable travelling wave. Five saddle points, up to constants due to translational invariance, are identified based on the information given by the minimum action path. Heteroclinic orbits between the saddle points are identified. Relations between noise-induced transitions and the saddle points are examined.

Time-varying perturbations can distinguish among integrate-to-threshold models for perceptual decision making in reaction time tasks

Several integrate-to-threshold models with differing temporal integration mechanisms have been proposed to describe the accumulation of sensory evidence to a prescribed level prior to motor response in perceptual decision-making tasks. An experiment and simulation studies have shown that the introduction of time-varying perturbations during integration may distinguish among some of these models. Here, we present computer simulations and mathematical proofs that provide more rigorous comparisons among one-dimensional stochastic differential equation models. Using two perturbation protocols and focusing on the resulting changes in the means and standard deviations of decision times, we show that for high signal-to-noise ratios, drift-diffusion models with constant and time-varying drift rates can be distinguished from Ornstein-Uhlenbeck processes, but not necessarily from each other. The protocols can also distinguish stable from unstable Ornstein-Uhlenbeck processes, and we show that a nonlinear integrator can be distinguished from these linear models by changes in standard deviations. The protocols can be implemented in behavioral experiments.

An Iterative Minimization Formulation for Saddle Point Search

This paper proposes and analyzes an iterative minimization formulation for searching index-1 saddle points of an energy function. We give a general and rigorous description of eigenvector-following...

Subcritical bifurcation in spatially extended systems

A theory for noise-driven subcritical instabilities in spatially extended systems is put forward. The theory allows one to calculate the critical bifurcation parameter for a first-order phase transition in such non-equilibrium systems in the thermodynamic limit and analyse the mechanism of phase transition. Two examples with distinctive features are studied in detail to demonstrate the usefulness of the theory and the different scenarios that can occur in the thermodynamic limit of non-equilibrium systems.

A Cross-Entropy Scheme for Mixtures

We discuss how to generalize the classic cross-entropy method in the case where a family of mixture distributions, such as the mixture of multiple Gaussian modes, is used as an importance sampling distribution. A new iterative cross-entropy scheme, based on the idea of the EM method, is proposed to overcome the challenge of deciding the optimal weights for each mode in the mixture. Detailed studies of this new algorithm and its applications to the estimation of rainbow option prices are presented to demonstrate the efficiency of the scheme.

Finding Transition Pathways on Manifolds

When a randomly perturbed dynamical system is subject to some constraints, the trajectories of the system and the noise-induced most probable transition pathways are restricted on the manifold associated with the given constraints. We present a constrained minimum action method to compute the optimal transition pathways on manifolds. By formulating the constrained stochastic dynamics in a Stratonovich stochastic differential equation of the projection form, we consider the system as embedded in the Euclidean space and present the Freidlin--Wentzell action functional via large deviation theory. We then reformulate it as a minimization problem in the space of curves through Maupertuis principle. Furthermore we show that the action functionals are intrinsically defined on the manifold. The constrained minimum action method is proposed to compute the minimum action path with the assistance of the constrained optimization scheme. The examples of conformational transition paths for both single and double rod molecules in polymeric fluid are numerically investigated.

Convex splitting method for the calculation of transition states of energy functional

Among numerical methods for partial differential equations arising from steepest descent dynamics of energy functionals (e.g., Allen-Cahn and Cahn-Hilliard equations), the convex splitting method is well-known to maintain unconditional energy stability for a large time step size. In this work, we show how to use the convex splitting idea to find transition states, i.e., index-1 saddle points of the same energy functionals. Based on the iterative minimization formulation (IMF) for saddle points (SIAM J. Numer. Anal., vol. 53, p1786, 2015), we introduce the convex splitting method to minimize the auxiliary functional at each cycle of the IMF. We present a general principle of constructing convex splitting forms for these auxiliary functionals and show how to avoid solving nonlinear equations. The new numerical scheme based on the convex splitting method allows for large time step sizes. The new methods are tested for the one dimensional Ginzburg-Landau energy functional in the search of the Allen-Cahn or Cahn-Hilliard types of transition states. We provide the numerical results of transition states for the two dimensional Landau-Brazovskii energy functional for diblock copolymers.