Small White Noise Research Articles

Optimal control for sampling the transition path process and estimating rates

Many processes in nature such as conformal changes in biomolecules and clusters of interacting particles, genetic switches, mechanical or electromechanical oscillators with added noise, 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. The direct simulation of rare transitions is difficult due to long waiting times. Transition path theory is a mathematical framework for the quantitative description of rare events. Its crucial component is the committor function, the solution to a boundary value problem for the backward Kolmogorov equation. The key fact exploited in this work is that the optimal controller constructed from the committor leads to the generation of transition trajectories exclusively. We prove this fact for a broad class of stochastic differential equations. Moreover, we demonstrate that the committor computed for a dimensionally reduced system and then lifted to the original phase space still allows us to construct an effective controller and estimate the transition rate with reasonable accuracy. Furthermore, we propose an all-the-way-through scheme for computing the committor via neural networks, sampling the transition trajectories, and estimating the transition rate without meshing the space. We apply the proposed methodology to four test problems: the overdamped Langevin dynamics with Mueller’s potential and the rugged Mueller potential in 10D, the noisy bistable Duffing oscillator, and Lennard-Jones-7 in 2D.

A General Deep Learning Method for Computing Molecular Parameters of a Viscoelastic Constitutive Model by Solving an Inverse Problem.

Prediction of molecular parameters and material functions from the macroscopic viscoelastic properties of complex fluids are of great significance for molecular and formulation design in fundamental research as well as various industrial applications. A general learning method for computing molecular parameters of a viscoelastic constitutive model by solving an inverse problem is proposed. The accuracy, convergence and robustness of a deep neural network (DNN)-based numerical solver have been validated by considering the Rolie-Poly model for modeling the linear and non-linear steady rheometric properties of entangled polymer solutions in a wide range of concentrations. The results show that as long as the DNN could be trained with a sufficiently high accuracy, the DNN-based numerical solver would rapidly converge to its solution in solving an inverse problem. The solution is robust against small white noise disturbances to the input stress data. However, if the input stress significantly deviates from the original stress, the DNN-based solver could readily converge to a different solution. Hence, the resolution of the numerical solver for inversely computing molecular parameters is demonstrated. Moreover, the molecular parameters computed by the DNN-based numerical solver not only reproduce accurately the steady viscoelastic stress of completely monodisperse linear lambda DNA solutions over a wide range of shear rates and various concentrations, but also predict a power law concentration scaling with a nearly same scaling exponent as those estimated from experimental results.

A non-stochastic control with admissible probabilities for SDDEs, application to linear reactors

We will develop a time-averaging method to compute an approximate control for a stochastic differential equation. The obtained control is nonstochastic, easy to compute, and has an absolute advantage in exactness with admissible probabilities compared to direct approximate control. The results are applied to a linear reactor under small white noise and are verified through numerical simulations.

Fluctuation analysis for a class of nonlinear systems with fast periodic sampling and small state-dependent white noise

We consider a nonlinear differential equation under the combined influence of small state-dependent Brownian perturbations of size ε, and fast periodic sampling with period δ; 0<ε,δ≪1. State samples (measurements) are taken every δ time units, and the instantaneous rate of change of the state depends on both the current value and most recent sample. For the resulting stochastic process indexed by ε,δ, we obtain asymptotic approximations for the mean behavior and fluctuations about the mean. The former is described by an ordinary differential equation, while the latter is governed by a stochastic differential equation (sde). This sde varies depending on the exact rates at which ε,δ↘0. The key contribution involves computing the effective drift term capturing the interplay between noise and sampling in the limiting sde. Connections with control systems with sampling are discussed and illustrated numerically through a simple example.

A large deviation principle for fluids of third grade

ABSTRACT This article establishes a large deviation principle for a non-Newtonian fluid of differential type, filling a two-dimensional non-axisymmetric bounded domain with slip boundary conditions. More precisely, we show that the solutions of small stochastic white noise perturbations of the third grade fluid equations converges to the deterministic solution, as the intensity of the noise goes to zero. Moreover, this convergence has an exponential rate given by a suitable rate function. To establish such asymptotic result, we follow the weak convergence approach introduced by Budhiraja, Dupuis and Ellis.

The deterministic core of stochastically perturbed nonlinear mechanical systems

Invariant manifolds, such as normally hyperbolic invariant manifolds and spectral submanifolds, are the key to understanding the dynamical behaviour of many nonlinear mechanical systems and serve as natural candidates for model-order reduction. While numerous related invariant manifold results are available for unforced and periodically forced nonlinear mechanical systems, their applicability to random external forcing remains to be established. Here, we clarify the continued role of deterministic invariant manifolds, more specifically normally hyperbolic invariant manifolds and spectral submanifolds under the addition of small white noise excitation. Subsequently, we demonstrate our results on several mechanical systems.

Precise Laplace asymptotics for singular stochastic PDEs: The case of 2D gPAM

We implement a Laplace method for the renormalised solution to the generalised 2D Parabolic Anderson Model (gPAM) driven by a small spatial white noise. Our work rests upon Hairer's theory of regularity structures which allows to generalise classical ideas of Azencott and Ben Arous on path space as well as Aida and Inahama and Kawabi on rough path space to the space of models. The technical cornerstone of our argument is a Taylor expansion of the solution in the noise intensity parameter: We prove precise bounds for its terms and the remainder and use them to estimate asymptotically irrevelant terms to arbitrary order. While most of our arguments are not specific to gPAM, we also outline how to adapt those that are.

Dynamic analysis of a stochastic toxin-mediated predator-prey model in aquatic environments

This paper considers a stochastic aquatic toxin-mediated predator-prey model. Our main aim is to investigate the effects of environmental toxins and white noise on the population dynamics of this model. First, we prove that there is a global positive solution for any given initial value. Then we show the existence of a unique stationary distribution applying the Lyapunov function method and establish extinction conditions using the invariant density of a one-dimensional diffusion process. In addition, by solving the corresponding Fokker-Plauck equation, we obtain the explicit probability density function around the quasi-stable equilibrium of the transformed stochastic system without pollution. Finally, we use numerical tests to illustrate the main results and reveal some interesting biological phenomena, such as small white noise can reduce the risk of extinction, and the same toxin concentration has different effects on predators and their prey, etc.

Comparative observer effects in 2D and 3D localization tasks.

.Purpose: Three-dimensional “volumetric” imaging methods are now a common component of medical imaging across many imaging modalities. Relatively little is known about how human observers localize targets masked by noise and clutter as they scroll through a 3D image and how it compares to a similar task confined to a single 2D slice.Approach: Gaussian random textures were used to represent noisy volumetric medical images. Subjects were able to freely inspect the images, including scrolling through 3D images as part of their search process. A total of eight experimental conditions were evaluated (2D versus 3D images, large versus small targets, power-law versus white noise). We analyze performance in these experiments using task efficiency and the classification image technique.Results: In 3D tasks, median response times were roughly nine times longer than 2D, with larger relative differences for incorrect trials. The efficiency data show a dissociation in which subjects perform with higher statistical efficiency in 2D tasks for large targets and higher efficiency in 3D tasks with small targets. The classification images suggest that a critical mechanism behind this dissociation is an inability to integrate across multiple slices to form a 3D localization response. The central slices of 3D classification images are remarkably similar to the corresponding 2D classification images.Conclusions: 2D and 3D tasks show similar weighting patterns between 2D images and the central slice of 3D images. There is relatively little weighting across slices in the 3D tasks, leading to lower task efficiency with respect to the ideal observer.

The effect of a small bounded noise on the hyperbolicity for autonomous semilinear differential equations

In this work, we study permanence of hyperbolicity for autonomous differential equations under nonautonomous random/stochastic perturbations. For the linear case, we study robustness and existence of exponential dichotomies for nonautonomous random dynamical systems. Next, we establish a result on the persistence of hyperbolic equilibria for nonlinear differential equations. We show that for each nonautonomous random perturbation of an autonomous semilinear problem with a hyperbolic equilibrium there exists a bounded random hyperbolic solution for the associated nonlinear nonautonomous random dynamical systems. Moreover, we show that these random hyperbolic solutions converge to the autonomous equilibrium. As an application, we consider a semilinear differential equation with a small nonautonomous multiplicative white noise, and as an example, we apply the abstract results to a strongly damped wave equation.

Dynamics of a stochastic tuberculosis transmission model with treatment at home

In this paper, we consider a stochastic tuberculosis model with two kinds of treatments, that is, treatment at home and treatment in hospital. Firstly, we obtain sufficient conditions for extinction and persistence in the mean of the diseases, then in the case of persistence, we establish sufficient conditions for the existence and uniqueness of an ergodic stationary distribution of the positive solutions to the model by constructing a suitable stochastic Lyapunov function. The theoretical results show that large white noise is beneficial for eliminating TB endemic while small white noise is not.

Asymptotic Accuracy in Estimation of a Fractional Signal in a Small White Noise

The problem of estimating the fractional Ornstein—Uhlenbeck process observed in a linear channel with white noise of small intensity is considered. Exact asymptotic formulas of the error of the filtering and interpolation estimates are obtained. The accuracy analysis is based on approximations of the eigenvalues and eigenfunctions of the signal’s covariance operator.

Stationary distribution of a stochastic cholera model between communities linked by migration

In this paper, we develop and study a stochastic cholera model between communities linked by migration. Firstly, we show that there exists a unique global positive solution of the considered system with any positive initial value. Then by using the stochastic Lyapunov analysis method, we obtain sufficient conditions for the existence and uniqueness of an ergodic stationary distribution of the positive solutions to the model. Our result shows that the migration and the small white noise are not beneficial for eliminating cholera. Finally, some examples together with numerical simulations are introduced to support the theoretical result.

Exponential loss of memory for the 2-dimensional Allen\u2013Cahn equation with small noise

We prove an asymptotic coupling theorem for the 2-dimensional Allen–Cahn equation perturbed by a small space-time white noise. We show that with overwhelming probability two profiles that start close to the minimisers of the potential of the deterministic system contract exponentially fast in a suitable topology. In the 1-dimensional case a similar result was shown in Martinelli et al. (Commun Math Phys 120(1):25–69, 1988; J Stat Phys 55(3–4):477–504, 1989). It is well-known that in two or more dimensions solutions of this equation are distribution-valued, and the equation has to be interpreted in a renormalised sense. Formally, this renormalisation corresponds to moving the minima of the potential infinitely far apart and making them infinitely deep. We show that despite this renormalisation, solutions behave like perturbations of the deterministic system without renormalisation: they spend large stretches of time close to the minimisers of the (un-renormalised) potential and the exponential contraction rate of different profiles is given by the second derivative of the potential in these points. As an application we prove an Eyring–Kramers law for the transition times between the stable solutions of the deterministic system for fixed initial conditions.

Modulation Equation and SPDEs on Unbounded Domains

We consider the approximation via modulation equations for nonlinear SPDEs on unbounded domains with additive space time white noise. Close to a bifurcation an infinite band of eigenvalues changes stability, and we study the impact of small space-time white noise on this bifurcation. As a first example we study the stochastic Swift-Hohenberg equation on the whole real line. Here due to the weak regularity of solutions the standard methods for modulation equations fail, and we need to develop new tools to treat the approximation. As an additional result we sketch the proof for local existence and uniqueness of solutions for the stochastic Swift-Hohenberg and the complex Ginzburg Landau equations on the whole real line in weighted spaces that allow for unboundedness at infinity of solutions, which is natural for translation invariant noise like space-time white noise. Moreover we use energy estimates to show that solutions of the Ginzburg-Landau equation are H\"older continuous and have moments in those functions spaces. This gives just enough regularity to proceed with the error estimates of the approximation result.

Computing the quasipotential for nongradient SDEs in 3D

Nongradient SDEs with small white noise often arise when modeling biological and ecological time-irreversible processes. If the governing SDE were gradient, the maximum likelihood transition paths, transition rates, expected exit times, and the invariant probability distribution would be given in terms of its potential function. The quasipotential plays a similar role for nongradient SDEs. Unfortunately, the quasipotential is the solution of a functional minimization problem that can be obtained analytically only in some special cases. We propose a Dijkstra-like solver for computing the quasipotential on regular rectangular meshes in 3D. This solver results from a promotion and an upgrade of the previously introduced ordered line integral method with the midpoint quadrature rule for 2D SDEs. The key innovations that have allowed us to keep the CPU times reasonable while maintaining good accuracy are (i) a new hierarchical update strategy, (ii) the use of Karush–Kuhn–Tucker theory for rejecting unnecessary simplex updates, and (iii) pruning the number of admissible simplexes and a fast search for them. An extensive numerical study is conducted on a series of linear and nonlinear examples where the quasipotential is analytically available or can be found at transition states by other methods. In particular, the proposed solver is applied to Tao's examples where the transition states are hyperbolic periodic orbits, and to a genetic switch model by Lv et al. (2014) [21]. The C source code implementing the proposed algorithm is available at M. Cameron's web page.

Modulation on Discrete Nonlinear Spectrum: Perturbation Sensitivity and Achievable Rates

Multi-soliton pulses are studied under small perturbations. By means of mutual information, we show that the nonlinear spectrum becomes correlated when a 64-PSK modulated 2-eigenvalues soliton is perturbed by a small additive white Gaussian noise (AWGN). Transmission over two channels are considered: an AWGN channel (back-to-back scenario) and an ideal optical fiber link with distributed noise. We numerically show that the Jost coefficients ${b}({\lambda }_{k})$ are less correlated than the discrete spectral amplitudes ${q}_{d}({\lambda }_{k})$ while either of them can be alternatively used for modulation. In both channels, we compare the achievable information rates (AIRs) of joint detection and separate detection if the modulation is over either ${q}_{d}({\lambda }_{k})$ or ${b}({\lambda }_{k})$ . We show that if the length of the link is relatively short, the separate detection of ${b}({\lambda }_{k})$ gives almost the same total information rate as the joint detection of ${b}({\lambda }_{k})$ or ${q}_{d}({\lambda }_{k})$ . Moreover, the AIRs are compared with one of a first-order soliton with the same modulation.

Ordered Line Integral Methods for Computing the Quasi-Potential

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, 2012), the new methods are 1.5 to 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 (i) have an optimized strategy for the use of computationally expensive { triangle} updates leading to a notable speed-up, and (ii) 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.

Dynamics of the stochastic chemostat with Monod-Haldane response function

The stochastic chemostat model with Monod-Haldane response function is perturbed by environmental white noise. This model has a global positive solution. We demonstrate that there is a stationary distribution of the stochastic model and the system is ergodic under appropriate conditions, on the basis of Khasminskii’s theory on ergodicity. Sufficient criteria for extinction of the microbial population in the stochastic system are established. These conditions depend strongly on the Brownian motion. We find that even small scale white noise can promote the survival of microorganism populations, while large scale noise can lead to extinction. Numerical simulations are carried out to illustrate our theoretical results.

Noise-induced stabilization of collective dynamics.

We illustrate a counterintuitive effect of an additive stochastic force, which acts independently on each element of an ensemble of globally coupled oscillators. We show that a very small white noise not only broadens the clusters, wherever they are induced by the deterministic forces, but can also stabilize a linearly unstable collective periodic regime: self-consistent partial synchrony. With the help of microscopic simulations we are able to identify two noise-induced bifurcations. A macroscopic analysis, based on a perturbative solution of the associated nonlinear Fokker-Planck equation, confirms the numerical studies and allows determining the eigenvalues of the stability problem. We finally argue about the generality of the phenomenon.