Presence Of Multiplicative Noise Research Articles

Weak error analysis for the stochastic Allen–Cahn equation

AbstractWe prove strong rate resp. weak rate $${{\mathcal {O}}}(\tau )$$ O ( τ ) for a structure preserving temporal discretization (with $$\tau $$ τ the step size) of the stochastic Allen–Cahn equation with additive resp. multiplicative colored noise in $$d=1,2,3$$ d = 1 , 2 , 3 dimensions. Direct variational arguments exploit the one-sided Lipschitz property of the cubic nonlinearity in the first setting to settle first order strong rate. It is the same property which allows for uniform bounds for the derivatives of the solution of the related Kolmogorov equation, and then leads to weak rate $${{\mathcal {O}}}(\tau )$$ O ( τ ) in the presence of multiplicative noise. Hence, we obtain twice the rate of convergence known for the strong error in the presence of multiplicative noise.

The Delayed Effect of Multiplicative Noise on the Blow-Up for a Class of Fractional Stochastic Differential Equations

We investigated the blow-up of the weak solution to a class of fractional nonlinear stochastic differential equations driven by multiplicative noise in this paper. The a priori estimates and Galerkin method were applied to demonstrate the existence and uniqueness of the weak solution. Underlying the hypotheses of the nonlinear function and the initial data, for finite time, we prove that the solution does not blow up. Additionally, under further assumptions, we verified that the presence of multiplicative noise can delay the blow-up of the solution to infinity.

Group sparse representation and saturation-value total variation based color image denoising under multiplicative noise

<abstract><p>In this article, we propose a novel group-based sparse representation (GSR) model for restoring color images in the presence of multiplicative noise. This model consists of a convex data-fidelity term, and two regularizations including GSR and saturation-value-based total variation (SVTV). The data-fidelity term is suitable for handling heavy multiplicative noise. GSR enables the retention of textures and details while sufficiently removing noise in smooth regions without producing the staircase artifacts engendered by total variation-based models. Furthermore, we introduce a multi-color channel-based GSR that involves coupling between three color channels. This avoids the generation of color artifacts caused by decoupled color channel-based methods. SVTV further improves the visual quality of restored images by diminishing certain artifacts induced by patch-based methods. To solve the proposed nonconvex model and its subproblem, we exploit the alternating direction method of multipliers, which contributes to an efficient iterative algorithm. Numerical results demonstrate the outstanding performance of the proposed model compared to other existing models regarding visual aspect and image quality evaluation values.</p></abstract>

Nonlinear optical properties of doped GaAs quantum dot modulated by noise: Role of impurity spread

This paper explores the role of spatial extension of impurity (SEI) on two nonlinear optical (NLO) properties and two other related properties of impurity doped GaAs quantum dot (QD). The NLO properties include the electro-optic effect (EOE) and the third-order nonlinear optical susceptibility (TONOS) whereas the other two related properties being total optical dielectric function (TODF) and the intraband transition lifetime (ITL). The study also involves Gaussian white noise (GWN) which has been introduced to the doped QD through additive and multiplicative pathways. The investigation unveils the subtle interplay between GWN and SEI which ultimately fine-tunes the said NLO properties. TODF, EOE and TONOS manifest red-shift with enhancement of SEI both with and without GWN. The ITL, on the other hand, exhibits monotonic growth with increase in SEI under all conditions. Moreover, the application of GWN causes noticeable quenching of the NLO properties. And the quenching becomes more stringent in presence of multiplicative noise than its additive counterpart.

Robust filtering for spacecraft attitude estimation systems with multiplicative noises, unknown measurement disturbances and correlated noises

Unknown measurement disturbances and correlated noise caused by jitter or vibration phenomena during spacecraft operation and the complex external environment are important topics for spacecraft attitude estimation. This paper studies the problem of spacecraft attitude estimation for nonlinear systems with multiplicative noises, unknown measurement disturbances and correlated noises. In addition, a two-step prediction framework is used to complete the noise decoupling. This paper aims to design a robust filter that minimizes the upper bound of the prediction error covariance and estimation error covariance in the presence of multiplicative noises, correlated noises and unknown measurement disturbances. The prediction gain and filtering gain of the robust filter are designed to minimize the upper bound. Finally, the simulation results verify the effectiveness of the proposed filter.

Cooperative Coevolutionary CMA-ES With Landscape-Aware Grouping in Noisy Environments

Many real-world optimization tasks suffer from noise. So far, the research on noise-tolerant optimization algorithms is still restricted to low-dimensional problems with less than 100 decision variables. In reality, many problems are high-dimensional. Cooperative coevolutionary (CC) algorithms based on a divide-and-conquer strategy are promising in solving complex high-dimensional problems. However, noisy fitness evaluations pose a challenge in problem decomposition for CC. The state-of-the-art grouping methods, such as differential grouping and recursive differential grouping, are unable to work properly in noisy environments. Because it is impossible to distinguish whether the change of one variable’s difference value is caused by noise or the perturbation of its interacting variables. As a result, every pair of variables will be identified as nonseparable in these methods. In this paper, we study how to group decision variables with covariance matrix adaptation evolution strategy (CMA-ES) in noisy environments and subsequently propose a landscape-aware grouping (LAG) method. Instead of detecting pairwise interacting variables, we directly identify a nonseparable subcomponent. To this end, we propose to use two convergence features, variable convergence time and accumulative path, to describe variables’ fitness landscapes; then, variables are clustered according to these two features. Numerical experiments show that LAG can more effectively identify interactive decision variables in the presence of multiplicative noise than the differential grouping and some of its variants. Up to 500 dimensions, the performance of cooperative coevolutionary CMA-ES with landscape-aware grouping (CC-CMAES-LAG) is competitive compared with existing CC algorithms and uncertainty-handling CMA-ES (UH-CMA-ES).

Embedded solitons with [formula omitted] and [formula omitted] nonlinear susceptibilities having multiplicative white noise via Itô Calculus

The objective of this paper is to locate the embedded solitons with χ(2) and χ(3) nonlinear susceptibilities in presence of multiplicative noise. Itô Calculus was implemented to carry out the analysis of the corresponding stochastic differential equation. The unified Riccati equation expansion method and enhanced Kudryashov’s approach yielded dark, bright and singular solitons. The derived soliton solutions show that the effect of white noise is present only in the phase component.

The influence of the noise on the exact solutions of a Kuramoto-Sivashinsky equation

Abstract In this article, we take into account the stochastic Kuramoto-Sivashinsky equation forced by multiplicative noise in the Itô sense. To obtain the exact stochastic solutions of the stochastic Kuramoto-Sivashinsky equation, we apply the G ′ G \frac{{G}^{^{\prime} }}{G} -expansion method. Furthermore, we extend some previous results where this equation has not been previously studied in the presence of multiplicative noise. Also, we show the influence of multiplicative noise on the analytical solutions of the stochastic Kuramoto-Sivashinsky equation.

A proposal of edge detection in images with multiplicative noise using the Ant Colony System algorithm

Multiplicative noise is one of the most aggressive types of noise present in various types of images: Synthetic Aperture Radar, Ultrasound images, Ultrasonic Imaging, among others. Edges detectors such as Canny or Sobel are not very efficient for processing an image with multiplicative noise, they also require filtering algorithms, a preprocessing, which is why bio-inspired algorithms are an alternative for processing images with the presence of multiplicative noise, due to its efficiency in finding an approximate solution. This article proposes a method for the edges detection in images with multiplicative noise using the Ant Colony System algorithm. For which we must adapt the Ant Colony System algorithm to detect contours, this we define the calculation of a global pheromone matrix between several edge detection equations, gradient, and the coefficient of variation, these equations are compared for their edge detection performance using a visual inspection and a performance function. The results of the experiments show a correct implementation of the algorithm proposed to the images with multiplicative noise even for high noise levels.

Construction of Adaptive Consensus Gains for Multiagent Systems with Multiplicative Noise

The leader-following consensus of linear multiagent systems with multiplicative noise and adaptive gains is investigated under the fixed directed graph. Because of multiplicative noise, the agents interacting with the leader can only obtain the inaccurate information. Firstly, this paper designs the adaptive gains of multiagent systems, which can converge to some fixed values and force the states to synchronize. Different from the mandatory gains, the adaptive gains depend on the states of agents and can be adjusted with the change of states. Secondly, it is shown that the multiagent systems without time delay can obtain mean square consensus under the adaptive gains. Moreover, in the presence of multiplicative noise and communication delay, the mean square consensus is also acquired in the same adaptive gains. Finally, the validity of the conclusion is simulated by an example.

Stochastic models with multiplicative noise for economic inequality and mobility

Abstract In this article, we discuss a dynamical stochastic model that represents the time evolution of income distribution of a population, where the dynamics develops from an interplay of multiple economic exchanges in the presence of multiplicative noise. The model remit stretches beyond the conventional framework of a Langevin-type kinetic equation in that our model dynamics is self-consistently constrained by dynamical conservation laws emerging from population and wealth conservation. This model is numerically solved and analysed to evaluate the inequality of income in correlation to other relevant dynamical parameters like the mobility M and the total income μ. Inequality is quantified by the Gini index G. In particular, correlations between any two of the mobility index M and/or the total income μ with the Gini index G are investigated and compared with the analogous quantities resulting from an additive noise model.

Анализ влияния быстрых и медленных мультипликативных помех на искажения диаграммы направленности системы "решетка-приемник".

The effect of fast and slow multiplicative noise on the distortion of the beam pattern of the «array-receiver» system in the conditions of the smoothing effect of the receiver and taking into account the time of relative lag from the grating elements was analyzed. Expressions are obtained for instantaneous beam patterns of the «array-receiver» system in the absence and presence of multiplicative noise. It is shown that in the case of action of multiplicative noise the beam pattern of the «array-receiver» system is obtained by averaging the instantaneous pattern on the grating correctness at a time interval equal to the signal duration. Influence of periodic multiplicative noise on distortion of beam pattern of system «array-receiver» is investigted. It is shown that for all signals except frequency-modulated, the beam pattern expression is maximized at the time t0=0. It is also shown that the beam pattern distortion of the «array-receiver» system when exposed to periodic multiplicative noise is less than for the separately considered phased antenna array. The degree of reduction of distortion from smoothing action of matched receiver filter is determined; and the distortion is less, the greater the ratio of the spectrum width of the noise modulation function to the signal spectrum width. Analysis of the influence of fluctuation multiplicative noise on the distortion of beam pattern of «array-receiver» system for the case of matching with signal of receiver filter is performed. The power-average beam pattern of the «array-receiver» system is determined through the energy spectrum of the noise modulation function. It is noted that multiplicative noise, all other things being equal, has less effect on the beam-receiver pattern of the system when using broadband pulse signals without intra-pulse modulation and at the same time resolution in range. It is noted that in case of deep phase distortions for gratings with reversible phases, the distortion of the beam pattern of the «array-receiver» system caused by multiplicative noise is significantly more than those distortions associated with the extremity of the signal spectrum width. The newly smoothing effect of the receiver reduces the expansion of the beam lobe caused by multiplicative noise. An expression is obtained for the average power of the beam pattern of the «array-receiver» system with normally distributed phase distortions of the signal. The effect of the signal duration on beam distortions caused by multiplicative noise was quantified.

Искажения диаграмм направленности фазированных антенных решеток под влиянием периодических и флуктуационных помех.

Issues related to determination of distortions of directional patterns of phased antenna arrays under the influence of periodic and fluctuating multiplicative (modulating) noise are in the focus of the paper. Expressions are obtained for instantaneous directional patterns in terms of voltage and power, as well as the average value of the directional pattern in terms of voltage under the influence of periodic multiplicative noise. So far, since the phase ratios in these instantaneous diagrams change over time, the pattern of the grating fluctuates. The condition is defined under which the structure of the grating pattern «crumbles». At the same time, lower frequencies of acting multiplicative noise lead to distortion of the diagram: its main maximum expands, the level of side lobes increases and lateral reception directions are «smoothed». It is noted that in synthesized arrays, «scattering» of the directional pattern can occur at relatively small values​ of the frequency of multiplicative noise. In lattices with delay lines, periodic multiplicative noise does not cause significant distortion of the directional patterns. Expressions are obtained for determining average directional patterns by voltage and power under the influence of fluctuating multipath noise, including those obtained using correlation function of noise modulation. It has been shown that under the influence of stationary multiplicative noise, the beam pattern on the power of the grating with phase shifters is a convolution of the undistorted diagram on an undistorted scale and the energy spectrum of the noise modulation function. The upper limit of increasing the width of the main lobe of the beam pattern caused by multiplicative noise is determined. It is also shown that as the noise correlation time decreases and the phase distortion depth increases, the diagram expands and its side lobes smooth. It is noted that for distortions of the directional pattern of synthesized arrays, all the basic provisions for phased antenna arrays with phase shifters are true. For lattices with delay lines in the presence of multiplicative noise, the maximum of the average power of the beam pattern does not shift regardless of the shape of the energy spectrum of the noise modulation function. Multiplicative noise can cause only some expansion of the main lobe of the diagram and smoothing of the zero reception directions. Distortions of the grating pattern when exposed to slow fluctuation multiplicative noise are shown. These distortions are determined by changes in the phase and amplitude of the signal over time corresponding to the time shift between the signals in the individual elements of the array, and with slow multiplicative noise, the distortions are small. It is also shown that slow multiplicative noise leads to some reduction in the maximum beam pattern and to smoothing the pattern in the area of the side lobes. In the presence of such noise, the zero reception directions disappear.

On the dynamics and robustness of the chemostat with multiplicative noise

Abstract The stochastic dynamics of a two-state bioreactor model with random feed flow fluctuations and non-monotonic specific growth rate is analyzed. Using the Fokker-Planck equation approach for describing the probability density function (PDF) evolution the lack of stochastic robustness due to deterministic bifurcation phenomena for the open-loop reactor operating under optimal (maximum production) operation condition is established, and the associated stochastic stabilization problem is addressed. Inherent differences between the presence of multiplicative noise, due to the feed flow fluctuations, and additive background noise are analytically established. Numerical simulation results illustrate these inherent differences, the stochastic fragility of the open-loop operation yielding a stochastic extinction phenomenon, as well as the stochastic PDF stabilization with a proportional feedback control.

Estimation for power quality disturbances with multiplicative noises and correlated noises: a recursive estimation approach

ABSTRACTIn this paper, the recursive estimation problem is investigated for the power quality disturbances. A system model for the power quality disturbances with multiplicative noises and correlated noises is proposed based on engineering practice. The multiplicative noises are considered to account for the stochastic disturbances on the system states. The process noise and the measurement noise are assumed to be one-step autocorrelated. The process noise and the measurement noise are two-step cross-correlated. Attention is focused on the design of an unbiased and recursive estimation algorithm in the presence of multiplicative noises and correlated noises. First, the state covariance, one-step prediction error covariance and estimation error covariance are derived. Then, the estimation error covariance is minimised by appropriately designing the estimator gain. Finally, simulation experiments under three scenarios are carried out to illustrate the effectiveness of the proposed estimation scheme.

Profiles of Static Quadrupole Polarizability of Impurity‐Doped Quantum Dots Driven by Gaussian White Noise

Herein, the influence of Gaussian white noise on static quadrupole polarizability (SQP) of the impurity‐doped quantum dots (QDs) is studied. The SQP profiles are critically analyzed as several physical quantities are varied over a range in the absence and presence of noise. The introduction of noise to the system is accomplished via two different routes viz. “additive” and “multiplicative.” SQP profiles consist of noticeable features like maximization, minimization, and saturation in the absence of noise and presence of additive noise. However, on most occasions, the SQP profile becomes devoid of any noticeable characteristics in the presence of multiplicative noise. The multiplicative noise plays some noticeable role only during the direct variation in confinement energy. Herein, the exclusive role of additive noise having strength around a typical value for the generation of large SQP is highlighted. The findings bear mentionable importance on the nonlinear optical properties of QD‐based devices under the aegis of noise.

How Disorder Originates and Grows Inside Order

We present a study of disorder origination and growth inside an ordered phase processes induced by the presence of multiplicative noise within mean-field approximation. Our research is based on the study of solutions of the nonlinear self-consistent Fokker-Planck equation for a stochastic spatially extended model of a chemical reaction. We carried out numerical simulation of the probability distribution density dynamics and statistical characteristics of the system under study for varying noise intensity values and system parameter values corresponding to a spatially inhomogeneous ordered state in a deterministic case. Physical interpretation of the results obtained that determines the scenario of noise-induced order-disorder transition is given. Mean-field results are compared with numerical simulations of the evolution of the model under study. We find that beginning from some value of external noise intensity the "embryo" of disorder appears inside the ordered phase. Its lifetime is finite, and it increases with growth of noise intensity. At some second noise intensity value the ordered and disordered phases begin to alternate repeatedly and almost periodically. The frequency of intermittency grows with the increasing of noise intensity. Ordered and disordered phase intermittency affects the process of spatial pattern formation as a consequent change of spatial inhomogeneity configurations.

Modulating electro-absorption coefficient of impurity doped quantum dots driven by noise

Present study carries out an extensive inspection of the profiles of electro-absorption coefficient (EAC) of doped GaAs quantum dot (QD) under the control of Gaussian white noise. A large number of important physical parameters has been varied over a range and the consequent changes in the EAC profiles have been thoroughly scrutinized. The said physical parameters comprise of electric field, magnetic field, confinement potential, dopant location, dopant potential, noise strength, aluminium concentration (only for AlxGa1−xAs alloy QD), carrier density, relaxation time, position-dependent effective mass (PDEM), position-dependent dielectric screening function (PDDSF), anisotropy, hydrostatic pressure (HP) and temperature. The particular physical quantity being varied, presence of noise and its pathway of application, together, lead to the appearance of diverse features in the EAC profiles. As a technologically meaningful aspect we often find maximization of EAC for some typical combinations as mentioned above. Presence of multiplicative noise, quite often, causes greater departure and greater suppression of EAC profiles from a noise-free condition than its additive counterpart. The outcomes of the study indicate prolific scope of harnessing EAC of doped QD systems in presence of noise by delicate adjustment of several control parameters.

Exploring DC-Kerr effect of impurity doped quantum dots under the aegis of noise

Present study performs an extensive exploration of the profiles of DC-Kerr effect (DCKE) of doped GaAsquantum dot (QD) under the control of Gaussian white noise. A large number of important physical parameters have been varied over a range and the resultant changes in the DCKE profiles have been thoroughly analyzed. The said physical parameters comprise of electric field, magnetic field, confinement potential, dopant location, dopant potential, noise strength, aluminium concentration (only for AlxGa1−xAs alloy QD), carrier density, relaxation time, position-dependent effective mass (PDEM), position-dependent dielectric screening function (PDDSF), anisotropy, hydrostatic pressure (HP) and temperature. The particular physical quantity being varied, presence of noise and its pathway of application, in combination, lead to emergence of diverse features in the DCKE profiles. As a technologically significant aspect we often find maximization of DCKE for some typical combinations as mentioned above. Presence of multiplicative noise, in general, causes greater shift and greater augmentation of DCKE profiles from a noise-free condition than its additive counterpart. The outcomes of the study indicate ample scope of tailoring DCKE of doped QD systems in presence of noise by minute adjustment of several control parameters.

Phase transitions in distributed control systems with multiplicative noise

Contemporary technological challenges often involve many degrees of freedom in a distributed or networked setting. Three aspects are notable: the variables are usually associated with the nodes of a graph with limited communication resources, hindering centralized control; the communication is subject to noise; and the number of variables can be very large. These three aspects make tools and techniques from statistical physics particularly suitable for the performance analysis of such networked systems in the limit of many variables (analogous to the thermodynamic limit in statistical physics). Perhaps not surprisingly, phase-transition like phenomena appear in these systems, where a sharp change in performance can be observed with a smooth parameter variation, with the change becoming discontinuous or singular in the limit of infinite system size.In this paper, we analyze the so called network consensus problem, prototypical of the above considerations, that has previously been analyzed mostly in the context of additive noise. We show that qualitatively new phase-transition like phenomena appear for this problem in the presence of multiplicative noise. Depending on dimensions, and on the presence or absence of a conservation law, the system performance shows a discontinuous change at a threshold value of the multiplicative noise strength. In the absence of the conservation law, and for graph spectral dimension less than two, the multiplicative noise threshold (the stability margin of the control problem) is zero. This is reminiscent of the absence of robust controllers for certain classes of centralized control problems. Although our study involves a ‘toy’ model, we believe that the qualitative features are generic, with implications for the robust stability of distributed control systems, as well as the effect of roundoff errors and communication noise on distributed algorithms.