Multiplicative Gaussian Noise Research Articles

Temporal approximation of stochastic evolution equations with irregular nonlinearities

In this paper, we prove convergence for contractive time discretisation schemes for semi-linear stochastic evolution equations with irregular Lipschitz nonlinearities, initial values, and additive or multiplicative Gaussian noise on 2-smooth Banach spaces X. The leading operator A is assumed to generate a strongly continuous semigroup S on X, and the focus is on non-parabolic problems. The main result concerns convergence of the uniform strong errorEk∞:=(Esupj∈{0,…,Nk}‖U(tj)-Uj‖Xp)1/p→0(k→0),\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\begin{aligned} \ extrm{E}_{k}^{\\infty } {:}{=}\\Big (\\mathbb {E}\\sup _{j\\in \\{0, \\ldots , N_k\\}} \\Vert U(t_j) - U^j\\Vert _X^p\\Big )^{1/p} \\rightarrow 0\\quad (k \\rightarrow 0), \\end{aligned}$$\\end{document}where p∈[2,∞)\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$p \\in [2,\\infty )$$\\end{document}, U is the mild solution, Uj\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$U^j$$\\end{document} is obtained from a time discretisation scheme, k is the step size, and Nk=T/k\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$N_k = T/k$$\\end{document} for final time T>0\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$T>0$$\\end{document}. This generalises previous results to a larger class of admissible nonlinearities and noise, as well as rough initial data from the Hilbert space case to more general spaces. We present a proof based on a regularisation argument. Within this scope, we extend previous quantified convergence results for more regular nonlinearity and noise from Hilbert to 2-smooth Banach spaces. The uniform strong error cannot be estimated in terms of the simpler pointwise strong errorEk:=(supj∈{0,…,Nk}E‖U(tj)-Uj‖Xp)1/p,\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\begin{aligned} \ extrm{E}_k {:}{=}\\bigg (\\sup _{j\\in \\{0,\\ldots ,N_k\\}}\\mathbb {E}\\Vert U(t_j) - U^{j}\\Vert _X^p\\bigg )^{1/p}, \\end{aligned}$$\\end{document}which most of the existing literature is concerned with. Our results are illustrated for a variant of the Schrödinger equation, for which previous convergence results were not applicable.

A Boundary Control Problem for Stochastic 2D-Navier–Stokes Equations

AbstractWe study a stochastic velocity tracking problem for the 2D-Navier–Stokes equations perturbed by a multiplicative Gaussian noise. From a physical point of view, the control acts through a boundary injection/suction device with uncertainty, modeled by stochastic non-homogeneous Navier-slip boundary conditions. We show the existence and uniqueness of the solution to the state equation, and prove the existence of an optimal solution to the control problem.

A lower bound estimate of life span of solutions to stochastic 3D Navier–Stokes equations with convolution-type noise

In this paper, we investigate the stochastic 3D Navier–Stokes equations perturbed by linear multiplicative Gaussian noise of convolution type by transformation to random PDEs. We focus on obtaining bounds from below for the life span associated with regular initial data. The key point of the proof is the fixed point argument.

Unsupervised Denoising and Super-Resolution of Vascular Flow Data by Physics-Informed Machine Learning.

We present an unsupervised deep learning method to perform flow denoising and super-resolution without high-resolution labels. We demonstrate the ability of a single model to reconstruct three-dimensional stenosis and aneurysm flows, with varying geometries, orientations, and boundary conditions. Ground truth data was generated using computational fluid dynamics, and then corrupted with multiplicative Gaussian noise. Auto-encoders were used to compress the representations of the flow domain geometry and the (possibly noisy and low-resolution) flow field. These representations were used to condition a physics-informed neural network. A physics-based loss was implemented to train the model to recover lost information from the noisy input by transforming the flow to a solution of the Navier-Stokes equations. Our experiments achieved mean squared errors in the true flow reconstruction of O(1.0 × 10-4), and root mean squared residuals of O(1.0 × 10-2) for the momentum and continuity equations. Our method yielded correlation coefficients of 0.971 for the hidden pressure field and 0.82 for the derived wall shear stress field. By performing point-wise predictions of the flow, the model was able to robustly denoise and super-resolve the field to 20× the input resolution.

Residual Hyperbolic Graph Convolution Networks

Hyperbolic graph convolutional networks (HGCNs) have demonstrated representational capabilities of modeling hierarchical-structured graphs. However, as in general GCNs, over-smoothing may occur as the number of model layers increases, limiting the representation capabilities of most current HGCN models. In this paper, we propose residual hyperbolic graph convolutional networks (R-HGCNs) to address the over-smoothing problem. We introduce a hyperbolic residual connection function to overcome the over-smoothing problem, and also theoretically prove the effectiveness of the hyperbolic residual function. Moreover, we use product manifolds and HyperDrop to facilitate the R-HGCNs. The distinctive features of the R-HGCNs are as follows: (1) The hyperbolic residual connection preserves the initial node information in each layer and adds a hyperbolic identity mapping to prevent node features from being indistinguishable. (2) Product manifolds in R-HGCNs have been set up with different origin points in different components to facilitate the extraction of feature information from a wider range of perspectives, which enhances the representing capability of R-HGCNs. (3) HyperDrop adds multiplicative Gaussian noise into hyperbolic representations, such that perturbations can be added to alleviate the over-fitting problem without deconstructing the hyperbolic geometry. Experiment results demonstrate the effectiveness of R-HGCNs under various graph convolution layers and different structures of product manifolds.

Reaction-diffusion equations on metric graphs with edge noise

We investigate stochastic reaction-diffusion equations on finite metric graphs. On each edge in the graph a multiplicative cylindrical Gaussian noise driven reaction-diffusion equation is given. The vertex conditions are the standard continuity and generalized, non-local Neumann-Kirchhoff-type law in each vertex. The reaction term on each edge is assumed to be an odd degree polynomial, not necessarily of the same degree on each edge, with possibly stochastic coefficients and negative leading term. The model is a generalization of the problem in [14] where polynomials with much more restrictive assumptions are considered and no first order differential operator is involved. We utilize the semigroup approach from [15] to obtain existence and uniqueness of solutions with sample paths in the space of continuous functions on the graph.

Stochastic resonance in Hindmarsh-Rose neural model driven by multiplicative and additive Gaussian noise

This paper investigates the occurrence of stochastic resonance in the three-dimensional Hindmarsh-Rose (HR) neural model driven by both multiplicative and additive Gaussian noise. Firstly, the three-dimensional HR neural model is transformed into the one-dimensional Langevin equation of the HR neural model using the adiabatic elimination method, and the effects of HR neural model parameters on the potential function are analyzed. Secondly the Steady-state Probability Density (SPD), the Mean First-Passage Time (MFPT), and the Signal-to-Noise Ratio (SNR) of the HR neural model are derived, based on two-state theory. Then, the effects of different parameters (a, b, c, s), noise intensity, and the signal amplitude on these metrics are analyzed through theoretical simulations, and the behavior of particles in a potential well is used to analyze how to choose the right parameters to achieve high-performance stochastic resonance. Finally, numerical simulations conducted with the fourth-order Runge–Kutta algorithm demonstrate the superiority of the HR neural model over the classical bistable stochastic resonance (CBSR) in terms of performance. The peak SNR of the HR neural model is 0.63 dB higher than that of the CBSR system. Simulation results indicate that the occurrence of stochastic resonance occur happens in HR neural model under different values of parameters. Furthermore, under certain conditions, there is a ‘suppress’ phenomenon that can be produced by changes in noise, which provides great feasibilities and practical value for engineering application.

Inferring nonlinear fractional diffusion processes from single trajectories

We present a method to infer the arbitrary space-dependent drift and diffusion of a nonlinear stochastic model driven by multiplicative fractional Gaussian noise from a single trajectory. Our method, fractional Onsager-Machlup optimisation (fOMo), introduces a maximum likelihood estimator by minimising a field-theoretic action which we construct from the observed time series. We successfully test fOMo for a wide range of Hurst exponents using artificial data with strong nonlinearities, and apply it to a data set of daily mean temperatures. We further highlight the significant systematic estimation errors when ignoring non-Markovianity, underlining the need for nonlinear fractional inference methods when studying real-world long-range (anti-)correlated systems.

The Stochastic Klausmeier System and A Stochastic Schauder-Tychonoff Type Theorem

On the one hand, we investigate the existence and pathwise uniqueness of a nonnegative martingale solution to the stochastic evolution system of nonlinear advection-diffusion equations proposed by Klausmeier with Gaussian multiplicative noise. On the other hand, we present and verify a general stochastic version of the Schauder-Tychonoff fixed point theorem, as its application is an essential step for showing existence of the solution to the stochastic Klausmeier system. The analysis of the system is based both on variational and semigroup techniques. We also discuss additional regularity properties of the solution.

Large deviations for the two-time-scale stochastic convective Brinkman-Forchheimer equations

The convective Brinkman-Forchheimer (CBF) equations are employed to characterize the motion of incompressible fluid in a saturated porous medium. This work investigates the small noise asymptotic of two-time-scale stochastic CBF equations in two and three dimensional bounded domains. More precisely, we establish a Wentzell-Freidlin type large deviation principle for stochastic partial differential equations that have slow and fast time-scales. The slow component is the stochastic CBF equations in two or three dimensions perturbed by a small multiplicative Gaussian noise, while the fast component is a stochastic reaction-diffusion equation with damping. The results are obtained by using a variational method (based on weak convergence approach) developed by Budhiraja and Dupuis, Khasminkii's time discretization approach and stopping time arguments. In particular, the findings from this study are also applicable to two-dimensional stochastic Navier-Stokes equations.

Convergence of densities of spatial averages of the parabolic Anderson model driven by colored noise

In this paper, we present a rate of convergence in the uniform norm for the densities of spatial averages of the solution to the d-dimensional parabolic Anderson model driven by a Gaussian multiplicative noise, which is white in time and has a spatial covariance given by the Riesz kernel. The proof is based on the combination of Malliavin calculus techniques and the Stein's method for normal approximations.

ON MODELLING WATER QUALITY WITH STOCHASTIC DIFFERENTIAL EQUATIONS

AbstractBased on biochemical kinetics, a stochastic model to characterize wastewater treatment plants and dynamics of river water quality under the influence of random fluctuations is proposed in this paper. This model describes the interaction between dissolved oxygen (DO) and biochemical oxygen demand (BOD), and is in the form of stochastic differential equations driven by multiplicative Gaussian noises. The stochastic persistence problem for the model of the system is analysed. Further, a numerical simulation of the stationary probability distributions of BOD and OD by approximations of the stochastic process solution is presented. These results have implications for the prediction and control of pollutants.

Climate change for global warming 1.5∘C under the influence of multiplicative Gaussian noise

We propose a stochastic energy balance differential equation with multiplicative Gaussian noise, which better captures the influence of the atmosphere as external noise on climate change. Meanwhile, we apply the Milstein method with stronger convergence to approximate the stochastic climate change trajectory. Further, the numerical approach can efficiently calculate two exit concepts: the mean first exit time and the first escape probability. These two quantities are used to examine the transition of the climate from one state to another. For global warming of [Formula: see text]C and [Formula: see text]C, we find that the enhanced noise intensity is more likely to induce a shorter mean first exit time or a higher first escape probability. These results provide important insights into the underlying mechanisms of climate change for global warming.

Hyperbolic Anderson Model 2: Strichartz Estimates and Stratonovich Setting

Abstract We study a wave equation in dimension $d\in \{1,2\}$ with a multiplicative space-time Gaussian noise. The existence and uniqueness of the Stratonovich solution is obtained under some conditions imposed on the Gaussian noise. The strategy is to develop some Strichartz-type estimates for the wave kernel in weighted Besov spaces, by which we can prove the well-posedness of an associated Young-type equation. Those Strichartz bounds are of independent interest.

Signal Modulation Recognition Algorithm Based on Improved Spatiotemporal Multi-Channel Network

Automatic modulation recognition (AMR) plays an essential role in modern communication systems. In recent years, various modulation recognition algorithms based on deep learning have been emerging, but the problem of low recognition accuracy has not been solved well. To solve this problem, based on the existing MCLDNN algorithm, in this paper, we proposed an improved spatiotemporal multi-channel network (IQ-related features Multi-channel Convolutional Bi-LSTM with Gaussian noise, IQGMCL). Firstly, dividing the input IQ signals into three channels, time sequence feature extraction is carried out for route I, route Q, and route IQ, respectively. For route IQ, convolution kernel (2,1) is first used to extract relevant features. Two layers of the small convolution kernel (1,3) are used to extract time sequence features further, and the three channels are used to extract features further. Then, a two-layer short-length memory network is used to extract features from time and space more effectively. Through comparison experiments, Bi-LSTM is introduced to replace one layer of LSTM, and a fully connected layer is removed to prevent overfitting. Finally, multiplicative Gaussian noise is introduced to naturally corrode the feature parameters, further improving the robustness and accuracy of the model. Experiments are carried out on three public datasets RML2016.10a, RML2016.10b, and RML2016.04C. The experiments show that the IQGMCL network has higher recognition accuracies on all datasets, especially on the RML2016.10a dataset. When the SNR is 4 dB, the recognition accuracy reaches 93.52%. When the SNR is greater than 0 dB, the average recognition accuracy reaches 92.3%, 1.31%, and 1.2% higher than the original MCLDNN network, respectively.

Stochastic resonance in gene transcriptional regulatory system driven by Gaussian noise and Lévy noise

The impacts of noise parameters and input signal parameters on stochastic resonance are explored in a gene transcriptional regulatory model driven by a combination of multiplicative Gaussian noise and additive Lévy noise. The response time series and signal-to-noise ratios are obtained by numerical simulations to quantify the occurrence of stochastic resonance. It is shown that the Gaussian noise intensity can cause the stochastic resonance, while the Lévy noise intensity can cause both the stochastic resonance and inverse stochastic resonance. The increases of Gaussian noise intensity inhibits these two phenomena and enhances system stability. The changes of the Lévy noise intensity parameter can cause the system to switch between low and high concentration state. In addition, the stability index and skewness parameter of Lévy noise can also result in stochastic resonance. Therefore, the optimum stochastic resonance can be achieved and useful genetic information can be easily acquired by adjusting the noise parameters. Our findings should contribute in the selection of appropriate parameters to achieve stochastic resonance in gene transcriptional regulation models, laying the foundation for the selection of stochastic resonance parameter ranges in actual gene transcriptional engineering.

Existence and regularity results for semilinear stochastic time-tempered fractional wave equations with multiplicative Gaussian noise and additive fractional Gaussian noise

To model wave propagation in inhomogeneous media with frequency-dependent power-law attenuation, it is needed to use the fractional powers of symmetric coercive elliptic operators in space and the Caputo tempered fractional derivative in time. The model studied in this paper is semilinear stochastic space-time fractional wave equations driven by infinite dimensional multiplicative Gaussian noise and fractional Gaussian noise, because of the potential fluctuations of the external sources. We first give a representation of the mild solution and some stability estimates for the homogeneous problem, and then prove the existence and uniqueness of the mild solution by using fixed point theorem. Finally, the decay and regularity theory of the solution are provided.

Noise Impact on a Recurrent Neural Network with a Linear Activation Function

In this paper we analyze an echo state neural network (ESN) in the presence of uncorrelated additive and multiplicative white Gaussian noise. Here we consider the case where artificial neurons have a linear activation function with different slope coefficients. We consider the influence of the input signal, memory and connection matrices on the accumulation of noise. We have found that the general view of variance and the signal-to-noise ratio of the ESN output signal is similar to only one neuron. The noise is less accumulated in ESN with a diagonal reservoir connection matrix with a large “blurring” coefficient. This is especially true of uncorrelated multiplicative noise.

Almost sure central limit theorems for stochastic wave equations

In this paper, we study almost sure central limit theorems for the spatial average of the solution to the stochastic wave equation in dimension d≤2 over a Euclidean ball, as the radius of the ball diverges to infinity. This equation is driven by a general Gaussian multiplicative noise, which is temporally white and colored in space including the cases of the spatial covariance given by a fractional noise, a Riesz kernel, and an integrable function that satisfies Dalang’s condition.

Gaussian fluctuations of a nonlinear stochastic heat equation in dimension two

We study the Gaussian fluctuations of a nonlinear stochastic heat equation in spatial dimension two. The equation is driven by a Gaussian multiplicative noise. The noise is white in time, smoothed in space at scale \(\varepsilon \), and tuned logarithmically by a factor \(\frac{1}{\sqrt{\log \varepsilon ^{-1}}}\) in its strength. We prove that, after centering and rescaling, the solution random field converges in distribution to an Edwards-Wilkinson limit as \(\varepsilon \downarrow 0\). The tool we used here is the Malliavin-Stein’s method. We also give a functional version of this result.