Small Random Perturbations Research Articles

Smoothed analysis of dynamic networks

In this paper, we generalize the technique of smoothed analysis to apply to distributed algorithms in dynamic networks in which the network graph can change from round to round. Whereas standard smoothed analysis studies the impact of small random perturbations of input values on algorithm performance metrics, our proposed dynamic network version of smoothed analysis studies the impact of random perturbations of the underlying changing network topologies. Similar to the original application of smoothed analysis, our goal is to study whether known strong lower bounds in these models are robust or fragile: do they withstand small (random) perturbations, or do such deviations push the graphs far enough from a precise pathological instance to enable much better performance? Fragile lower bounds are likely not relevant for real-world deployment, while robust lower bounds represent a true difficulty caused by dynamic behavior. We apply this technique to three standard dynamic network problems with known strong worst-case lower bounds: random walks, flooding, and aggregation. We prove that these bounds provide a spectrum of robustness when subjected to smoothing—some are fragile (random walks), some are moderately fragile (flooding), and some are robust (aggregation).

Contributions of imprecision in PET-MRI rigid registration to imprecision in amyloid PET SUVR measurements.

Quantitative measurement of β‐amyloid from amyloid PET scans typically relies on localizing target and reference regions by image registration to MRI. In this work, we present a series of simulations where 50 small random perturbations of starting location and orientation were applied to each subject's PET scan, and rigid registration using spm_coreg was performed between each perturbed PET scan and its corresponding MRI. We then measured variation in the output PET‐MRI registrations and how this variation affected the resulting SUVR measurements. We performed these experiments using scans of 1196 participants, half using 18F florbetapir and half using 11C PiB. From these experiments, we measured the magnitude of the imprecision in the rigid registration steps used to localize measurement regions, and how this contributes to the overall imprecision in SUVR measurements. Unexpectedly, we found for both tracers that the imprecision in these measurements depends on the degree of amyloid tracer uptake, and thus also indirectly on Alzheimer's disease clinical status. We then examined common choices of reference regions, and we show that SUVR measurements using supratentorial white matter references are relatively resistant to this source of error. We also show that the use of partial volume correction further magnifies the effects of registration imprecision on SUVR measurements. Together, these results suggest that this rigid registration step is an attractive target for future work in improving measurement techniques. Hum Brain Mapp 38:3323–3336, 2017. © 2017 The Authors Human Brain Mapping Published by Wiley Periodicals, Inc.

Strong Averaging Along Foliated Lévy Diffusions with Heavy Tails on Compact Leaves

This article shows a strong averaging principle for diffusions driven by discontinuous heavy-tailed Levy noise, which are invariant on the compact horizontal leaves of a foliated manifold subject to small transversal random perturbations. We extend a result for such diffusions with exponential moments and bounded, deterministic perturbations to diffusions with polynomial moments of order \(p\geqslant 2\), perturbed by deterministic and stochastic integrals with unbounded coefficients and polynomial moments. The main argument relies on a result of the dynamical system for each individual jump increments of the corresponding canonical Marcus equation. The example of Levy rotations on the unit circle subject to perturbations by a planar Levy-Ornstein-Uhlenbeck process is carried out in detail.

The Standing Accretion Shock Instability: Enhanced Growth in Rotating Progenitors

Abstract We investigate the effect of progenitor rotation on the standing accretion shock instability (SASI) using two- and three-dimensional hydrodynamic simulations. We find that the growth rate of the SASI is a near-linearly increasing function of the specific angular momentum in the accreting gas. Both the growth rate and the angular frequency in the two-dimensional model with cylindrical geometry agree well with previous linear stability analyses. When excited by very small random perturbations, a one-armed spiral mode dominates the small rotation rates predicted by current stellar evolution models, while progressively higher-order modes are seen as the specific angular momentum increases.

Modulation of localized solutions in quadratic-cubic nonlinear Schrödinger equation with inhomogeneous coefficients

We study the presence of exact localized solutions in a quadratic-cubic nonlinear Schrödinger equation with inhomogeneous nonlinearities. Using a specific ansatz, we transform the nonautonomous nonlinear equation into an autonomous one, which engenders composed states corresponding to solutions localized in space, with an oscillating behavior in time. Direct numerical simulations are employed to verify the stability of the modulated solutions against small random perturbations.

Errors, correlations and fidelity for noisy Hamilton flows. Theory and numerical examples

We analyse the asymptotic growth of the error for Hamiltonian flows due to small random perturbations. We compare the forward error with the reversibility error, showing their equivalence for linear flows on a compact phase space. The forward error, given by the root mean square deviation of the noisy flow, grows according to a power law if the system is integrable and according to an exponential law if it is chaotic. The autocorrelation and the fidelity, defined as the correlation of the perturbed flow with respect to the unperturbed one, exhibit an exponential decay as . Some numerical examples such as the anharmonic oscillator and the Hénon Heiles model confirm these results. We finally consider the effect of the observational noise on an integrable system, and show that the decay of correlations can only be observed after a sequence of measurements and that the multiplicative noise is more effective if the delay between two measurements is large.

High-order numerical method for the derivative nonlinear Schrödinger equation

In this work, a fourth-order numerical scheme in space and two second-order numerical schemes in both time and space are proposed for the derivative nonlinear Schrödinger equation. We verify the mass conservation for the two-level implicit scheme. The influence on the soliton solution by adding a small random perturbation to the initial condition is discussed. The numerical experiments are given to test the accuracy order for different schemes, respectively. We also test the conservative property of mass and Hamiltonian for these schemes from the numerical point of view.

A Coordinate Descent Method for Total Variation Minimization

Total variation (TV) is a well-known image model with extensive applications in various images and vision tasks, for example, denoising, deblurring, superresolution, inpainting, and compressed sensing. In this paper, we systematically study the coordinate descent (CoD) method for solving general total variation (TV) minimization problems. Based on multidirectional gradients representation, the proposed CoD method provides a unified solution for both anisotropic and isotropic TV-based denoising (CoDenoise). With sequential sweeping and small random perturbations, CoDenoise is efficient in denoising and empirically converges to optimal solution. Moreover, CoDenoise also delivers new perspective on understanding recursive weighted median filtering. By incorporating with the Augmented Lagrangian Method (ALM), CoD was further extended to TV-based image deblurring (ALMCD). The results on denoising and deblurring validate the efficiency and effectiveness of the CoD-based methods.

Lyapunov exponents for random perturbations of some area-preserving maps including the standard map

We consider a large class of 2D area-preserving diffeomorphisms that are not uniformly hyperbolic but have strong hyperbolicity properties on large regions of their phase spaces. A prime example is the Standard map. Lower bounds for Lyapunov exponents of such systems are very hard to estimate, due to the potential switching of "stable" and "unstable" directions. This paper shows that with the addition of (very) small random perturbations, one obtains with relative ease Lyapunov exponents reflecting the geometry of the deterministic maps.

Langevin Equations for Landmark Image Registration with Uncertainty

Registration of images parameterized by landmarks provides a useful method of describing shape variations by computing the minimum-energy time-dependent deformation field that flows from one landmark set to the other. This is sometimes known as the geodesic interpolating spline and can be solved via a Hamiltonian boundary-value problem to give a diffeomorphic registration between images. However, small changes in the positions of the landmarks can produce large changes in the resulting diffeomorphism. We formulate a Langevin equation for looking at small random perturbations of this registration. The Langevin equation and three computationally convenient approximations are introduced and used as prior distributions. A Bayesian framework is then used to compute a posterior distribution for the registration, and also to formulate an average of multiple sets of landmarks.

White noise perturbation of locally stable dynamical systems

The influence of small random perturbations on a deterministic dynamical system with a locally stable equilibrium is considered. The perturbed system is described by the Itô stochastic differential equation. It is assumed that the noise does not vanish at the equilibrium. In this case the trajectories of the stochastic system may leave any bounded domain with probability one. We analyze the stochastic stability of the equilibrium under a persistent perturbation by white noise on an asymptotically long time interval [Formula: see text], where [Formula: see text] is a perturbation parameter.

On stochastic perturbations of slowly changing dynamical systems

In this paper we consider a diffusion process obtained as a small random perturbation of a dynamical system attracted to a stable equilibrium point. The drift and the diffusive perturbation are assumed to evolve slowly in time. We describe the asymptotics of the time it takes the process to exit a given domain and the limiting distribution of the exit point.

A dynamic-solver–consistent minimum action method: With an application to 2D Navier–Stokes equations

This paper discusses the necessity and strategy to unify the development of a dynamic solver and a minimum action method (MAM) for a spatially extended system when employing the large deviation principle (LDP) to study the effects of small random perturbations. A dynamic solver is used to approximate the unperturbed system, and a minimum action method is used to approximate the LDP, which corresponds to solving an Euler–Lagrange equation related to but more complicated than the unperturbed system. We will clarify possible inconsistencies induced by independent numerical approximations of the unperturbed system and the LDP, based on which we propose to define both the dynamic solver and the MAM on the same approximation space for spatial discretization. The semi-discrete LDP can then be regarded as the exact LDP of the semi-discrete unperturbed system, which is a finite-dimensional ODE system. We achieve this methodology for the two-dimensional Navier–Stokes equations using a divergence-free approximation space. The method developed can be used to study the nonlinear instability of wall-bounded parallel shear flows, and be generalized straightforwardly to three-dimensional cases. Numerical experiments are presented.

The Precise Shape of the Eigenvalue Intensity for a Class of Non-Self-Adjoint Operators Under Random Perturbations

We consider a non-self-adjoint h-differential model operator $$P_h$$ in the semiclassical limit ( $$h\rightarrow 0$$ ) subject to small random perturbations. Furthermore, we let the coupling constant $$\delta $$ be $$\mathrm {e}^{-\frac{1}{Ch}}\le \delta \ll h^{\kappa }$$ for constants $$C,\kappa >0$$ suitably large. Let $$\Sigma $$ be the closure of the range of the principal symbol. Previous results on the same model by Hager, Bordeaux-Montrieux and Sjostrand show that if $$\delta \gg \mathrm {e}^{-\frac{1}{Ch}}$$ there is, with a probability close to 1, a Weyl law for the eigenvalues in the interior of the of the pseudospectrum up to a distance $$\gg \left( -h\ln {\delta h}\right) ^{\frac{2}{3}}$$ to the boundary of $$\Sigma $$ . We study the intensity measure of the random point process of eigenvalues and prove an h-asymptotic formula for the average density of eigenvalues. With this we show that there are three distinct regions of different spectral behavior in $$\Sigma $$ : the interior of the pseudospectrum is solely governed by a Weyl law, close to its boundary there is a strong spectral accumulation given by a tunneling effect followed by a region where the density decays rapidly.

Nonlinear dynamic response analysis on gear-rotor-bearing transmission system

A coupled lateral-torsional nonlinear dynamic model with 16-degree-of-freedom (16-DOF) of gear-rotor-bearing transmission system (GRBTS) is developed after comprehensive considering the nonlinear features associated with time-varying meshing stiffness, backlash, transmission error, friction force, input/output load, gravity and gear eccentricity. Based on the nonlinear differential equations, the coupled multi-body dynamic responses of the GRBTS are demonstrated using the Runge-Kutta numerical method, and the effects of friction coefficient and mean load on the dynamic characteristics are investigated. The results show that the friction force could enlarge the vibration amplitude and affect the low frequency components seriously. The mean load excitation has a complicated influence on the coupled GRBTS, and the torsional vibration is the dominate response. Whereas the mean load excitation has a certain extent vibration suppression, and light load and heavy load could no longer effectively control the nonlinear vibration of the GRBTS. With the increasing of rotational speed, the friction coefficient and mean load ranges of the chaotic behavior widen and the chaotic characteristics strengthens. It is shown that small parameter random perturbation might be propagated in the vibration system and lead to relatively large vibration of the system. The contribution to provide a reference for the design and study of gear system.

Experimental and numerical investigation of a strongly-forced precessing cylinder flow

Planar particle image velocimetry (PIV) and direct numerical simulations (DNS) of the flow in a precessing cylinder of height-to-radius ratio of 1.835 are presented for a nutation angle of 15° and a Reynolds number 6430 based on the cylinder’s angular frequency. We find excellent agreement in terms of time-averaged velocity profiles and Kelvin mode amplitudes during the initial development and in the asymptotic state. In the experiment, a rapid transition to a disordered state follows a brief appearance of azimuthal structures with a predominant wavenumber m=9. In the DNS, the transition occurs later than in the experiment, and shows dominance of a triad of azimuthal wavenumbers m=1,4,5 during the transient evolution. Adding a small random perturbation to the simulation after the forced mode is established accelerates transition to flow dominated by m=9 which eventually transitions to a disordered state, consistent with observations from PIV. The breaking of rotoreflection (inversion) symmetry of the system is found to be critical in establishing evolution paths to the disordered and asymmetric asymptotic state.

Solution of the Problem of Stochastic Stability of an Integral Manifold by the Second Lyapunov Method

By the method of Lyapunov functions, we establish sufficient conditions of stability and asymptotic stability in probability for the integral manifold of the Ito differential equations in the presence of random perturbations from the class of processes with independent increments. Theorems on stochastic stability of the analytically given integral manifold of differential equations are proved in the first approximation and under the permanent action of small (in the mean) random perturbations.

Linear and nonlinear compact modes in quasi-one-dimensional flatband systems

We examine analytically and numerically the spectral properties of three quasi-one-dimensional lattices, namely, kagome, Lieb, and stub lattices, which are characterized for having flatbands in their spectrum. It is observed that the degenerate eigenmodes modes of these flatbands form a Starklike ladder where each mode is shifted by one lattice site. Their combination can give rise to compact modes that do not diffract due to a geometrical phase cancellation. For all three cases we computed the stability of the fundamental band mode against perturbation of their amplitude and phase, the effect of possible anisotropy of the couplings, and the presence of small random perturbations of the coupling. For the Lieb and stub ribbon, the compact mode turns out to be quite robust and the flatband survives, while for the kagome ribbon, the compact mode is destroyed and the flatband is lost. When adding nonlinear effects, the compact mode turns out to be also a nonlinear eigenvector, with a power curve that is proportional to the eigenvalue and exists for any eigenvalue, in marked contrast to the usual case of discrete solitons, which can exist only outside the linear bands. These properties look promising for a future design of a robust system for long-distance propagation of information.

Asymptotic behaviors for functionals of random dynamical systems

ABSTRACTIn this article, we consider asymptotic behaviors for functionals of dynamical systems with small random perturbations. First, we present a deviation inequality for Gaussian approximation of dynamical systems with small random perturbations under Hölder norms and establish the moderate deviation principle and the central limit theorem for the dynamical systems by the deviation inequality. Then, applying these results to forward-backward stochastic differential equations and diffusions in small time intervals, combining the delta method in large deviations, we give a moderate deviation principle for solutions of forward-backward stochastic differential equations with small random perturbations, and obtain the central limit theorem, the moderate deviation principle and the iterated logarithm law for functionals of diffusions in small time intervals.

Transmission and localisation in ordered and randomly-perturbed structured flexural systems

The paper presents a novel analysis of localisation and transmission properties of randomly-perturbed flexural systems. Attention is given to the study of propagation regimes and the connection with transmission resonances following perturbations of ordered stacks. The analytical study is complemented with numerical simulations relevant to different discrete systems. Applications are in the design of efficient vibration isolation systems and filters of elastic waves considered as ordered structures subjected to small random perturbations.