Fractional Noise Research Articles

Study of a fractional stochastic heat equation

In this article, we study a d-dimensional stochastic nonlinear heat equation (SNLH) with a quadratic nonlinearity, forced by a fractional space-time white noise: ∂ t u − ∆u = ρ 2 u 2 +Ḃ , t ∈ [0, T ] , x ∈ R d , u 0 = φ. Two types of regimes are exhibited, depending on the ranges of the Hurst index H = (H 0 , ..., H d) ∈ (0, 1) d+1. In particular, we show that the local well-posedness of (SNLH) resulting from the Da Prato-Debussche trick, is easily obtained when 2H 0 + d i=1 H i > d. On the contrary, (SNLH) is much more difficult to handle when 2H 0 + d i=1 H i ≤ d. In this case, the model has to be interpreted in the Wick sense, thanks to a time-dependent renormalization. Helped with the regularising effect of the heat semigroup, we establish local well-posedness results for (SNLH) for all dimension d ≥ 1. Contents

Regularisation by fractional noise for one-dimensional differential equations with distributional drift

Regularisation by fractional noise for one-dimensional differential equations with distributional drift

Stochastic heat equation with Burgers term driven by fractional noises with two reflecting walls

In this article, we study stochastic heat equation with Burgers term with two reflecting walls h1 and h2, driven by space-time fractional noises. We will first show the existence and uniqueness of the solution by Picard iteration scheme, then we will consider the penalized approximating equations with respect to the reflecting terms to obtain the Hölder continuity of this solution. Furthermore, the large deviation principle for the law of the solution to this equation with a small perturbation will be established through a classical method.

Bifurcations analysis of a multiple attractors energy harvesting system with fractional derivative damping under random excitation

The multiple attractors energy harvester, which can convert mechanical energy into electrical energy, is a typical birhythmic system. This paper investigates the stochastic bifurcation of the system with fractional derivative damping under different noise excitations. Variable transformation and conversion mechanism for decoupling the electromechanical equations are utilized to obtain the approximate equivalent system. Probability density functions of the system are generated via the stochastic averaging method. Numerical results are presented to verify the effectiveness of the method. In the deterministic case, the fractional order can effectively control the birhythmic properties. In the stochastic case, the effects of fractional derivative damping and different noises on the stochastic P-bifurcations of the system are discussed separately. It concludes that appropriately changing noise parameters and fractional derivative parameters can increase the harvested energy from vibrations. This study highlights the theoretical contribution in energy harvesting and hopes to provides ideas for optimizing the energy harvesting performance.

On highly skewed fractional log‐stable noise sequences and their application

Considering log‐LFSN (log‐linear fractional stable noise) sequences , driven by non‐Gaussian one‐sided LFSN with constant skewness intensity , for any and , we show that the auto‐covariance function (ACVF) exists if and only if is persistent, with stability index , Hurst exponent and extreme skewness (if ) or (if ). Within that range of existence, and in short, we calculate explicitly and establish persistence of too, by showing asymptotic proportionality of , as . We discuss explicit links of to a generalized co‐difference function of the driving one‐sided LFSN , and to the ACVF's of fractional Gaussian noise (FGN) and log‐FGN. The results are numerically demonstrated via ensemble simulation of synthetic time series generated by the considered log‐LFSN model fitted to time series of spatio‐temporal accumulations of rain rate data.

Mitigating Threshold Effects in Human Control by Stochastic Resonance With Fractional Colored Noise

In industrial applications, mechanical and physiological thresholds may limit the capability of human manipulating machine via control devices, such as joysticks and steering wheels. These thresholds can result in loss of information in the control signals that are kept below the threshold of detection of the device or the human operator. One approach to mitigate these effects is <italic xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">stochastic resonance</i> , i.e., by injecting additive noise into a signal to raise its energy content over the threshold of detection. Though this noise partially corrupts the signal, it can increase the detectability of the signal by the control device. This article provides, for the first time, research towards using <italic xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">stochastic resonance</i> to improve human performance in control tasks. In particular, it shows that using adaptive colored noise can improve the detectability of the steering control signals recorded from human participants. The approach converts a signal processing task to an optimization problem, where <italic xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">particle swarm optimization</i> is employed to obtain the optimal color (or spectral exponent) of the injected additive noise, generated through an intelligent technique with fractional order filters. The results have shown that the proposed method improves the detectability of subthreshold steering control signals. This method can be widely applicable to other industrial domains, such as energy harvesting and enhancing sensory perception.

Strong convergence rate of the Euler scheme for SDEs driven by additive rough fractional noises

The strong convergence rate of the Euler scheme for stochastic differential equations driven by additive fractional Brownian motions is studied, where the fractional Brownian motion has Hurst parameter H∈(13,12) and the drift coefficient is not required to be bounded. The Malliavin calculus, the rough path theory and the 2D Young integral are utilized to overcome the difficulties caused by the low regularity of the fractional Brownian motion and the unboundedness of the drift coefficient. The Euler scheme is proved to have strong order 2H for the case that the drift coefficient has bounded derivatives up to order three and have strong order H+12 for linear cases.

Iterative computed tomography reconstruction of void fraction in a fuel bundle on unstructured mesh

The experimental investigation of internal two-phase flow structures in high pressure/high temperature systems is technologically challenging and limited to a few instrumentation techniques. For complex fuel bundle geometries, the use of Computed Tomography (CT) allows for relatively high spatial resolution (0.2 to 1 mm) and non-intrusive measurement of time-averaged void fraction. One of the main issues using standard structured reconstruction mesh is due to the loss of accuracy near the test section boundaries, for instance within the pixels overlapping both the two-phase coolant and the heated rods. In this work, a void fraction reconstruction methodology based on unstructured mesh is developed to accurately bound relevant attenuation regions within the considered test section. This method eliminates any overlapping reconstruction pixels and allows straightforward applications of knowledge-based, low noise, algebraic iterative reconstruction techniques (ART, SIRT, etc.). Within the same consistent framework, CT scan forward projections can be simulated for various selected cases (single-phase, arbitrary-two-phase void fraction distributions, etc.) and reconstructed to assess the performance of both the CT scan setup (e.g., projection data sampling and number of projections) and the reconstruction technique. The method is applied to the void fraction reconstruction of gamma CT data in a 5 × 5 fuel bundle geometry measured at the Westinghouse FRIGG thermal–hydraulic test facility operating under prototypical conditions of Boiling Water Reactor. The unstructured mesh is designed to match the features of the test section (isolation, pressure vessel, water gap, fuel channel, heated rods and coolant) with high-resolution in the domain of interest (coolant) and very low-resolution in all other domains with known attenuation coefficients. As compared to previous void fraction reconstruction attempts for this data set, the new reconstruction methodology allows significant improvements in void fraction resolution and noise reduction.

An Integration by Parts Formula for Stochastic Heat Equations with Fractional Noise

An Integration by Parts Formula for Stochastic Heat Equations with Fractional Noise

Nontrivial Equilibrium Solutions and General Stability for Stochastic Evolution Equations with Pantograph Delay and Tempered Fractional Noise

Nontrivial Equilibrium Solutions and General Stability for Stochastic Evolution Equations with Pantograph Delay and Tempered Fractional Noise

Euler Approximation for A Class of Singular Multi-Dimensional SDEs Driven by an Additive Fractional Noise

We consider a class of multi-dimensional stochastic differential equations driven by fractional Brownian motion with Hurst index . In particular, the drift coefficient blows up at 0. We first prove that this equation has a unique positive solution, and then propose a semi-implicit Euler approximation scheme for the equation, and finally show that it is also positive, and study its rate of convergence.&#x0D;

Regularization of differential equations by two fractional noises

In this paper, we show the existence and uniqueness of a solution for a stochastic differential equation driven by an additive noise which is the sum of two fractional Brownian motions with different Hurst parameters. The proofs are based on the techniques of fractional calculus and Girsanov theorem. In particular, we show that the regularization effect of the fractional Brownian motion with the smaller Hurst index dominates. A key challenge in this paper is to extend and apply the Girsanov theorem for two noises given by the sum of two (dependent) fractional Brownian motions by using profound techniques of fractional operator theory.

Parameter estimation of stochastic differential equation driven by small fractional noise

We study the problem of parametric estimation for continuously observed stochastic processes driven by additive small fractional Brownian motion with the Hurst index . Under some assumptions on the drift coefficient, we obtain the asymptotic normality and moment convergence of maximum likelihood estimator of the drift parameter when a small dispersion coefficient .

Hybrid neural networks for noise reductions of integrated navigation complexes

The necessity of integrated navigation complexes (INC) construction is substantiated. It is proposed to include in the complex the following inertial systems: inertial, satellite and visual. It helps to increase the accuracy of determining the coordinates of unmanned aerial vehicles. It is shown that in unfavorable cases, namely the suppression of external noise of the satellite navigation system, an increase in the errors of the inertial navigation system (INS), including through the use of accelerometers and gyroscopes manufactured using MEMS technology, the presence of bad weather conditions, which complicates the work of the visual navigation system. In order to ensure the operation of the navigation complex, it is necessary to ensure the suppression of interference (noise). To improve the accuracy of the INS, which is part of the INC, it is proposed to use the procedure for extracting noise from the raw signal of the INS, its prediction using neural networks and its suppression. To solve this problem, two approaches are proposed, the first of which is based on the use of a multi-row GMDH algorithm and single-layer networks with sigm_piecewise neurons, and the second is on the use of hybrid recurrent neural networks, when neural networks were used, which included long-term and short-term memory (LSTM) and Gated Recurrent Units (GRU). Various types of noise, that are inherent in video images in visual navigation systems are considered: Gaussian noise, salt and pepper noise, Poisson noise, fractional noise, blind noise. Particular attention is paid to blind noise. To improve the accuracy of the visual navigation system, it is proposed to use hybrid convolutional neural networks.

Comparison of Radiocarbon‐ and Background Location‐Corrections on Soil‐Gas CO2 Flux‐Based NSZD Rate Measurements at Petroleum Impacted Sites

AbstractThe measurement of contaminant natural source zone depletion (NSZD) rates has become an important tool to manage petroleum contaminated sites. Most NSZD rate measurement methods rely on a balance on the biodegradation by‐products (either carbon or heat). Carbon balance‐based methods stoichiometrically convert measured soil‐gas CO2 flux related to contaminant degradation to equivalent contaminant mass loses. CO2 flux‐based methods require separating the fraction of the total CO2 flux produced by NSZD from the fraction of CO2 flux produced by natural soil processes (due to modern carbon turnover). Two method corrections are available to distinguish the nature of the measured CO2 flux: (1) The background location correction subtracts a measurement from an unimpacted location from measurements collected at impacted locations, and (2) a location‐specific radiocarbon (14C) analysis that differentiates the modern and old carbon fractions of each total CO2 measurement. This work evaluates both correction approaches using 36 measurements at impacted locations from five sites. The 14C‐corrected data shows that the magnitude and variability of the modern carbon‐related fraction (noise) and the NSZD related fossil fuel signal are similar, suggesting a location‐specific correction is more valid. Only one sparsely vegetated arid site showed close agreement between both corrections (14C‐ and background location). These results highlight the strong effect of the correction method used on NSZD rate measurements, and the importance of considering data quality on the subsequent data management process.

Linear Filtering with Fractional Noises: Large Time and Small Noise Asymptotics

The classical state-space approach to optimal estimation of stochastic processes is efficient when the driving noises are generated by martingales. In particular, the weight function of the optimal linear filter, which solves a complicated operator equation in general, simplifies to the Riccati ordinary differential equation in the martingale case. This reduction lies in the foundations of the Kalman--Bucy approach to linear optimal filtering. In this paper we consider a basic Kalman--Bucy model with noises, generated by independent fractional Brownian motions, and develop a new method of asymptotic analysis of the integro-differential filtering equation arising in this case. We establish existence of the steady-state error limit and find its asymptotic scaling in the high signal-to-noise regime. Closed form expressions are derived in a number of important cases.

Parameter-dependent filtering of Gaussian processes in Hilbert spaces

The filtering problem for non-Markovian Gaussian processes on rigged Hilbert spaces is considered. Continuous dependence of the filter and observation error on parameters which may be present both in the signal and observation processes is proved. The general results are applied to signals governed by stochastic heat equations driven by distributed or pointwise fractional noise. The observation process may be a noisy observation of the signal at given points in the domain, the position of which may depend on the parameter.

Modelling cycles in climate series: The fractional sinusoidal waveform process

The paper proposes a novel model for time series displaying persistent stationary cycles, the fractional sinusoidal waveform process. The underlying idea is to allow the parameters that regulate the amplitude and phase to evolve according to fractional noise processes. Its advantages with respect to popular alternative specifications, such as the Gegenbauer process, are twofold: the autocovariance function is available in closed form, which opens the way to exact maximum likelihood estimation; secondly, the model encompasses deterministic cycles, so that discrete spectra arise as a limiting case. A generalization of the process, featuring multiple components, an additive ‘red noise’ component and exogenous variables, provides the basic model for climate time series with mixed spectra. Our illustrations deal with the change in amplitude and phase of the intra-annual component of carbon dioxide concentrations in Mauna Loa, and with the estimation and the quantification of the contribution of orbital cycles to the variability of paleoclimate time series.

Low-loss superconducting aluminum microwave coplanar waveguide resonators on sapphires for the qubit readouts

The coplanar waveguide (CPW) microwave resonators have been widely applied for solid-state quantum computation and single-photon detection. Based on the physical analysis for the high fidelity readouts of the qubit(s), in this paper we design and then fabricate accordingly the desired aluminum (Al) film quarter-wavelength resonators on sapphire substrates. The ultra-low temperature measurement results show that the linearity of the internal quality factor under the proper driving power is satisfied well for the high fidelity readouts of the qubit(s); and the phase- and amplitude fractional frequency noises are significantly less for avoiding the potential information false alarms of the detected qubit during the readouts. With the demonstrated qubit-resonator experimental parameters, we argue that the designed and fabricated quarter-wavelength CPW resonators, with optimized parameters, can be used to implement the desired weakly perturbing readout measurements of the solid-state qubits on-chip.

On ill-posedness of nonlinear stochastic wave equations driven by rough noise

We highlight a fundamental ill-posedness issue for nonlinear stochastic wave equations driven by a fractional noise. Namely, if the noise becomes too rough (i.e., the sum of its Hurst indexes becomes too small), then there is essentially no hope to provide a systematic interpretation of the model, whether directly or through a Wick-type renormalization procedure. This phenomenon can be compared with the situation of a general SDE driven by a two-dimensional fractional noise of index H≤14.Our results clarify and extend previous similar properties exhibited in Deya (2020) or in Oh and Okamoto (2021).