Analytical probability density functions for LMS adaptive filters using the Fokker-Planck equation

1991 ◽  
Author(s):  
S.T. Alexander ◽  
V.L. Stonic
Keyword(s):  
Probability Density ◽  
Adaptive Filters ◽  
Planck Equation ◽  
Probability Density Functions ◽  
Density Functions ◽  
Fokker Planck Equation ◽  
Fokker Planck

scholarly journals On universal features of the turbulent cascade in terms of non-equilibrium thermodynamics

2018 ◽  
Vol 848 ◽  
pp. 117-153 ◽  
Author(s):  
Nico Reinke ◽  
André Fuchs ◽  
Daniel Nickelsen ◽  
Joachim Peinke
Keyword(s):  
Probability Density ◽  
Entropy Production ◽  
Turbulent Flows ◽  
Planck Equation ◽  
Small Scale ◽  
Equilibrium Thermodynamics ◽  
Fokker Planck Equation ◽  
Velocity Increments ◽  
Turbulent Cascade ◽  
Fokker Planck

Features of the turbulent cascade are investigated for various datasets from three different turbulent flows, namely free jets as well as wake flows of a regular grid and a cylinder. The analysis is focused on the question as to whether fully developed turbulent flows show universal small-scale features. Two approaches are used to answer this question. First, two-point statistics, namely structure functions of longitudinal velocity increments, and, second, joint multiscale statistics of these velocity increments are analysed. The joint multiscale characterisation encompasses the whole cascade in one joint probability density function. On the basis of the datasets, evidence of the Markov property for the turbulent cascade is shown, which corresponds to a three-point closure that reduces the joint multiscale statistics to simple conditional probability density functions (cPDFs). The cPDFs are described by the Fokker–Planck equation in scale and its Kramers–Moyal coefficients (KMCs). The KMCs are obtained by a self-consistent optimisation procedure from the measured data and result in a Fokker–Planck equation for each dataset. Knowledge of these stochastic cascade equations enables one to make use of the concepts of non-equilibrium thermodynamics and thus to determine the entropy production along individual cascade trajectories. In addition to this new concept, it is shown that the local entropy production is nearly perfectly balanced for all datasets by the integral fluctuation theorem (IFT). Thus, the validity of the IFT can be taken as a new law of the turbulent cascade and at the same time independently confirms that the physics of the turbulent cascade is a memoryless Markov process in scale. The IFT is taken as a new tool to prove the optimal functional form of the Fokker–Planck equations and subsequently to investigate the question of universality of small-scale turbulence in the datasets. The results of our analysis show that the turbulent cascade contains universal and non-universal features. We identify small-scale intermittency as a universality breaking feature. We conclude that specific turbulent flows have their own particular multiscale cascades, in other words, their own stochastic fingerprints.

scholarly journals Linking Nonlinearity and Non-Gaussianity of Planetary Wave Behavior by the Fokker–Planck Equation

2005 ◽  
Vol 62 (7) ◽  
pp. 2098-2117 ◽  
Author(s):  
Judith Berner
Keyword(s):  
Stochastic Model ◽  
Probability Density ◽  
Planetary Wave ◽  
Multiplicative Noise ◽  
Planck Equation ◽  
Fokker Planck Equation ◽  
State Dependent ◽  
Nonlinear Stochastic Model ◽  
Non Gaussian ◽  
Fokker Planck

Abstract To link prominent nonlinearities in the dynamics of 500-hPa geopotential heights to non-Gaussian features in their probability density, a nonlinear stochastic model of atmospheric planetary wave behavior is developed. An analysis of geopotential heights generated by extended integrations of a GCM suggests that a stochastic model and its associated Fokker–Planck equation call for a nonlinear drift and multiplicative noise. All calculations are carried out in the reduced phase space spanned by the leading EOFs. It is demonstrated that this nonlinear stochastic model of planetary wave behavior captures the non-Gaussian features in the probability density function of atmospheric states to a remarkable degree. Moreover, it not only predicts global temporal characteristics, but also the nonlinear, state-dependent divergence of state trajectories. In the context of this empirical modeling, it is discussed on which time scale a stochastic model is expected to approximate the behavior of a continuous deterministic process. The reduced model is then used to determine the importance of the nonlinearities in the drift and the role of the multiplicative noise. While the nonlinearities in the drift are crucial for a good representation of planetary wave behavior, multiplicative (i.e., state dependent) noise is not absolutely essential. It is found that a major contributor to the stochastic component is the Branstator–Kushnir oscillation, which acts as a fluctuating force for physical processes with even longer time scales, like those that project on the Arctic Oscillation pattern. In this model, the oscillation is represented by strongly correlated noise.

Cooperative target localization using multiple UAVs with out-of-sequence measurements

2017 ◽  
Vol 89 (1) ◽  
pp. 112-119 ◽  
Author(s):  
Xiaogang Wang ◽  
Wutao Qin ◽  
Yuliang Bai ◽  
Naigang Cui
Keyword(s):  
Time Delay ◽  
Probability Density ◽  
Gaussian Noise ◽  
Nonlinear Filtering ◽  
Planck Equation ◽  
Target Localization ◽  
Content Type ◽  
Fokker Planck Equation ◽  
Delay Measurement ◽  
Fokker Planck

Purpose The time delay would occurs when the measurements of multiple unmanned aerial vehicles (UAVs) are transmitted to the date processing center during cooperative target localization. This problem is often named as the out-of-sequence measurement (OOSM) problem. This paper aims to present a nonlinear filtering based on solving the Fokker–Planck equation to address the issue of OOSM. Design/methodology/approach According to the arrival time of measurement, the proposed nonlinear filtering can be divided into two parts. The non-delay measurement would be fused in the first part, in which the Fokker–Planck equation is utilized to propagate the conditional probability density function in the forward form. The time delay measurement is fused in the second part, in which the Fokker–Planck is used in the backward form approximately. The Bayes formula is applied in both parts during the measurement update. Findings Under the Bayesian filtering framework, this nonlinear filtering is not only suitable for the Gaussian noise assumption but also for the non-Gaussian noise assumption. The nonlinear filtering is applied to the cooperative target localization problem. Simulation results show that the proposed filtering algorithm is superior to the previous Y algorithm. Practical implications In this paper, the research shows that a better performance can be obtained by fusing multiple UAV measurements and treating time delay in measurement with the proposed algorithm. Originality/value In this paper, the OOSM problem is settled based on solving the Fokker–Planck equation. Generally, the Fokker–Planck equation can be used to predict the probability density forward in time. However, to associate the current state with the state related to OOSM, it would be used to propagate the probability density backward either.

scholarly journals Probability density function (PDF) models for particle transport in porous media

2020 ◽  
Vol 11 (1) ◽  
Author(s):  
Matteo Icardi ◽  
Marco Dentz
Keyword(s):  
Porous Media ◽  
Probability Density ◽  
Turbulent Flows ◽  
Concentration Field ◽  
Planck Equation ◽  
Transport Processes ◽  
Fokker Planck Equation ◽  
Mixing Problem ◽  
And Diffusion ◽  
Fokker Planck

AbstractMathematical models based on probability density functions (PDF) have been extensively used in hydrology and subsurface flow problems, to describe the uncertainty in porous media properties (e.g., permeability modelled as random field). Recently, closer to the spirit of PDF models for turbulent flows, some approaches have used this statistical viewpoint also in pore-scale transport processes (fully resolved porous media models). When a concentration field is transported, by advection and diffusion, in a heterogeneous medium, in fact, spatial PDFs can be defined to characterise local fluctuations and improve or better understand the closures performed by classical upscaling methods. In the study of hydrodynamical dispersion, for example, PDE-based PDF approach can replace expensive and noisy Lagrangian simulations (e.g., trajectories of drift-diffusion stochastic processes). In this work we derive a joint position-velocity Fokker–Planck equation to model the motion of particles undergoing advection and diffusion in in deterministic or stochastic heterogeneous velocity fields. After appropriate closure assumptions, this description can help deriving rigorously stochastic models for the statistics of Lagrangian velocities. This is very important to be able to characterise the dispersion properties and can, for example, inform velocity evolution processes in continuous time random walk dispersion models. The closure problem that arises when averaging the Fokker–Planck equation shows also interesting similarities with the mixing problem and can be used to propose alternative closures for anomalous dispersion.

scholarly journals OPTIMAL CONTROL OF PROBABILITY DENSITY FUNCTIONS OF STOCHASTIC PROCESSES

2010 ◽  
Vol 15 (4) ◽  
pp. 393-407 ◽  
Author(s):  
Mario Annunziato ◽  
Alfio Borzì
Keyword(s):  
Optimal Control ◽  
Stochastic Processes ◽  
Probability Density ◽  
Control Strategy ◽  
Time Windows ◽  
Probability Density Functions ◽  
Density Functions ◽  
Optimal Control Strategy ◽  
Control Laws ◽  
Fokker Planck

A Fokker‐Planck framework for the formulation of an optimal control strategy of stochastic processes is presented. Within this strategy, the control objectives are defined based on the probability density functions of the stochastic processes. The optimal control is obtained as the minimizer of the objective under the constraint given by the Fokker‐Planck model. Representative stochastic processes are considered with different control laws and with the purpose of attaining a final target configuration or tracking a desired trajectory. In this latter case, a receding‐horizon algorithm over a sequence of time windows is implemented.

scholarly journals Time-Dependent Probability Density Functions and Attractor Structure in Self-Organised Shear Flows

Entropy ◽  
2018 ◽  
Vol 20 (8) ◽  
pp. 613 ◽  
Author(s):  
Quentin Jacquet ◽  
Eun-jin Kim ◽  
Rainer Hollerbach
Keyword(s):  
Probability Density ◽  
Shear Flows ◽  
Numerical Solutions ◽  
Planck Equation ◽  
Probability Density Functions ◽  
Time Dependent ◽  
Parameter Study ◽  
Density Functions ◽  
Stochastic Forcing ◽  
Mean Values

We report the time-evolution of Probability Density Functions (PDFs) in a toy model of self-organised shear flows, where the formation of shear flows is induced by a finite memory time of a stochastic forcing, manifested by the emergence of a bimodal PDF with the two peaks representing non-zero mean values of a shear flow. Using theoretical analyses of limiting cases, as well as numerical solutions of the full Fokker–Planck equation, we present a thorough parameter study of PDFs for different values of the correlation time and amplitude of stochastic forcing. From time-dependent PDFs, we calculate the information length ( L ), which is the total number of statistically different states that a system passes through in time and utilise it to understand the information geometry associated with the formation of bimodal or unimodal PDFs. We identify the difference between the relaxation and build-up of the shear gradient in view of information change and discuss the total information length ( L ∞ = L ( t → ∞ ) ) which maps out the underlying attractor structures, highlighting a unique property of L ∞ which depends on the trajectory/history of a PDF’s evolution.

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