Case Of Multiplicative Noise Research Articles

On the stochastic Euler-Poincaré equations driven by pseudo-differential/multiplicative noise

The stochastic Euler-Poincaré equations with pseudo-differential/multiplicative noise are considered in this work. We first establish two new cancellation properties on pseudo-differential operators, which considerably extend the previous results for transport type noise only involving gradient operator. Then, we obtain results on local solution, blow-up criterion, and global existence. The interplay between stability on exiting times and continuous dependence of solution on initial data is also studied for the multiplicative noise case.

Performance analysis of mobile adaptive networks in VLC multiplicative SPAD channel noise conditions

Mobile diffusion adaptive networks are distributed sensor networks used for tracking and localization. They are considered as the network of several mobile nodes which are able to perform target localization by sharing their information among their neighboring nodes. Due to their low computational costs, they have found many applications. However, there are some hindering problems and disturbances preventing their smooth operation particularly for mobile networks. Among them, the link noise during data exchange between the nodes can have significant impact on the network performance, causing a large convergence error. Therefore, detailed and precise investigation of noise model and its impact on the mobile networks becomes important. In this paper the challenging case of multiplicative noise for a visible light communication (VLC) has been modeled, implemented, and the performance of mobile adaptive network analyzed. This is achieved by using single photon avalanche diodes (SPADS) for node communications within the mobile diffusion adaptive networks. The results show a completely different performance of the network compared with the case of additive noises.

Renormalized solutions for stochastic $ p $-Laplace equations with $ L^1 $-initial data: The case of multiplicative noise

<p style='text-indent:20px;'>We consider a <inline-formula><tex-math id="M3">\begin{document}$ p $\end{document}</tex-math></inline-formula>-Laplace evolution problem with multiplicative noise on a bounded domain <inline-formula><tex-math id="M4">\begin{document}$ D\subset\mathbb{R}^d $\end{document}</tex-math></inline-formula> with homogeneous Dirichlet boundary conditions for <inline-formula><tex-math id="M5">\begin{document}$ 1<p<\infty $\end{document}</tex-math></inline-formula>. The random initial data is merely integrable. Consequently, the key estimates are available with respect to truncations of the solution. We introduce the notion of renormalized solutions for multiplicative stochastic <inline-formula><tex-math id="M6">\begin{document}$ p $\end{document}</tex-math></inline-formula>-Laplace equations with <inline-formula><tex-math id="M7">\begin{document}$ L^1 $\end{document}</tex-math></inline-formula>-initial data and study existence and uniqueness of solutions in this framework.</p>

Metastability and exit problems for systems of stochastic reaction–diffusion equations

In this paper we develop a metastability theory for a class of stochastic reaction–diffusion equations exposed to small multiplicative noise. We consider the case where the unperturbed reaction–diffusion equation features multiple asymptotically stable equilibria. When the system is exposed to small stochastic perturbations, it is likely to stay near one equilibrium for a long period of time but will eventually transition to the neighborhood of another equilibrium. We are interested in studying the exit time from the full domain of attraction (in a function space) surrounding an equilibrium and, therefore, do not assume that the domain of attraction features uniform attraction to the equilibrium. This means that the boundary of the domain of attraction is allowed to contain saddles and limit cycles. Our method of proof is purely infinite dimensional, that is, we do not go through finite dimensional approximations. In addition, we address the multiplicative noise case, and we do not impose gradient type of assumptions on the nonlinearity. We prove large deviations logarithmic asymptotics for the exit time and for the exit shape, also characterizing the most probable set of shapes of solutions at the time of exit from the domain of attraction.

A discontinuous Galerkin method for systems of stochastic differential equations with applications to population biology, finance, and physics

In this paper, we propose a discontinuous Galerkin (DG) method for systems of stochastic differential equations (SDEs) driven by m-dimensional Brownian motion. We first construct a new approximate system of SDEs on each element using whose converges to the solution of the original system. The new system is then discretized using the standard DG method for deterministic ordinary differential equations (ODEs). For the case of additive noise, we prove that the proposed scheme is convergent in the mean-square sense. Our numerical experiments suggest that our results hold true for the case of multiplicative noise as well. Several linear and nonlinear test problems are presented to show the accuracy and effectiveness of the proposed method. In particular, the proposed scheme is illustrated by considering different examples arising in population biology, physics, and mathematical finance.

Thermal rectification in one-dimensional lattices with nonlinear system–reservoir coupling

In this work we study the thermal rectification properties of a one-dimensional, mass-graded oscillator lattice with nonlinear system–reservoir couplings. For the case of a harmonic oscillator chain the system presents no rectification at high temperatures and a very weak one for low temperatures as the asymmetric, nonlinear interaction with the thermal reservoirs is increased. The observed rectification rapidly degrades as either the thermal bias or mass asymmetry increase in magnitude. For an anharmonic oscillator Fermi–Pasta–Ulam lattice the rectification increases for the aforementioned conditions, and is maximized when only nonlinear system-reservoir interactions are considered. These latter ones correspond to the case of multiplicative noise affecting the boundaries of the lattice.

Well-posedness for nonlinear SPDEs with strongly continuous perturbation

AbstractWe consider the well-posedness of a stochastic evolution problem in a bounded Lipschitz domain D ⊂ ℝd with homogeneous Dirichlet boundary conditions and an initial condition in L2(D). The main technical difficulties in proving the result of existence and uniqueness of a solution arise from the nonlinear diffusion-convection operator in divergence form which is given by the sum of a Carathéodory function satisfying p-type growth associated with coercivity assumptions and a Lipschitz continuous perturbation. In particular, we consider the case 1 < p < 2 with an appropriate lower bound on p determined by the space dimension. Another difficulty arises from the fact that the additive stochastic perturbation with values in L2(D) on the right-hand side of the equation does not inherit the Sobolev spatial regularity from the solution as in the multiplicative noise case.

A support theorem for 3d-stochastic wave equations in Hölder norm with some general noises

ABSTRACT In this paper, we characterize the topological support in Hölder norm of the law of the solution to a stochastic wave equation with three-dimensional space variable. This note is an extension of Delgado-Vences and Sanz-Solé [Approximation of a stochastic wave equation in dimension three, with applications to a support theorem in Hölder norm, Bernoulli 20(4) (2014), pp. 2169–2216; Approximation of a stochastic wave equation in dimension three, with application to a support theorem in Hölder norm: The non-stationary case, Bernoulli 22(3) (2016), pp. 1572–1597]. The result presented here characterize a more general type of stochastic wave equations in 3-d space variable than those considered in Delgado-Vences and Sanz-Solé [Approximation of a stochastic wave equation in dimension three, with applications to a support theorem in Hölder norm, Bernoulli 20(4) (2014), pp. 2169–2216; Approximation of a stochastic wave equation in dimension three, with application to a support theorem in Hölder norm: The non-stationary case, Bernoulli 22(3) (2016), pp. 1572–1597]. Here we extend these two previous results in the folowing sense. The first extension is that we cover the case of multiplicative noise and non-zero initial conditions. The second extension is related to the covariance function associated with the noise, here we follow the approach of Hu, Huang and Nualart and ask conditions in terms of the mean Hölder continuity of such covariance function. As in Delgado-Vences and Sanz-Solé [Approximation of a stochastic wave equation in dimension three, with applications to a support theorem in Hölder norm, Bernoulli 20(4) (2014), pp. 2169–2216; Approximation of a stochastic wave equation in dimension three, with application to a support theorem in Hölder norm: The non-stationary case, Bernoulli 22(3) (2016), pp. 1572–1597] the result is a consequence of an approximation theorem, in the convergence of probability, for a sequence of evolution equations driven by a family of regularizations of the driving noise.

DoPAMINE: Double-Sided Masked CNN for Pixel Adaptive Multiplicative Noise Despeckling

We propose DoPAMINE, a new neural network based multiplicative noise despeckling algorithm. Our algorithm is inspired by Neural AIDE (N-AIDE), which is a recently proposed neural adaptive image denoiser. While the original NAIDE was designed for the additive noise case, we show that the same framework, i.e., adaptively learning a network for pixel-wise affine denoisers by minimizing an unbiased estimate of MSE, can be applied to the multiplicative noise case as well. Moreover, we derive a double-sided masked CNN architecture which can control the variance of the activation values in each layer and converge fast to high denoising performance during supervised training. In the experimental results, we show our DoPAMINE possesses high adaptivity via fine-tuning the network parameters based on the given noisy image and achieves significantly better despeckling results compared to SAR-DRN, a state-of-the-art CNN-based algorithm.

Well-posedness for a pseudomonotone evolution problem with multiplicative noise

Our aim is the study of well-posedness for the stochastic evolution equation $$\begin{aligned} \mathrm{d}u -\mathrm {div}\,(|\nabla u|^{p-2}\nabla u +F(u)) \,\mathrm{d} t = H(u) \,\mathrm{d}W, \end{aligned}$$ for $$T>0$$ , on a bounded Lipschitz domain $$D \subset \mathbb {R}^d$$ with homogeneous Dirichlet boundary conditions, initial values $$u_0\in L^2(D)$$ , $$p>2$$ and $$F:\mathbb {R}\rightarrow \mathbb {R}^d$$ Lipschitz continuous. W(t) is a cylindrical Wiener process in $$L^2(D)$$ with respect to a filtration $$(\mathcal {F}_t)$$ satisfying the usual assumptions on a complete, countably generated probability space $$(\varOmega ,\mathcal {F},\mathbb {P})$$ . We consider the case of multiplicative noise satisfying appropriate regularity conditions. By a semi-implicit time discretization, we obtain approximate solutions. Using the theorems of Skorokhod and Prokhorov, we are able to pass to the limit and show existence of martingale solutions. We establish pathwise $$L^1$$ -contraction and uniqueness and obtain existence and uniqueness of strong solutions.

Likelihood for transcriptions in a genetic regulatory system under asymmetric stable Lévy noise.

This work is devoted to investigating the evolution of concentration in a genetic regulation system, when the synthesis reaction rate is under additive and multiplicative asymmetric stable Lévy fluctuations. By focusing on the impact of skewness (i.e., non-symmetry) in the probability distributions of noise, we find that via examining the mean first exit time (MFET) and the first escape probability (FEP), the asymmetric fluctuations, interacting with nonlinearity in the system, lead to peculiar likelihood for transcription. This includes, in the additive noise case, realizing higher likelihood of transcription for larger positive skewness (i.e., asymmetry) index β, causing a stochastic bifurcation at the non-Gaussianity index value α = 1 (i.e., it is a separating point or line for the likelihood for transcription), and achieving a turning point at the threshold value β≈-0.5 (i.e., beyond which the likelihood for transcription suddenly reversed for α values). The stochastic bifurcation and turning point phenomena do not occur in the symmetric noise case (β = 0). While in the multiplicative noise case, non-Gaussianity index value α = 1 is a separating point or line for both the MFET and the FEP. We also investigate the noise enhanced stability phenomenon. Additionally, we are able to specify the regions in the whole parameter space for the asymmetric noise, in which we attain desired likelihood for transcription. We have conducted a series of numerical experiments in "regulating" the likelihood of gene transcription by tuning asymmetric stable Lévy noise indexes. This work offers insights for possible ways of achieving gene regulation in experimental research.

On Large Deviations of Stochastic Integrodifferential Equations with Brownian Motion

In this paper, a Freidlin-Wentzell type large deviation principle is established for the stochastic integrodifferential equation driven by finite dimensional Brownian motion. Both the additive and multiplicative noise cases are considered here. Large deviation principle for additive noise case is established via contraction principle whilst weak convergence approach is employed to obtain the same for the multiplicative noise case.

Global delay induced transition in a bistable system with multiplicative and additive noises

Global time delay is introduced to a bistable system driven by multiplicative and additive noises. Approximation of small delay and numerical simulations are employed to investigate the delay induced transition. The stationary probability distribution function \(P_{st}(x)\) and the first order moment \(\langle x\rangle _{st}\) are derived. Results indicate that with the increase of global time delay, \(P_{st}(x)\) undergoes a transition from a bimodal structure to a unimodal shape and \(\langle x\rangle _{st}\) as a function of the multiplicative noise intensity exhibits suppression-like and resonance-like behavior. For the case of multiplicative noise with delay, \(P_{st}(x)\) undergoes a transition from a monostable to a bistable system. These results illustrate that global delay can control the transition of a bistable system effectively.

Approximation of stationary solutions to SDEs driven by multiplicative fractional noise

In a previous paper, we studied the ergodic properties of an Euler scheme of a stochastic differential equation with a Gaussian additive noise in order to approximate the stationary regime of such an equation. We now consider the case of multiplicative noise when the Gaussian process is a fractional Brownian motion with Hurst parameter H>1/2 and obtain some (functional) convergence properties of some empirical measures of the Euler scheme to the stationary solutions of such SDEs.

Optimality of Euler-type algorithms for approximation of stochastic differential equations with discontinuous coefficients

We consider the pointwise approximation of solutions of scalar stochastic differential equations with discontinuous coefficients. We assume the singularities of coefficients to be unknown. We show that any algorithm which does not locate the discontinuities of a diffusion coefficient has the error at least Ω(n−min{1/2, ϱ}), where ϱ∈(0, 1] is the Hölder exponent of the coefficient. In order to obtain better results, we consider algorithms that adaptively locate the unknown singularities. In the additive noise case, for a single discontinuity of a diffusion coefficient, we define an Euler-type algorithm based on adaptive mesh which obtains an error of order n−ϱ. That is, this algorithm preserves the optimal error known from the Hölder continuous case. In the case of multiple discontinuities we show, both for the additive and the multiplicative noise case, that the optimal error is Θ(n−min{1/2, ϱ}), even for the algorithms locating unknown singularities.

The variational approach to Hamilton–Jacobi equations driven by a Gaussian noise

The global existence and uniqueness of viscosity solutions to the Cauchy problem for the Hamilton–Jacobi equations in RN driven by additive and multiplicative Wiener processes are studied for convex Hamiltonians via variational techniques. The finite speed of propagation is also established in the multiplicative noise case for equations with Lipschitzian Hamiltonians.

Effect of Noise on Gradient Systems

 Abstract—Recently, we have analyzed the convergence properties and derived the objective functions of the weight noise injection-based algorithms. In this paper, we generalize our work to perturbed gradient systems (PGS). For original gradient systems (GS), the change of a vector x(t) at time t is defined as the gradient vector of a function F(x) times a small negative constant. For PGS, it is assumed that the vector x(t) is perturbed by mean zero Gaussian noise. The noise can either be multiplicative or additive. In this paper, the corresponding energy functions for both cases are derived. It is found that their energy functions are very difference from F(x). For the case of multiplicative noise, the energy function consists of three terms: (i) F(x), (ii) a regularization term and (iii) a de-regularization term. For the case of additive noise, the energy function consists of only two terms: (i) F(x) and (ii) a regularization term. Note that de-regularization could lead to divergence behavior while regularization will improve the convergence behavior of a system. Our results suggest that special caution should be done to a gradient system with multiplicative noise.

Speckle noise reduction in SAS imagery

Synthetic aperture sonar (SAS) is actively used in sea bed imagery. Indeed high resolution images provided by SAS are of great interest, especially for the detection, localization and eventually classification of objects lying on sea bed. SAS images are highly corrupted by a granular multiplicative noise, called speckle noise which reduces spatial and radiometric resolutions. The purpose of this article is to present a new adaptive processing that allows image filtering, for both the additive and multiplicative noise case. This new process is based on the marriage between a multi-resolution transformation and a filtering method. The filtering technique used here is based on the two-dimensional stochastic matched filtering method, which maximizes the signal-to-noise ratio after processing and minimizes mean square error between the signal's approximation and the original one. Results obtained on real SAS data are presented and compared with those obtained using classical processing.

Harmonics in Gaussian multiplicative and additive noise: Cramer-Rao bounds

The concern here is retrieval of single and multiple tone harmonics observed in white Gaussian multiplicative and additive noise. Computable Cramer-Rao bound (CRB) expressions are derived on the frequency and phase estimates as well as on the sample mean or variance of the multiplicative noise processes. The zero- and nonzero-mean multiplicative noise cases are addressed separately and are shown to yield distinct CRBs on the frequency and phase estimates. Tight lower and upper bounds on the CRBs themselves are developed, which, relative to the CRBs, are intuitively more appealing and easier to implement. Well-established formulas on the achievable accuracy for estimates of constant amplitude harmonics turn put to be special cases of our results. Numerical studies support our claims. >

Statistical properties of single-mode oscillations in photorefractive ring resonators

We investigate the statistical properties of the optical field that arises from two-wave mixing in a single-mode unidirectional photorefractive ring resonator. Two cases are considered: additive noise produced by generation-recombination fluctuations in the photorefractive material's ionized donor concentration and multiplicative noise produced by fluctuations in the cavity detuning and losses. Like the homogeneously broadened two-level laser near threshold, the photorefractive resonator has Gaussian and Rayleigh-like intensity distributions in the additive and multiplicative noise cases, respectively. The equations describing the linewidths associated with the intensity fluctuations for the two cases and those describing the linewidth and frequency shift associated with the amplitude fluctuation in the multiplicative noise case also have the same form as the laser equations. The descriptions of the linewidth and frequency shift associated with the amplitude fluctuation in the additive noise case are, in general, different from those of the laser.< <ETX xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">&gt;</ETX>