Effects of Multiplicative Noise on the Hartree-type Schrödinger Equation

Effect of multiplicative noise on parametric instabilities

Issues related to estimating the influence of quasi-deterministic and fluctuating multiplicative noise on the delay resolution and frequency resolution of systems processing radio signals based on the Woodward criterion for narrow-band and broadband signals are considered. The problem of resolution when detecting or measuring signal parameters can be considered both for useful signals alone and for cases when interfering signals are present as well. It is pointed out that the effect of multiplicative noise on the signal almost always leads to a resolution problem. It is shown, that under the influence of significant broadband multiplicative noise, the time resolution interval is determined only by the signal envelope and does not depend on its phase structure. For instance, for signals with a rectangular or bell-shaped envelope, it is equal to the equivalent signal duration. Examples of calculating resolution intervals under the influence of multiplicative noise are given. Taking into account the effect of multiplicative noise on signal resolution leads to an increase in the efficiency of radio systems, detection of objects being an example.

<abstract><p>We consider in this paper the stochastic nonlinear Schrödinger equation forced by multiplicative noise in the Itô sense. We use two different methods as sine-cosine method and Riccati-Bernoulli sub-ODE method to obtain new rational, trigonometric and hyperbolic stochastic solutions. These stochastic solutions are of a qualitatively distinct nature based on the parameters. Moreover, the effect of the multiplicative noise on the solutions of nonlinear Schrödinger equation will be discussed. Finally, two and three-dimensional graphs for some solutions have been given to support our analysis.</p></abstract>

The effect of multiplicative noise on a system described by two modes close to a bifurcation point is investigated. The bifurcation is assumed stationary and noise acts as random coupling between these modes. An analytic formula that predicts the onset of instability is derived, and the domain of existence of on-off intermittency is calculated based on an eigenvalue problem. This approach, confirmed by numerical simulations of the Langevin equations, allows quantifying the possible effects of the noise. The stability and the on-off behavior are shown to be very sensitive to deviations of the deterministic system from the case where both modes grow with equal rate and the system displays a continuous symmetry associated to rotation in phase space. In general, a noise term that breaks this continuous symmetry will increase the domain of instability of the system while a noise term that preserves the symmetry reduces the domain of instability.

The effect of multiplicative noise on a signal with a constant pulse-modulated frequency and a bell-shaped envelope, on a signal with linear frequency modulation, and on a phase-shift signal with a rectangular envelope is analyzed. During the calculation, it was assumed that the spectrum of fluctuations in the noise modulation function has a bell-shaped shape. It was shown, that the effect of multiplicative noise on the delay resolution interval when using linearly frequency-modulated signals does not depend on the shape of the signal envelope and is determined only by the spectrum width of the noise modulation function. It was shown that the Woodward criterion cannot generally be used to estimate the resolution interval for the arrival time of phase-shift signals in the presence of multiplicative noise. Mathematical expressions are obtained that allow calculating the resolution intervals in time and frequency.

Effects of multiplicative noise on fluctuation-induced transport

We investigate noise-induced phase transitions in globally coupled active rotators with multiplicative and additive noises. In the system there are four phases, stationary one-cluster, stationary two-cluster, moving one-cluster, and moving two-cluster phases. It is shown that multiplicative noise induces a bifurcation from one-cluster phase to two-cluster phase. Pinning force also induces a bifurcation from moving phase to stationary phase suppressing the multiplicative noise effect. Additive noise reduces both effects of multiplicative noise and pinning force urging the system to the stationary one-cluster phase. The frustrated effects of pinning force and additive and multiplicative noises lead to a reentrant transition at intermediate additive noise intensity. Nature of the transition is also discussed.

We analyzed the effect of multiplicative noise on signal resolution characteristics when a statistical criterion for processing is used in the receiving device. The receiver is optimal for resolving two signals in the sense of detecting against the background of additive white noise. It is shown that the complex envelope of the signal at the output of the analyzed receiver can be considered as a flat vector with correlated components having different variances. However, with fast multiplicative noise, the named components become uncorrelated, and their variances become equal. It is shown that the delay time resolution intervals and frequency shift resolution intervals with known statistical characteristics of signals, additive and multiplicative noise are uniquely determined by the probabilities of correct and false resolution. It is pointed out that the use of a statistical criterion allows us to determine quantitative values of resolution intervals in a fairly wide range of changes in the parameters of additive and multiplicative noise, significantly exceeding the Woodward criterion for example. Thus, the use of statistical criteria leads to an increase in the efficiency of radio systems.

We analyzed the effect of multiplicative noise on signal resolution characteristics when a statistical criterion for processing is used in the receiving device. The receiver is optimal for resolving two signals in the sense of detecting against the background of additive white noise. It is shown that the complex envelope of the signal at the output of the analyzed receiver can be considered as a flat vector with correlated components having different variances. However, with fast multiplicative noise, the named components become uncorrelated, and their variances become equal. It is shown that the delay time resolution intervals and frequency shift resolution intervals with known statistical characteristics of signals, additive and multiplicative noise are uniquely determined by the probabilities of correct and false resolution. It is pointed out that the use of a statistical criterion allows us to determine quantitative values of resolution intervals in a fairly wide range of changes in the parameters of additive and multiplicative noise, significantly exceeding the Woodward criterion for example. Thus, the use of statistical criteria leads to an increase in the efficiency of radio systems.

Issues related to estimating the influence of quasi-deterministic and fluctuating multiplicative noise on the delay resolution and frequency resolution of systems processing radio signals based on the Woodward criterion for narrow-band and broadband signals are considered. The problem of resolution when detecting or measuring signal parameters can be considered both for useful signals alone and for cases when interfering signals are present as well. It is pointed out that the effect of multiplicative noise on the signal almost always leads to a resolution problem. It is shown, that under the influence of significant broadband multiplicative noise, the time resolution interval is determined only by the signal envelope and does not depend on its phase structure. For instance, for signals with a rectangular or bell-shaped envelope, it is equal to the equivalent signal duration. Examples of calculating resolution intervals under the influence of multiplicative noise are given. Taking into account the effect of multiplicative noise on signal resolution leads to an increase in the efficiency of radio systems, detection of objects being an example.

Purpose. This study aims to establish the characteristics of noise propagation and accumulation in convolutional neural networks. The article investigates how the accuracy of a trained convolutional network varies depending on the type and intensity of noise exposure. Methods. White Gaussian noise sources were used as the basis for noise exposure. Two types of noise exposure were applied to artificial neurons: additive and multiplicative. Additionally, the effects of correlated and uncorrelated noise on the layers of neurons were examined. Results. The findings indicate that additive noise (both correlated and uncorrelated) accumulates more significantly in networks with convolutional layers compared to those without. The relationship between network accuracy and the intensity of multiplicative correlated noise is similar for both types of networks. However, the impact of multiplicative uncorrelated noise is more favorable for networks with convolutional layers. The study also considered pooling layers, specifically MaxPooling and MeanPooling, which significantly enhance accuracy in the presence of additive noise within the convolutional layer. The decline in accuracy due to increasing intensity of multiplicative correlated noise is nearly identical for networks with and without pooling layers. Conversely, networks employing MaxPooling demonstrate reduced resilience to uncorrelated multiplicative noise. Conclusion. The study demonstrates that additive noise severely degrades network performance when a convolutional layer is present, though this negative effect can be mitigated by including a pooling layer immediately following the convolutional layer. In contrast, the effects of multiplicative noise are less clear-cut. In most cases, its impact remains consistent regardless of the presence of convolution and pooling layers. However, the use of MaxPooling in the pooling layer may compromise the network’s robustness against multiplicative uncorrelated noise.

In this paper, we consider the stochastic Burgers' equation, which is forced by multiplicative noise in the Stratonovich sense. To get a new trigonometric and hyperbolic stochastic solutions, we apply exp ⁡ ( − φ ( μ ) ) -expansion method. In addition, we demonstrate the effect of multiplicative noise on the exact solutions of the Burgers' equation by introducing some graphic representations.

This article investigates optical soliton solutions within the stochastic resonant nonlinear Schrödinger equation (SRNLSE). This equation incorporates both spatio-temporal dispersion (STD) and inter-modal dispersion (IMD), along with multiplicative white noise and generalized Kudryashov's law non-linearity. By applying the extended auxiliary equation method, we identify a range of soliton solutions, including bright, dark, and singular solitons. Additionally, we derive solutions that take the form of Jacobi elliptic functions, Weierstrass elliptic functions, and periodic wave functions. The study provides significant insights into soliton dynamics within nonlinear optical systems affected by stochastic influences and complex dispersion interactions. Specifically, it highlights how the interplay of STD and IMD, coupled with the presence of multiplicative noise, shapes the behavior of solitons. Moreover, we delve into the effects of multiplicative noise on the exact solutions of the NLSE using the Maple software. Our analysis reveals that multiplicative noise, interpreted in the Ito sense, plays a crucial role in stabilizing the soliton solutions, particularly maintaining their stability around the zero state. This finding underscores the importance of noise in influencing the stability and dynamics of solitons in optical systems.

The effect of multiplicative noise on a signal when compared with that of additive noise is very large. In this paper, we address the problem of suppressing multiplicative noise in one-dimensional signals. To deal with signals that are corrupted with multiplicative noise, we propose a denoising algorithm based on minimization of an unbiased estimator (MURE) of mean-square error (MSE). We derive an expression for an unbiased estimate of the MSE. The proposed denoising is carried out in wavelet domain (soft thresholding) by considering time-domain MURE. The parameters of thresholding function are obtained by minimizing the unbiased estimator MURE. We show that the parameters for optimal MURE are very close to the optimal parameters considering the oracle MSE. Experiments show that the SNR improvement for the proposed denoising algorithm is competitive with a state-of-the-art method.

We address the problem of denoising images corrupted by multiplicative noise. The noise is assumed to follow a Gamma distribution. Compared with additive noise distortion, the effect of multiplicative noise on the visual quality of images is quite severe. We consider the mean-square error (MSE) cost function and derive an expression for an unbiased estimate of the MSE. The resulting multiplicative noise unbiased risk estimator is referred to as MURE. The denoising operation is performed in the wavelet domain by considering the image-domain MURE. The parameters of the denoising function (typically, a shrinkage of wavelet coefficients) are optimized for by minimizing MURE. We show that MURE is accurate and close to the oracle MSE. This makes MURE-based image denoising reliable and on par with oracle-MSE-based estimates. Analogous to the other popular risk estimation approaches developed for additive, Poisson, and chi-squared noise degradations, the proposed approach does not assume any prior on the underlying noise-free image. We report denoising results for various noise levels and show that the quality of denoising obtained is on par with the oracle result and better than that obtained using some state-of-the-art denoisers.