Random Manifolds Research Articles

A Wong-Zakai approximation for random invariant manifolds

Random invariant manifolds are geometric objects useful for understanding complex dynamics under stochastic influences. But these random objects are difficult to be visualized geometrically or computed numerically. The current work provides a perturbation approach to approximate these random invariant manifolds. We first discuss the existence of a random invariant manifold for a class of stochastic evolutionary equations. Then, we approximate the random invariant manifold by the invariant manifold of a new system with smooth colored noise (i.e., integrated Ornstein-Uhlenbeck processes). The convergence in a pathwise Wong-Zakai sense is shown.

Convergence of the reach for a sequence of Gaussian-embedded manifolds

Motivated by questions of manifold learning, we study a sequence of random manifolds, generated by embedding a fixed, compact manifold M into Euclidean spheres of increasing dimension via a sequence of Gaussian mappings. One of the fundamental smoothness parameters of manifold learning theorems is the reach, or critical radius, of M. Roughly speaking, the reach is a measure of a manifold’s departure from convexity, which incorporates both local curvature and global topology. This paper develops limit theory for the reach of a family of random, Gaussian-embedded, manifolds, establishing both almost sure convergence for the global reach, and a fluctuation theory for both it and its local version. The global reach converges to a constant well known both in the reproducing kernel Hilbert space theory of Gaussian processes, as well as in their extremal theory.

Approximate the dynamical behavior for stochastic systems by Wong–Zakai approaching

Random invariant manifolds and foliations play an important role in the study of the qualitative dynamical behaviors for nonlinear stochastic partial differential equations. In a general way, these random objects are difficult to be visualized geometrically or computed numerically. The current work provides a perturbation approach to approximate these random invariant manifolds and foliations. After briefly discussing the existence of random invariant manifolds and foliations for a class of stochastic systems driven by additive noises, the corresponding Wong–Zakai type of convergence result in path-wise sense is established.

Slow manifold for a nonlocal stochastic evolutionary system with fast and slow components

This work is devoted to investigating invariant manifolds for a fast–slow stochastic evolutionary system with nonlocal diffusion. We establish the slow reduction via a random slow manifold, which captures slow dynamics of the original stochastic fast–slow system. A simple example is shown to illustrate this slow reduction method.

Ends of unimodular random manifolds

We describe the space of ends of a unimodular random manifold, a random object that generalizes finite volume manifolds, random leaves of measured foliations and invariant random subgroups of isometry groups of Riemannian manifolds. As an application, we give a quick proof of a variant of a theorem of E. Ghys (1995) on the topology of random leaves.

The Intrinsic geometry of some random manifolds

We study the a.s. convergence of a sequence of random embeddings of a fixed manifold into Euclidean spaces of increasing dimensions. We show that the limit is deterministic. As a consequence, we show that many intrinsic functionals of the embedded manifolds also converge to deterministic limits. Particularly interesting examples of these functionals are given by the Lipschitz-Killing curvatures, for which we also prove unbiasedness, using the Gaussian kinematic formula.

Approximation of random invariant manifolds for a stochastic Swift-Hohenberg equation

Random invariant manifolds are considered for a stochastic Swift-Hohenberg equation with multiplicative noise in the Stratonovich sense. Using a stochastic transformation and a technique of cut-off function, existence of random invariant manifolds and attracting property of the corresponding random dynamical system are obtained by Lyaponov-Perron method. Then in the sense of large probability, an approximation of invariant manifolds has been investigated and this is further used to describe the geometric shape of the invariant manifolds.

Approximation for random stable manifolds under multiplicative correlated noises

The usual Wong-Zakai approximation is about simulating individual solutions of stochastic differential equations(SDEs). From the perspective of dynamical systems, it is also interesting to approximate random invariant manifolds which are geometric objects useful for understanding how complex dynamics evolve under stochastic influences. We study a Wong-Zakai type of approximation for the random stable manifold of a stochastic evolutionary equation with multiplicative correlated noise. Based on the convergence of solutions on the invariant manifold, we approximate the random stable manifold by the invariant manifolds of a family of perturbed stochastic systems with smooth correlated noise (i.e., an integrated Ornstein-Uhlenbeck process). The convergence of this approximation is established.

Spatial Homogenization of Stochastic Wave Equation with Large Interaction

AbstractA dynamical approximation of a stochastic wave equation with large interaction is derived. A random invariant manifold is discussed. By a key linear transformation, the random invariant manifold is shown to be close to the random invariant manifold of a second-order stochastic ordinary differential equation.

Singular fold with real noise

We study the effect of small real noise on the jump behavior near a singular fold point, which is an important step in understanding the burst-spike behavior in many biological models. We show by the theory of center manifolds and random invariant manifolds that if the order of the noise is high enough, trajectories essentially pass the fold point in the manner as though there is no noise.

Modulation equation for SPDEs in unbounded domains with space–time white noise — Linear theory

We study the approximation of SPDEs on the whole real line near a change of stability via modulation or amplitude equations, which acts as a replacement for the lack of random invariant manifolds on extended domains. Due to the unboundedness of the underlying domain a whole band of infinitely many eigenfunctions changes stability. Thus we expect not only a slow motion in time, but also a slow spatial modulation of the dominant modes, which is described by the modulation equation.As a first step towards a full theory of modulation equations for nonlinear SPDEs on unbounded domains, we focus, in the results presented here, on the linear theory for one particular example, the Swift–Hohenberg equation. These linear results are one of the key technical tools to carry over the deterministic approximation results to the stochastic case with additive forcing. One technical problem for establishing error estimates rises from the spatially translation invariant nature of space–time white noise on unbounded domains, which implies that at any time we can expect the error to be always very large somewhere in space.

Brownian motion on stationary random manifolds

We introduce the notion of a stationary random manifold and develop the basic entropy theory for it. Examples include manifolds admitting a compact quotient under isometries and generic leaves of a compact foliation. We prove that the entropy of an ergodic stationary random manifold is zero if and only if the manifold satisfies the Liouville property almost surely, and is positive if and only if it admits an infinite dimensional space of bounded harmonic functions almost surely. Upper and lower bounds for the entropy are provided in terms of the linear drift of Brownian motion and average volume growth of the manifold. Other almost sure properties of these random manifolds are also studied.

Dynamics and invariant manifolds for a nonlocal stochastic Swift-Hohenberg equation

The dynamics and invariant manifolds for a nonlocal stochastic Swift-Hohenberg equation with multiplicative noise are investigated. Using a stochastic transformation process, a nonlocal stochastic Swift-Hohenberg equation is studied with either a positive kernel or a non-negative kernel. Then the dynamics, existence, and uniqueness of a global random attractor for the nonlocal stochastic Swift-Hohenberg equation is shown. Moreover, the existence of a local random invariant manifold of the corresponding random dynamical system for the nonlocal stochastic Swift-Hohenberg equation with multiplicative noise is obtained using the technique of a cut-off function and the Lyapunov-Perron method.

Semi-supervised manifold learning with affinity regularization for Alzheimer's disease identification using positron emission tomography imaging.

Dementia, Alzheimer's disease (AD) in particular is a global problem and big threat to the aging population. An image based computer-aided dementia diagnosis method is needed to providing doctors help during medical image examination. Many machine learning based dementia classification methods using medical imaging have been proposed and most of them achieve accurate results. However, most of these methods make use of supervised learning requiring fully labeled image dataset, which usually is not practical in real clinical environment. Using large amount of unlabeled images can improve the dementia classification performance. In this study we propose a new semi-supervised dementia classification method based on random manifold learning with affinity regularization. Three groups of spatial features are extracted from positron emission tomography (PET) images to construct an unsupervised random forest which is then used to regularize the manifold learning objective function. The proposed method, stat-of-the-art Laplacian support vector machine (LapSVM) and supervised SVM are applied to classify AD and normal controls (NC). The experiment results show that learning with unlabeled images indeed improves the classification performance. And our method outperforms LapSVM on the same dataset.

Geometric singular perturbation theory with real noise

We prove the existence of families of random invariant manifolds for singularly perturbed systems of ordinary differential equations with sufficiently small real noise. We use these invariant manifolds to prove a random version of the inclination theorem or exchange lemma.

Approximation of Random Slow Manifolds and Settling of Inertial Particles Under Uncertainty

A method is provided for approximating random slow manifolds of a class of slow–fast stochastic dynamical systems. Thus approximate, low dimensional, reduced slow systems are obtained analytically in the case of sufficiently large time scale separation. To illustrate this dimension reduction procedure, the impact of random environmental fluctuations on the settling motion of inertial particles in a cellular flow field is examined. It is found that noise delays settling for some particles but enhances settling for others. A deterministic stable manifold is an agent to facilitate this phenomenon. Overall, noise appears to delay the settling in an averaged sense.

Approximation of Random Slow Manifolds and Settling of Inertial Particles Under Uncertainty

Approximation of Random Slow Manifolds and Settling of Inertial Particles Under Uncertainty

A parameter estimation method based on random slow manifolds

A parameter estimation method is devised for a slow–fast stochastic dynamical system, where only the slow component is observable. By using available observations on the slow component, a system parameter is estimated by studying the slow system on the random slow manifold. This offers a benefit of dimension reduction in quantifying parameters in stochastic dynamical systems. An example is presented to illustrate this method, and to verify that the parameter estimator based on the lower dimensional, reduced slow system is a good approximation of the parameter estimator for the original slow–fast stochastic dynamical system.

Approximation of the random inertial manifold of singularly perturbed stochastic wave equations

By applying Rohlin's result on the classification of homomorphisms of Lebesgue space, the random inertial manifold of a stochastic damped nonlinear wave equations with singular perturbation is proved to be approximated almost surely by that of a stochastic nonlinear heat equation which is driven by a new Wiener process depending on the singular perturbation parameter. This approximation can be seen as the Smolukowski–Kramers approximation as time goes to infinity. However, as time goes infinity, the approximation changes with the small parameter, which is different from the approximation on a finite time interval.

Invariant foliations for random dynamical systems

We prove the existence of invariant foliations of stable and unstable manifolds of anormally hyperbolic random invariant manifold. The normally hyperbolic random invariant manifoldreferred to here is that which was shown to exist in a previous paper when a deterministic dynamicalsystem having a normally hyperbolic invariant manifold is subjected to a small random perturbation.