Method Of Stochastic Averaging Research Articles

Revisiting a Drag Partition Model For Canopy-Like Roughness Elements

Turbulent flows over a large surface area (S) covered by n obstacles experience an overall drag due to the presence of the ground and the protruding obstacles into the flow. The drag partition between the roughness obstacles and the ground is analyzed using an analytical model proposed by Raupach (Boundary-Layer Meteorol 60:375-395, 1992) and is hereafter referred to as R92. The R92 is based on the premise that the wake behind an isolated roughness element can be described by a shelter area A and a shelter volume V. The individual sizes of A and V without any interference from other obstacles can be determined from scaling analysis for the spread of wakes. To upscale from an individual roughness element to n/S elements where wakes may interact, R92 adopted a background stress re-normalizing instead of reducing A or V with each element addition. This work demonstrates that R92’s approach results in a linear background stress reduction in A and V only when the ratio of n/S is small, due to a low probability of wake interactions. This probabilistic nature suggests that up-scaling from individual to multiple roughness elements can be re-formulated using stochastic averaging methods proposed here. The two approaches are shown to recover R92 under plausible conditions. An alternative scaling for the shelter volume is also proposed here using thermodynamic arguments of work and dissipation though the final outcome remains similar to R92. Comparisons between R92 and available data spanning more than two decades after R92 on blocks and vegetation-like roughness elements confirm the practical utility of R92. The agreement between R92 and this updated databases of experiments and simulations confirm the potential use of R92 in large-scale models provided that the relevant parameters accommodate certain features of the roughness element type (cube versus vegetation-like) and, to a lesser extent, their configuration throughout S. Last, a comparison between R92 and models based on first-order closure principles with constant mixing length suggests that R92 can outperform such models when evaluated across a wide range of roughness densities.

Higher-order stochastic averaging for investigating a vehicle suspension system with nonlinear damping and stiffness

The paper deals with a quarter-car model with nonlinear damping and stiffness under white noise base excitation using the higher-order stochastic averaging method for analyzing approximate responses. Recently, a novel higher-order averaging procedure has been developed to find analytically the first-, second-, and third-order stationary joint probability density functions (PDF) of amplitude and full phase by solving the corresponding Fokker-Planck-Kolmogorov (FPK) equation, and it will be extended to the nonlinear quarter-car model. Accordingly, the mean-square responses such as the displacement, and velocity of the sprung mass can be obtained analytically. The influences of excitation intensity on the dynamic responses, as well as the variations of linear and nonlinear damping, are analyzed. A very satisfactory agreement is found between the accuracy of the solutions corresponding to higher-order stochastic averaging and that of Monte Carlo simulation.

Can specific THz fields induce collective base-flipping in DNA? A stochastic averaging and resonant enhancement investigation based on a new mesoscopic model.

We study the metastability, internal frequencies, activation mechanism, energy transfer, and the collective base-flipping in a mesoscopic DNA via resonance with specific electric fields. Our new mesoscopic DNA model takes into account not only the issues of helicity and the coupling of an electric field with the base dipole moments, but also includes environmental effects, such as fluid viscosity and thermal noise. Also, all the parameter values are chosen to best represent the typical values for the opening and closing dynamics of a DNA. Our study shows that while the mesoscopic DNA is metastable and robust to environmental effects, it is vulnerable to certain frequencies that could be targeted by specific THz fields for triggering its collective base-flipping dynamics and causing large amplitude separation of base pairs. Based on applying the Freidlin-Wentzell method of stochastic averaging and the newly developed theory of resonant enhancement to our mesoscopic DNA model, our semi-analytic estimates show that the required fields should be THz fields with frequencies around 0.28 THz and with amplitudes in the order of 450 kV/cm. These estimates compare well with the experimental data of Titova et al., which have demonstrated that they could affect the function of DNA in human skin tissues by THz pulses with frequencies around 0.5 THz and with a peak electric field at 220 kV/cm. Moreover, our estimates also conform to a number of other experimental results, which appeared in the last couple years.

Stochastic Analysis for the Embedded Single-Walled Carbon Nanotube Under Random Vibrations

It is inevitable that carbon nanotubes (CNTs) are subjected to random vibrations due to environmental noise, which may impair their mechanical properties. The nonlinear dynamics associated with single-walled carbon nanotubes (SWCNTs) under random noise has been investigated in this work. First, based on the nonlocal theory and the Euler–Bernoulli beam model with fixed supports at both ends, the governed equations and boundary conditions are established according to Hamilton’s principle. Second, the Galerkin method is used to approximate the differential equations of motion and the stochastic averaging method (SAM) is applied to predict the approximate response of the original system. Subsequently, a detailed parametric study is conducted to analyze the nonlinear random vibration response of the SWCNT with nonlocal effects, and the effectiveness of the proposed method is verified by comparing the analytical results with those from the Runge–Kutta method. Finally, by solving the backward Kolmogorov (BK) equation and the generalized Pontryagin (GP) equation simultaneously, the conditional reliability function (CRF) and mean first-passage time (MFPT) for determining the reliability of SWCNT systems are obtained.

Response of Gaussian white noise excited oscillators with inertia nonlinearity based on the RBFNN method

Although stochastic averaging methods have proven effective in solving the responses of nonlinear oscillators with a strong stiffness term under broadband noise excitations, these methods appear to be ineffective when dealing with oscillators that have a strong inertial nonlinearity term (also known as coordinate-dependent mass) or multiple potential wells. To address this limitation, a radial basis function neural network (RBFNN) algorithm is applied to predict the responses of oscillators with both a strong inertia nonlinearity term and multiple potential wells. The well-known Gaussian functions are chosen as radial basis functions in the model. Then, the approximate stationary probability density function (PDF) is expressed as the sum of Gaussian basis functions (GBFs) with weights. The squared error of the approximate solution for the Fokker-Plank-Kolmogorov (FPK) function is minimized using the Lagrange multiplier method to determine optimal weight coefficients. Three examples are presented to demonstrate how inertia nonlinearity terms and potential wells affect the responses. The mean square errors between Monte Carlo simulations (MCS) and RBFNN predictions are provided. The results indicate that RBFNN predictions align perfectly with those obtained from MCS.

Stochastic analysis and vibration suppression of a time-delayed system with nonlinear energy sink

Nonlinear energy sink (NES) is of great significance in attenuating structural vibration. Time-delay phenomenon in a mechanical system is inevitable. Different from existing works, the influences of time-delayed factors in a nonlinear oscillator attached with NES are considered in this paper under Gaussian white noise (GWN). Firstly, the time-delayed term is transformed into a linear combination of stiffness and damping terms, and an equivalent system without the time-delayed term is obtained. Stationary probability density function (SPDF) is deduced by leveraging stochastic averaging method. Secondly, good agreement is observed between analytical and numerical results, which verifies the effectiveness of the method. Furthermore, bifurcation conditions for the amplitude of SPDF of the primary system are obtained, indicating that delayed time, feedback gain and damping coefficients can cause stochastic P-bifurcation of the primary system. Finally, the vibration suppression performance of NES is analyzed via energy variation and the time–history diagram of displacement.

A novel approach for dimensionality reduction of high-dimensional stochastic dynamical systems using symbolic regression

This paper introduces an innovative data-driven approach for dimensionality reduction in stochastic dynamical systems. The method combines stochastic averaging theory with symbolic regression. First, this approach generates simulation data from the high-dimensional stochastic dynamical system that needs to be reduced. Then, a loss function is designed based on stochastic averaging theory, and symbolic regression is used to obtain a parameterized one-dimensional equation from the simulation data. Finally, the effectiveness of the approach proposed in this paper is demonstrated through validation on Hamiltonian systems with 5, 10, and 20 degrees of freedom (DOF) subjected to noisy excitation. The primary advantage of this method is that it ultimately provides a closed form rather than a black-box process. Furthermore, it overcomes the numerical integration challenges that stochastic averaging methods often face when dealing with high-dimensional systems, therefore it performs effectively with high-dimensional stochastic dynamical systems.

Stochastic bifurcation and chaos study for nonlinear ship rolling motion with random excitation and delayed feedback controls

In this paper, we investigate the stochastic dynamics of a class of nonlinear ship rolling motion with multiplicative noise under both displacement and velocity delay feedback controls. The Itoˆ-stochastic differential equation for the amplitude and phase of the roll motion is derived using the stochastic center manifold method and stochastic average method. Subsequently, the stochastic stability and bifurcation behaviors of the system are analyzed. Furthermore, using the stationary probability density method, we derive the parameter conditions for the occurrence of stochastic D-bifurcation and stochastic P-bifurcation. We also analyze the properties and shape changes of the system’s probability density function under different parameters through numerical simulation. It has been determined that the system exhibits stochastic bifurcation behavior, specifically P-bifurcation and D-bifurcation. The validity of the method is verified by a numerical model. The theoretical chaos threshold of the system is determined using the random Melnikov method, and the impact of delayed feedback parameters on the chaotic motion of the system is analyzed by combining the bifurcation diagram, phase portrait, and time series.

Theoretical and experimental study of a bi-stable piezoelectric energy harvester under hybrid galloping and band-limited random excitations

In the practical environment, it is very common for the simultaneous occurrence of base excitation and crosswind. Scavenging the combined energy of vibration and wind with a single energy harvesting structure is fascinating. For this purpose, the effects of the wind speed and random excitation level are investigated with the stochastic averaging method (SAM) based on the energy envelope. The results of the analytical prediction are verified with the Monte-Carlo method (MCM). The numerical simulation shows that the introduction of wind can reduce the critical excitation level for triggering an inter-well jump and make a bi-stable energy harvester (BEH) realize the performance enhancement for a weak base excitation. However, as the strength of the wind increases to a particular level, the influence of the random base excitation on the dynamic responses is weakened, and the system exhibits a periodic galloping response. A comparison between a BEH and a linear energy harvester (LEH) indicates that the BEH demonstrates inferior performance for high-speed wind. Relevant experiments are conducted to investigate the validity of the theoretical prediction and numerical simulation. The experimental findings also show that strong random excitation is favorable for the BEH in the range of low wind speeds. However, as the speed of the incoming wind is up to a particular level, the disadvantage of the BEH becomes clear and evident.

Resonant behaviors of two coupled fluctuating-frequency oscillators with tempered Mittag-Leffler memory kernel

The dynamics of coupled oscillators and their collective behaviors are of great significance in various fields. This paper explores the resonant behaviors in the system composed of two coupled fluctuating-frequency oscillators with tempered Mittag-Leffler (M-L) memory kernel. The study begins by demonstrating that the two particles synchronize in the long-time limit, indicating that the mean-field behavior aligns with that of each individual particle. Accordingly, we derive an exact analytical expression for the steady-state output amplitude gain (OAG) of the mean field using the stochastic average method. Furthermore, the research investigates the diverse phenomena of generalized stochastic resonance (GSR) and examines how various system parameters influence GSR. To validate the theoretical analysis, numerical simulations are conducted to support and verify our findings.

An underdamped and delayed tri-stable model-based stochastic resonance

Stochastic resonance (SR) is investigated in an underdamped tri-stable potential system driven by Gaussian colored noise and a periodic excitation, where both displacement and velocity time-delayed states feedback are considered. It is challenging to study SR in a second-order delayed multi-stable system analytically. In this paper, the improved energy envelope stochastic average method is developed to derive the analytical expressions of stationary probability density (SPD) and spectral amplification. The effects of noise intensity, damping coefficient, and time delay on SR are analyzed. The results show that the shapes of joint SPD can be adjusted to the desired structure by choosing the time delay and feedback gains. For fixed time delay, the SR peak is increased for negative displacement or velocity feedback gain. Meanwhile, the SR peak is decreased while the optimal noise intensity increases with increasing correlation time of noise. The Monte Carlo simulations (MCS) confirm the effectiveness of the theoretical results.

Stochastic approximation based confidence regions for stochastic variational inequalities

The sample average approximation (SAA) and the stochastic approximation (SA) are two popular schemes for solving the stochastic variational inequalities problem (SVIP). In the past decades, theories on the consistency of the SAA solutions and SA solutions have been well studied. More recently, the asymptotic confidence regions of the true solution to SVIP have been constructed when the SAA scheme is implemented. It is of fundamental interest to develop confidence regions of the true solution to the SVIP when the SA scheme is employed. In this paper, we discuss the framework of constructing asymptotic confidence regions for the true solution of SVIP with a focus on stochastic dual average method. We first establish the asymptotic normality of the SA solutions both in ergodic sense and non-ergodic sense. Then the online methods of estimating the covariance matrices in the normal distributions are studied. Finally, practical procedures of building the asymptotic confidence regions of solutions to SVIP with numerical simulations are presented.

Design and analysis of a galloping energy harvester with V-shape spring structure under Gaussian white noise

This paper presents the design of an innovative galloping energy harvester (GEH) that incorporates an elastic structure to reconstructed the traditional GEH. The mathematical model of the GEH is derived based on the Euler–Bernoulli beam theory and Kirchhoff’s law. Due to the nonlinear properties of the model, the partial linearization technique and Fokker–Planck–Kolmogorov equation are used to further explore the joint probability density function (PDF) and the stationary PDF of displacement and velocity. Subsequently, Monte Carlo simulation is used to illustrate the behavior of the theoretical results. The simulation results show that, under different conditions, the stationary PDF exhibits double-peak or single-peak shape, which is caused by the compression or elongation state. Meanwhile, higher wind speeds correspond to larger displacement amplitudes. Then, the stationary PDF of displacement amplitude is obtained using the stochastic average method, and the expressions for mean square voltage and mean output power are derived. Numerical results indicate that when the wind speed is below the critical wind speed for galloping phenomenon, the vibration of the GEH is mainly driven by noise. When the wind speed exceeds the critical wind speed, the GEH is mainly affected by the wind speed. Higher wind speeds correspond to higher mean output power. An optimal electromechanical coupling coefficient and the ratio of the mechanical and electrical can be derived based on the GEH parameters when galloping occurs.

A Dual Averaging Algorithm for Online Modeling of Infinite Memory Nonlinear Systems

An on-line modelling algorithm is derived from a generic stochastic Dual Averaging (DA) method. It employs a negative entropy as a distance-generating function and the Volterra series expansion as a dictionary. Assuming that the measurement data are not <italic xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">i.i.d.</i> but generated by a nonlinear dynamical system with an infinite, exponentially fading memory, the error bounds are established for both the generic DA method and for the proposed modelling algorithm. The experiments performed on a set of benchmark systems confirm the applicability of the algorithm in real-world scenarios and demonstrate its low computational complexity.

A Dynamic Behavior Analysis of a Rolling Mill’s Main Drive System with Fractional Derivative and Stochastic Disturbance

Taking the random factors into account, a fractional main drive system of a rolling mill with Gaussian white noise is developed. First, the potential deterministic bifurcation is investigated by a linearized stability analysis. The results indicate that the fractional order changes the system from a stable point to a limit cycle with symmetric phase trajectories. Then, the stochastic response is obtained with the aid of the equivalent transformation of the fractional derivative and stochastic averaging methods. It is found that the joint stationary probability density function appears to have symmetric distribution. Finally, the influence of the fractional order and noise intensity on system dynamics behavior is discussed. The study is beneficial to understand the intrinsic mechanisms of vibration abatement.

Performance enhancement of snap-through vibration energy harvester with displacement amplifier

Vibration energy harvesters (VEH) based on bistable and snap-through mechanisms are being widely employed to harvest energy from different vibration sources. However, under weak ambient excitation, the performance of these systems is marginal. Multi-stability and enhancement techniques, such as displacement amplifier, are being used with bistable VEH to improve its performance. Similar to this, multi-stable snap-through VEH has also been developed; however, the usage of a displacement amplifier in the context of snap-through VEH has not yet been investigated. In this novel study, a displacement amplifier/ displacement amplification mechanism (DAM) is integrated into a snap-through VEH, and the enhanced performance under weak excitation is investigated. A generalized transformation is applied to develop a reduced-order model of the proposed system, and its dynamics and performance under harmonic and random excitation are investigated analytically and numerically. The frequency–amplitude relationship is derived under harmonic excitation, and a detailed investigation of the influence of system parameters on the frequency and force responses is performed. It is observed that by adjusting the mass and stiffness ratio between the VEH and DAM, it is possible to harvest more energy under harmonic excitation of low intensity. In the case of Gaussian white noise excitation, effective potential, and stochastic averaging methods are used to obtain marginal, joint, and stationary probability density functions and mean square power. The influence of noise intensity and other system parameters on the aforementioned measures is thoroughly investigated. It is observed that for specific parameter values, the proposed VEH can harvest 3.3 times more power under harmonic excitation and four times more power under random excitation compared to the snap-through VEH without DAM.

LoSAC: An Efficient Local Stochastic Average Control Method for Federated Optimization

Federated optimization (FedOpt), which targets at collaboratively training a learning model across a large number of distributed clients, is vital for federated learning. The primary concerns in FedOpt can be attributed to the model divergence and communication efficiency, which significantly affect the performance. In this article, we propose a new method, i.e., LoSAC, to learn from heterogeneous distributed data more efficiently. Its key algorithmic insight is to locally update the estimate for the global full gradient after each regular local model update. Thus, LoSAC can keep clients’ information refreshed in a more compact way. In particular, we have studied the convergence result for LoSAC. Besides, the bonus of LoSAC is the ability to defend the information leakage from the recent technique Deep Leakage Gradients (DLG). Finally, experiments have verified the superiority of LoSAC comparing with state-of-the-art FedOpt algorithms. Specifically, LoSAC significantly improves communication efficiency by more than 100% on average, mitigates the model divergence problem, and equips with the defense ability against DLG.

Stochastic weight averaging enhanced temporal convolution network for EEG-based emotion recognition

Emotion plays an important role in human–computer interactions, in which emotion recognition is the key problem. Electroencephalogram (EEG) has been widely used in emotion recognition due to its high temporal resolution and reliability. In this study, we propose a strategy in which stochastic weight averaging is introduced into an improved temporal convolutional network for emotion recognition. The temporal convolution network not only is suitable for sequence models such as the recurrent neural networks, but also retains the characteristics of parallel computing similar to convolutional neural networks. Considering that the traditional softmax loss does not explicitly encourage discriminative learning of features, we further introduce an improved version of the loss function that explicitly encourages intraclass compactness and interclass separability between learned features. Moreover, the method of stochastic weight averaging is introduced into the network framework to make the network find the point closest to the global optimum by adjusting the learning rate and updating the weight, which can effectively alleviate the local optimum problem. We test the performance of the proposed strategy on two open emotion EEG datasets: DEAP and SEED. The experimental results show that the new strategy has higher recognition accuracy than the state-of-the-art approaches. Moreover, to investigate the general pattern of brain functional connectivity in different individuals, we select the key electrodes and analyze the roles of various brain regions in processing different emotions.

An Alternating Efficient Approach for Determination of the Non-Stationary Responses of Strongly Nonlinear Systems Driven by Random Excitations

Abstract An alternating efficient approach for predicting non-stationary response of randomly excited nonlinear systems is proposed by a combination of radial basis function neural network (RBFNN) and stochastic averaging method (SAM). First, the n-degree-of-freedom quasi-non-integrable-Hamiltonian (QNIH) system is reduced to a one-dimensional averaged Itô differential equation within the framework of SAM for QNIH. Subsequently, the associated Fokker–Planck–Kolmogorov (FPK) equation is solved with the RBFNN. Specifically, the solution of the associated FPK equation is expressed in a linear combination of a series of basis functions with time-correlation weights. These time-depended weights are solved by minimizing a loss function, which involves the residual of the differential equations and the constraint conditions. Three typical nonlinear systems are studied to verify the applicability of the developed scheme. Comparisons to the data generated by simulation technique indicate that the approach yields reliable results with high efficiency.

Forecast analysis and sliding mode control on a stochastic epidemic model with alertness and vaccination

In this paper, a stochastic SEIR epidemic model is studied with alertness and vaccination. The goal is to stabilize the infectious disease system quickly. The dynamic behavior of the model is analyzed and an integral sliding mode controller with distributed compensation is designed. By using Lyapunov function method, the sufficient conditions for the existence and uniqueness of global positive solutions and the existence of ergodic stationary distributions are obtained. The stochastic center manifold and stochastic average method are used to simplify the system into a one-dimensional Markov diffusion process. The stochastic stability and Hopf bifurcation are analyzed using singular boundary theory. An integral sliding mode controller with non-parallel distributed compensation is designed by linear matrix inequality (LMI) method, which realizes the stability of system and prevents the outbreak of epidemic disease. The correction of theoretical analysis and the effectiveness of controller are validated using numerical simulation performed in MATLAB/Simulink.