Multiplicative Noise Research Articles

Approximations in Mean Square Analysis of Stochastically Forced Equilibria for Nonlinear Dynamical Systems

Motivated by important applications to the analysis of complex noise-induced phenomena, we consider a problem of the constructive description of randomly forced equilibria for nonlinear systems with multiplicative noise. Using the apparatus of the first approximation systems, we construct an approximation of mean square deviations that explicitly takes into account the presence of multiplicative noises, depending on the current system state. A spectral criterion of existence and exponential stability of the stationary second moments for the solution of the first approximation system is presented. For mean square deviation, we derive an expansion in powers of the small parameter of noise intensity. Based on this theory, we derive a new, more accurate approximation of mean square deviations in a general nonlinear system with multiplicative noises. This approximation is compared with the widely used approximation based on the stochastic sensitivity technique. The general mathematical results are illustrated with examples of the model of climate dynamics and the van der Pol oscillator with hard excitement.

Resonance behaviour for a bistable system driven by random-phase square-wave signal-modulated noise and multiplicative noise

Resonance behaviour for a bistable system driven by random-phase square-wave signal-modulated noise and multiplicative noise

On the local linearization of the one-dimensional stochastic wave equation with a multiplicative space-time white noise forcing

On the local linearization of the one-dimensional stochastic wave equation with a multiplicative space-time white noise forcing

Highly dispersive eighth-order embedded solitons with cubic–quartic χ(2) and χ(3) nonlinear susceptibilities under the influence of multiplicative white noise using Itô calculus

This research explores the intriguing realm of eighth-order embedded solitons in highly dispersive media with cubic–quartic nonlinear susceptibilities [Formula: see text] and [Formula: see text], all within the dynamic context of multiplicative white noise and the framework of Itô calculus. Two different approaches are used in this study. The new auxiliary equation approach produces the bright soliton and singular soliton solutions, while the addendum Kudryashov’s approach produces the bright soliton, singular soliton and combo bright–singular soliton solutions. The system under investigation and the results documented within this work stand as pioneering and original contributions to the field of nonlinear optics. Furthermore, a collection of 2D, 3D and contour plots is produced to visually represent the spatial distribution and progression of different solutions. This not only contributes to the advancement of nonlinear equations in theory but also offers valuable insights for practical applications.

Physics-guided deep learning-based inversion for airborne electromagnetic data

SUMMARY The Earth's subsurface structure provides critical insights into sustainable resource management and geologic evolution. The airborne electromagnetic (AEM) method is an efficient data acquisition technique and can be used to image the underground resistivity structure with high spatial resolution. However, inversion of the increasingly huge volume of AEM data poses a heavy computational burden. In this study, we develop a hybrid deep learning-based approach by using the physics-guided neural network (PGNN) which incorporates the governing physical laws into the loss function to solve the AEM inverse problem. The PGNN integrates the strength of data-driven method for representation learning with electromagnetic laws and allows for the underlying physical constraints to be strictly satisfied. We validate the effectiveness of our approach using both synthetic and field datasets. Compared with the classic Gauss–Newton method, our PGNN inversion system shows strong robustness against multiple noise sources and reduces the risk of being trapped in local extrema. Moreover, the PGNN-inverted results are physically more consistent with the AEM observations compared to the purely data-driven approach. Application to the field AEM data from Northern Australia demonstrates that the PGNN-based inversion framework effectively estimates the subsurface electrical properties with considerable lateral continuity and significantly higher efficiency, completing the inversion of more than 2734000 AEM soundings taking only minutes on a common PC. Our proposed PGNN-based method shows great promise for large-scale underground resistivity imaging, and the well-identified subsurface resistivity structure can effectively improve our understanding of resource distributions and geological hazards.

A conservative stochastic Dirac-Klein-Gordon system

Considered herein is a particular nonlinear dispersive stochastic system consisting of Dirac and Klein-Gordon equations. They are coupled by nonlinear terms due to the Yukawa interaction. We consider a case of homogeneous multiplicative noise that seems to be very natural from the perspective of the least action formalism. We are able to show existence and uniqueness of a corresponding Cauchy problem in Bourgain spaces. Moreover, the regarded model implies charge conservation, known for the deterministic analogue of the system, and this is used to prove a global existence result for suitable initial data.

Convergence uniform on compacts in probability with applications to stochastic analysis in duals of nuclear spaces

Let Φ′ denotes the strong dual of a nuclear space Φ. In this article, we introduce sufficient conditions for the convergence uniform on compacts in probability for a sequence of Φ′-valued processes with continuous or càdlàg paths. We illustrate the usefulness of our results by considering two applications to stochastic analysis. First, we introduce a topology on the space of Φ′-valued semimartingales which are good integrators and show that this topology is complete and that the stochastic integral mapping is continuous on the integrators. Second, we introduce sufficient conditions for the convergence uniform on compacts in probability of the solutions to a sequence of stochastic evolution equations; we do this both in the case of linear equations with semimartingale noise and nonlinear equations with multiplicative cylindrical martingale-valued measure noise.

Stochastic nonlinear model for nanoparticles and viruses escape from endosomes

AbstractA stochastic nonlinear model for Rab5 and Rab7 proteins describing the transformation of early endosomes into late endosomes was formulated. This model consists of two stochastic nonlinear differential equations for Rab5 and Rab7 protein levels on the endosome surface. The primary goal of this paper is to understand the impact of multiplicative noise on the nonlinear dynamics of Rab5 and Rab7. The main idea is to introduce the stochastic variable T, which defines the random time when the conversion from Rab5 to Rab7 occurs. It follows from the dynamics of pH level that T can also be considered as the escape time of pH-sensitive nanoparticles and viruses from endosomes. The probability density function for T was obtained numerically. It was shown that the average conversion time T is shifted to the right when compared to the deterministic one, potentially influencing the pH distribution function and, consequently, the average escape time of viruses and nanoparticles.

An efficient method for solving high-dimension stationary FPK equation of strongly nonlinear systems under additive and/or multiplicative white noise

Engineering structures may suffer from drastic nonlinear random vibrations in harsh environments. Random vibration has been extensively studied since 1960s, but is still an open problem for large-scale strongly nonlinear systems. In this paper, a random vibration analysis method based on the Neural Networks for large-scale strongly nonlinear systems under additive and/or multiplicative Gaussian white noise (GWN) excitations is proposed. In the proposed method, the high-dimensional steady-state Fokker–Planck-Kolmogorov (FPK) equation governing the state’s probability density function (PDF) is firstly reduced to low-dimensional FPK equation involving only the interested state variables, generally one or two dimensions. The equivalent drift coefficients (EDCs) and diffusion coefficients (EDFs) in the low-dimensional FPK equation are proven to be the conditional mean of the coefficients given the interested variables. Furthermore, it is shown that the conditional mean can be optimally estimated by regression. Subsequently, the EDCs and EDFs, as functions of the retained variables, are approximated by the semi-analytical Radial Basis Functions Neural Networks trained with samples generated by a few deterministic analyses. Finally, the Physics Informed Neural Network is employed to solve the reduced steady-state FPK equation, and the PDF of the system responses is obtained. Four typical examples under additive and/or multiplicative GWN excitations are used to examine the accuracy and efficiency of the proposed method by comparing its results with the exact solution (if available) or Monte Carlo simulations. The proposed method also exhibits greater accuracy than the globally-evolving-based generalized density evolution equation scheme, a similar method of its kind, especially for strongly nonlinear systems. Notably, even though steady-state systems are applied in this paper, there is no obstacle to extending the proposed framework to transient systems.

Adaptive Fault-Tolerant Tracking Control for Discrete-Time Nonstrict-Feedback Nonlinear Systems With Stochastic Noises

This paper discusses the problem of fault-tolerant tracking control for discrete-time nonstrict-feedback nonlinear systems in the presence of stochastic noises and actuator faults. The system is characterized by a discrete-time nonstrict-feedback structure and multiplicative stochastic noise related to the states, which poses a challenge for the design and analysis of the fault-tolerant controller. Moreover, the considered actuator faults include multiplicative and additive faults without prior information. By integrating the properties of the backstepping framework and neural networks, and by introducing an adaptive fault compensation term, a novel fault-tolerant tracking control strategy is proposed. The effects of faults are compensated and the difficulties caused by the system structure are overcome, avoiding the algebraic loop problem and overcoming the causal contradiction. Given specific parameters, the designed fault-tolerant controller can ensure that the tracking error converges to an adjustable region regarding the origin and that all system signals are uniformly bounded concerning the mean-square sense. The simulation examples illustrate the effectiveness of the designed fault-tolerant control method. <italic xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">Note to Practitioners</i> —As system complexity has been increasing, practical systems are often affected by faults when performing tracking control. Fault-tolerant control can provide acceptable robustness and improve system safety. On the one hand, many practical systems are nonlinear, difficult to model mathematically and accurately, and affected by stochastic noises. On the other hand, with digital control, the object is a discrete-time system. In practical applications, the upper bounds of the faults and the noises are often unknown. In this paper, a novel adaptive fault-tolerant control method is proposed for the more general uncertain discrete-time nonlinear systems, i.e., nonstrict-feedback nonlinear systems, with stochastic noises and actuator faults. Neural networks and an adaptive fault compensation term are integrated into the fault-tolerant controller. By adjusting the controller parameters properly, it is ensured that the tracking error converges to a small neighborhood of zero in the event of a fault occurring. In future research, sensor faults, input constraints, optimal control design, and applications in multi-agent systems will be considered.

Analysis of a positivity-preserving splitting scheme for some semilinear stochastic heat equations

We construct a positivity-preserving Lie–Trotter splitting scheme with finite difference discretization in space for approximating the solutions to a class of semilinear stochastic heat equations with multiplicative space-time white noise. We prove that this explicit numerical scheme converges in the mean-square sense, with rate 1/4 in time and rate 1/2 in space, under appropriate CFL conditions. Numerical experiments illustrate the superiority of the proposed numerical scheme compared with standard numerical methods which do not preserve positivity.

Pathwise synchronization of global coupled system with linear multiplicative rough noise.

This paper focuses on achieving pathwise synchronization in stochastic differential equations with linear multiplicative rough noises, which are fractional Brownian rough paths with Hurst parameter H∈(13,12). Using rough paths theory, a useful transformation is introduced to convert the equations into random differential equations. Stability and dynamical behavior of the solutions to the equations are discussed, and pathwise synchronization of the solutions to the coupled system is proven. Also we have verified the synchronization results in Hölder space. And at the end, two alternative forms of noises are considered, and synchronization results are presented. Moreover, numerical simulations are provided to illustrate the results.

Synchronization transition in space–time chaos in the presence of quenched disorder

Synchronization of two replicas of coupled map lattices for continuous maps is known to be in the multiplicative noise universality class. We study this transition in the presence of quenched disorder in coupling. The disorder is identical in both replicas. We study one-dimensional, two-dimensional, and globally coupled logistic and tent maps. We observe a clear second-order transition with new exponents. The order parameter decays as t−δ and δ depends on the map and its parameters. The asymptotic order parameter for Δ distance from a critical point grows as Δβ with β∼δ. The quenched disorder in coupling is a relevant perturbation for the replica synchronization of coupled map lattices. The critical exponents are different from those of the multiplicative noise universality class. However, it does not depend on dimensionality in the cases studied if the transition is continuous.

Fast bicriteria streaming algorithms for submodular cover problem under noise models

The Submodular Cover (SC) problem has attracted the attention of researchers because of its wide variety of applications in many domains. Previous studies on this problem have focused on solving it under the assumption of a non-noise environment or using the greedy algorithm to solve it under noise. However, in some applications, the data is often large-scale and brings a noisy version, so the existing solutions are ineffective or not applicable to large and noisy data. Motivated by this phenomenon, we study the Submodular Cover under Noises (SCN) problem and propose two efficient streaming algorithms, which provide a solution with theoretical bounds under two common noise models, multiplicative and additive noises. The experimental results indicate that our proposed algorithms not only provide the solution with a high objective function value but also outperform the state-of-the-art algorithm in terms of both the number of queries and the running time.

Stochastic modulation of the Montgomery’s rings to generate the self-imaging effect with revival features

The synthesis of optical fields that follows a stochastic process and whose statistical mean values generate the self-imaging-revivals effect is analyzed. A sufficient condition for optical fields to exhibit the self-imaging effect occurs when the frequency representation is located on the Montgomery’s rings. The study is performed by implementing stationary random fluctuations on the Montgomery’s rings of additive and multiplicative types. The additive noise is of the stochastic radial walk type with zero mean. Multiplicative noise generates stochastic angular fluctuations in the rings and is implemented using the Karhunen–Loève theorem. The modal representation implicit in the theorem is obtained by interpreting the self-imaging planes as an optical cavity, assuring the statistical periodicity of the noise. A time consonance of the random fluctuations for each ring allows to determine the revivals period for the self-imaging optical fields. The model is corroborated by computer simulations.

Martingale solutions to stochastic nonlocal Cahn–Hilliard- Navier–Stokes systems with singular potentials driven by multiplicative noise of jump type

Abstract In the diffuse interface theory, the motion of two incompressible viscous fluids and the evolution of their interface are described by the well-known model H. The model consists of the Navier–Stokes equations, nonlinearly coupled with a convective Cahn–Hilliard type equation. Here we consider a stochastic version of the model driven by a noise of Lévy type, and where the standard Cahn–Hilliard equation is replaced by its nonlocal version with a singular (e.g., logarithmic) potential. The case of smooth potentials with arbitrary polynomial growth has been already analyzed in [G. Deugoué, A. Ndongmo Ngana and T. Tachim Medjo, Martingale solutions to stochastic nonlocal Cahn–Hilliard–Navier–Stokes equations with multiplicative noise of jump type, Phys. D 398 2019, 23–68]. Taking advantage of this previous result, we investigate this more challenging and physically relevant case. Global existence of a martingale solution is proved with no-slip and no-flux boundary conditions in both 2D and 3D bounded domains. In the two-dimensional case, we prove the uniqueness of weak solutions when the viscosity is constant.

Direct Numerical Solutions to Stochastic Differential Equations with Multiplicative Noise.

Inspired by path integral solutions to the quantum relaxation problem, we develop a numerical method to solve classical stochastic differential equations with multiplicative noise that avoids averaging over trajectories. To test the method, we simulate the dynamics of a classical oscillator multiplicatively coupled to non-Markovian noise. When accelerated using tensor factorization techniques, it accurately estimates the transition into the bifurcation regime of the oscillator and outperforms trajectory-averaging simulations with a computational cost that is orders of magnitude lower.

Adaptive Bayesian speckle reduction in the 2D DOST domain using 2D MC-GARCH-M model: Preserving image edges and textural information

Image denoising and image quality enhancement are major issues in image processing. Additive and multiplicative noise removal is one of the main image enhancement approaches. In this paper, we propose a novel 2D Merged Complex Generalized Autoregressive Conditional Heteroscedasticity Mixture (MC-GARCH-M) model for 2D complex stochastic processes. This model effectively captures non-Gaussian statistics and all three types of dependencies within the processes (RRD, IID, and RID) using the proposed linear mapping transformation T. Our statistical analysis shows that the 2D MC-GARCH-M model provides the best fit to the non-Gaussian statistics of the mapped matrix of 2D DOST coefficients compared to Gaussian, Generalized Gaussian, and Stable distributions. We confirm the presence of heteroscedasticity in the mapped matrix using the Engle hypothesis test, verifying ARCH/GARCH effects. The 2D MC-GARCH-M model, with its location-dependent conditional variances, provides a flexible approach for noise removal and modeling complex stochastic processes. Experimental results on artificially speckled aerial and actual SAR images demonstrate that our method outperforms state-of-the-art approaches in terms of speckle removal and preserving the edges and textural information of the image. Key innovations include: a new statistical analysis using linear mapping T showing non-Gaussian heavy-tailed distributions; an adaptive Bayesian estimator, MCMAP, based on the heteroscedastic model for speckle noise removal; and the method's applicability to images with varying mutual information (MI) between real and imaginary parts of 2D DOST coefficients. Our method is adaptive and robust to initial parameter settings, effectively utilizing magnitude and phase information, providing an optimal closed-form solution, reducing memory and computational requirements, and demonstrating robustness to speckle power levels. While requiring more computational resources compared to thresholding methods, our approach is less demanding than other Bayesian algorithms, making it suitable for offline processing.

Global well-posedness of the 2D nonlinear Schrödinger equation with multiplicative spatial white noise on the full space

We consider the nonlinear Schrödinger equation with multiplicative spatial white noise and an arbitrary polynomial nonlinearity on the two-dimensional full space domain. We prove global well-posedness by using a gauge-transform introduced by Hairer and Labbé (Electron Commun Probab 20(43):11, 2015) and constructing the solution as a limit of solutions to a family of approximating equations. This paper extends a previous result by Debussche and Martin (Nonlinearity 32(4):1147–1174, 2019) with a sub-quadratic nonlinearity.

Optical dromions with polarization–mode dispersion having parabolic law of self–phase modulation with multiplicative white noise

Optical dromions with polarization–mode dispersion having parabolic law of self–phase modulation with multiplicative white noise