Large deviations for multiscale diffusion via weak convergence methods

We study the large deviations principle for locally periodic SDEs with small noise and fast oscillating coefficients. There are three regimes depending on how fast the intensity of the noise goes to zero relative to homogenization parameter. We use weak convergence methods which provide convenient representations for the action functional for all regimes. Along the way, we study weak limits of controlled SDEs with fast oscillating coefficients. We derive, in some cases, a control that nearly achieves the large deviations lower bound at prelimit level. This control is useful for designing efficient importance sampling schemes for multiscale small noise diffusion.     

Heuristic Parameter Choice Rules for Tikhonov Regularization with Weakly Bounded Noise

We study the choice of the regularization parameter for linear ill-posed problems in the presence of noise that is possibly unbounded but only finite in a weaker norm, and when the noise-level is unknown. For this task, we analyze several heuristic parameter choice rules, such as the quasi-optimality, heuristic discrepancy, and Hanke-Raus rules and adapt the latter two to the weakly bounded noise case. We prove convergence and convergence rates under certain noise conditions. Moreover, we analyze and provide conditions for the convergence of the parameter choice by the generalized cross-validation and predictive mean-square error rules.     

Efficent Generation of Fractional Brownian Motion for Simulation of Infrared Focal-plane Array Calibration Drift

Infrared focal-plane arrays suffer from a type of 1/f noise which leads to slow drifts in detected radiation levels following calibration. This noise can be modeled by fractional Brownian motion (FBM), with an empirical Hurst parameter (H) in the range 0 < H < 1 / 2. For such noise we examine the statistics of both the maximum deviation from the calibration point during a fixed time, and the time to reach a fixed deviation from the calibration point. We employ analytical and numerical means; for the latter, we provide a new algorithm for generating a discrete-time version of FBM with 0 < H 1 / 2 which is fast (order N log N), and exact. Statistics of the maximum deviation show the same qualitative behavior for different values of H, and rapidly approach a limit as the length N increases. Results for first passage times, in contrast, vary markedly with H, but not with N.     

Asymptotic properties of the MLE for the autoregressive process coefficients under stationary Gaussian noise

In this paper we study the Maximum Likelihood Estimator (MLE) of the vector parameter of an autoregressive process of order p with regular stationary Gaussian noise. We prove the large sample asymptotic properties of the MLE under very mild conditions. We do simulations for fractional Gaussian noise (fGn), autoregressive noise (AR(1)) and moving average noise (MA(1)).     

Stochastic Evolution Equations Driven by a Fractional White Noise

Abstract

In this paper we study stochastic evolution equations driven by a fractional white noise with arbitrary Hurst parameter in infinite dimension. We establish the existence and uniqueness of a mild solution for a nonlinear equation with multiplicative noise under Lipschitz condition by using a fixed point argument in an appropriate inductive limit space. In the linear case with additive noise, a strong solution is obtained. Those results are applied to stochastic parabolic partial differential equations perturbed by a fractional white noise.     

Large Deviations and Importance Sampling for Systems of Slow-Fast Motion

In this paper we develop the large deviations principle and a rigorous mathematical framework for asymptotically efficient importance sampling schemes for general, fully dependent systems of stochastic differential equations of slow and fast motion with small noise in the slow component. We assume periodicity with respect to the fast component. Depending on the interaction of the fast scale with the smallness of the noise, we get different behavior. We examine how one range of interaction differs from the other one both for the large deviations and for the importance sampling. We use the large deviations results to identify asymptotically optimal importance sampling schemes in each case. Standard Monte Carlo schemes perform poorly in the small noise limit. In the presence of multiscale aspects one faces additional difficulties and straightforward adaptation of importance sampling schemes for standard small noise diffusions will not produce efficient schemes. It turns out that one has to consider the so called cell problem from the homogenization theory for Hamilton-Jacobi-Bellman equations in order to guarantee asymptotic optimality. We use stochastic control arguments.     

An innovative fractional order LMS algorithm for power signal parameter estimation

Parameter estimation is an important issue for the quality monitoring and reliability assessment of power systems. In this study, an innovative fractional order least mean square (I-FOLMS) adaptive algorithm is presented for an effective parameter estimation. The I-FOLMS algorithm exploits the fractional gradient in its recursive parameter update mechanism, because its performance can be tuned by means of the fractional order. High values of the fractional order are good for fast convergence, but lead to steady state mis-adjustments. While, low values provide a smooth steady state behavior, but require a compromise in the convergence rate. The effective performance of I-FOLMS is verified and validated through two numerical examples of power signals estimation for different levels of noise variance and values of the fractional orders.     

Near optimality of stochastic control in systems with unknown parameter processes

Stochastic control for systems with an unknown parameter is considered in this paper. The underlying problem is to minimize a functional subject to a system described by a singularly perturbed differential equation with an unknown parameter process driven by fast fluctuating random disturbances. This problem arises in the context of stochastic adaptive control, adaptive signal processing, and failure-prone manufacturing systems. Due to the nature of the wide-bandwidth noise processes, identifying the parameter process for eacht is very hard since the driving noise changes very rapidly. An alternative approach is used, and an auxiliary control problem is introduced to overcome the difficulties. By means of weak convergence methods and comparison control techniques, nearly optimal controls are obtained.This research was supported in part by the National Science Foundation under Grant DMS-9022139 and DMS-9224372.     

Random attractor for the three-dimensional planetary geostrophic equations of large-scale ocean circulation with small multiplicative noise

The aim of this article is to study the asymptotical behavior, in terms of upper semi-continuous property of attractor, for small multiplicative noise of the three-dimensional planetary geostrophic equations of large-scale ocean circulation. In this article, we establish the existence of a random attractor for the three-dimensional planetary geostrophic equations of large-scale ocean circulation with small multiplicative noise by verifying the pullback flattening property and prove that the random attractor of the three-dimensional planetary geostrophic equations of large-scale ocean circulation with small multiplicative noise converges to the global attractor of the unperturbed three-dimensional planetary geostrophic equations of large-scale ocean circulation when the parameter of the perturbation tends to zero.     

Chaos Control by Nonfeedback Methods in the Presence of Noise

Nonfeedback methods of chaos control are suited for practical applications because of their speed, flexibility, no online monitoring and processing requirements. For applications where none, no real-time, or only highly restricted measurements of the system are available, or where the system behavior is to be altered more drastically, these schemes are quite useful. These methods convert the chaotic motion to any arbitrary fixed point or periodic orbit or quasiperiodic orbit. These attributes make them promising for controlling chaotic circuits, fast electro-optical systems, systems in which no parameter is accessible for control, and so on. For possible practical applications of the control methods, the robustness of the methods in the presence of noise is of special interest. The noise can be in the form of external disturbances to the system or in the form of uncertainties due to inexact modelling of the system. In this paper, we make an analysis of the control performance of various nonfeedback methods in controlling the chaotic behavior in the presence of noise in the chaotic system. The various nonfeedback methods considered for the analysis are: addition of (i) constant force, (ii) weak periodic force, (iii) periodic delta-pulses, (iv) rectangular-pulses. The examples considered for this study are the Murali–Lakshmanan–Chua Circuit, and Duffing–Ueda oscillator.     

Hunting French ducks in a noisy environment

We consider the effect of Gaussian white noise on fast–slow dynamical systems with one fast and two slow variables, containing a folded node singularity. In the absence of noise, these systems are known to display mixed-mode oscillations, consisting of alternating large- and small-amplitude oscillations. We quantify the effect of noise and obtain critical noise intensities beyond which the small-amplitude oscillations become hidden by fluctuations. Furthermore we prove that the noise can cause sample paths to jump away from so-called canard solutions with high probability before deterministic orbits do. This early-jump mechanism can drastically influence the local and global dynamics of the system by changing the mixed-mode patterns.     

Singular Hopf Bifurcation in Systems with Fast and Slow Variables

Summary. We study a general nonlinear ODE system with fast and slow variables, i.e., some of the derivatives are multiplied by a small parameter. The system depends on an additional bifurcation parameter. We derive a normal form for this system, valid close to equilibria where certain conditions on the derivatives hold. The most important condition concerns the presence of eigenvalues with singular imaginary parts, by which we mean that their imaginary part grows without bound as the small parameter tends to zero. We give a simple criterion to test for the possible presence of equilibria satisfying this condition. Using a center manifold reduction, we show the existence of Hopf bifurcation points, originating from the interaction of fast and slow variables, and we determine their nature. We apply the theory, developed here, to two examples: an extended Bonhoeffer—van der Pol system and a predator-prey model. Our theory is in good agreement with the numerical continuation experiments we carried out for the examples. Received October 24, 1996; revised October 31, 1997; accepted November 3, 1997     

Necessary and sufficient conditions for power convergence rate of approximations in Tikhonov’s scheme for solving ill-posed optimization problems

We investigate a rate of convergence of estimates for approximations generated by Tikhonov’s scheme for solving ill-posed optimization problems with smooth functionals under a structural nonlinearity condition in a Hilbert space, in the cases of exact and noisy input data. In the noise-free case, we prove that the power source representation of the desired solution is close to a necessary and sufficient condition for the power convergence estimate having the same exponent with respect to the regularization parameter. In the presence of a noise, we give a parameter choice rule that leads for Tikhonov’s scheme to a power accuracy estimate with respect to the noise level.     

Small Noise Asymptotics of the Bayesian Estimator in Nonidentifiable Models

We study the asymptotic behavior of the Bayesian estimator for a deterministic signal in additive Gaussian white noise, in the case where the set of minima of the Kullback–Leibler information is a submanifold of the parameter space. This problem includes as a special case the study of the asymptotic behavior of the nonlinear filter, when the state equation is noise-free, and when the limiting deterministic system is nonobservable. As the noise intensity goes to zero, the posterior probability distribution of the parameter asymptotically concentrates on the submanifold of minima of the Kullback–Leibler information. We give an explicit expression of the limit, and we study the rate of convergence. We apply these results to a practical example where nonidentifiability occurs.     

A “Nonconvex+Nonconvex” approach for image restoration with impulse noise removal

In this paper, we propose a new method for image restoration problems, which are degraded by impulsive noise, with nonconvex data fitting term and nonconvex regularizer.The proposed method possesses the advantages of nonconvex data fitting and nonconvex regularizer simultaneously, namely, robustness for impulsive noise and efficiency for restoring neat edge images.Further, we propose an efficient algorithm to solve the “Nonconvex+Nonconvex” structure problem via using the alternating direction minimization, and prove that the algorithm is globally convergent when the regularization parameter is known. However, the regularization parameter is unavailable in general. Thereby, we combine the algorithm with the continuation technique and modified Morozov’s discrepancy principle to get an improved algorithm in which a suitable regularization parameter can be chosen automatically. The experiments reveal the superior performances of the proposed algorithm in comparison with some existing methods.     

Parameter estimation for the Langevin equation with stationary-increment Gaussian noise

We study the Langevin equation with stationary-increment Gaussian noise. We show the strong consistency and the asymptotic normality with Berry–Esseen bound of the so-called second moment estimator of the mean reversion parameter. The conditions and results are stated in terms of the variance function of the noise. We consider both the case of continuous and discrete observations. As examples we consider fractional and bifractional Ornstein–Uhlenbeck processes. Finally, we discuss the maximum likelihood and the least squares estimators.     

A family of rules for parameter choice in Tikhonov regularization of ill-posed problems with inexact noise level

We consider Tikhonov regularization of linear ill-posed problems with noisy data. The choice of the regularization parameter by classical rules, such as discrepancy principle, needs exact noise level information: these rules fail in the case of an underestimated noise level and give large error of the regularized solution in the case of very moderate overestimation of the noise level. We propose a general family of parameter choice rules, which includes many known rules and guarantees convergence of approximations. Quasi-optimality is proved for a sub-family of rules. Many rules from this family work well also in the case of many times under- or overestimated noise level. In the case of exact or overestimated noise level we propose to take the regularization parameter as the minimum of parameters from the post-estimated monotone error rule and a certain new rule from the proposed family. The advantages of the new rules are demonstrated in extensive numerical experiments.     

Random attractor for the stochastic Cahn–Hilliard–Navier–Stokes system with small additive noise

The objective of this paper is to study the asymptotic behavior of solutions, in terms of the upper semi-continuous property of random attractor, of the Cahn–Hilliard–Navier–Stokes system with small additive noise. We prove the existence of a random attractor for the Cahn–Hilliard–Navier–Stokes system with small additive noise. Furthermore, we consider the stability of global attractor and prove the random attractor of the Cahn–Hilliard–Navier–Stokes system with small additive noise will convergent to the global attractor of the unperturbed Cahn–Hilliard–Navier–Stokes system when the parameter of the perturbation ε tends to zero.     

Regularization properties of the sequential discrepancy principle for Tikhonov regularization in Banach spaces

The stable solution of ill-posed non-linear operator equations in Banach space requires regularization. One important approach is based on Tikhonov regularization, in which case a one-parameter family of regularized solutions is obtained. It is crucial to choose the parameter appropriately. Here, a sequential variant of the discrepancy principle is analysed. In many cases, such parameter choice exhibits the feature, called regularization property below, that the chosen parameter tends to zero as the noise tends to zero, but slower than the noise level. Here, we shall show such regularization property under two natural assumptions. First, exact penalization must be excluded, and secondly, the discrepancy principle must stop after a finite number of iterations. We conclude this study with a discussion of some consequences for convergence rates obtained by the discrepancy principle under the validity of some kind of variational inequality, a recent tool for the analysis of inverse problems.     

基于BP神经网络的动态稳健参数设计

传统的动态稳健参数设计方法(田口方法)虽然在工业生产实践中展现了极大的方便,但是其本身也存在较大的改进空间.当调节变量不存在时,传统的田口方法难以实现;此外,田口方法只能根据所选取的参数水平得到最优参数组合,而这种所谓的最优结果有时并不符合实际的需要.首先构建BP神经网络模型,利用训练后的BP神经网络获得参数设计中质量特性、噪声因子以及各参数间的动态关系;然后,利用超拉丁方抽样,计算信号与特性参数间的斜率,并由此将动态稳健参数设计的寻优问题转化为相应的非线性规划问题;最后,利用次序二次规划(SQP)算法解决并优化动态稳健参数的设计。此外,我们选取了一个简单的数据案例对本文提出的方法的有效性进行了说明.