Fokker-Planck方程的非古典势对称群及新显式解

本文利用一种新方法对Fokker- Planck方程的非古典势对称群生成元进行研究,找到方程的几个非古典势对称群生成元,并采用非古典对称群方法由这些对称群生成元构造得到Fokker- Planck方程的相应显式解.这些新显式解不能由Fokker -Planck方程本身的Lie对称或Li-e B cklund对称来获得.在验证所求得显式解的过程中,还发现并得到了另外几个显式解.这些新显式解则不能由Fokker -Planck方程本身的Lie对称,Lie- B cklund对称或非古典势对称来获得.文章表明,通过偏微分方程的非古典势对称群生成元来寻找其显式解是可能的.     

Application of multi-scale finite element methods to the solution of the Fokker–Planck equation

This paper presents an application of multi-scale finite element methods to the solution of the multi-dimensional Fokker–Planck equation. The Fokker–Planck, or forward Kolmogorov, equation is a degenerate convective–diffusion equation arising in Markov-Process theory. It governs the evolution of the transition probability density function of the response of a broad class of dynamical systems driven by Gaussian noise, and completely describes the response process. Analytical solutions for the Fokker–Planck equation have been developed for only a limited number of low-dimensional systems, leading to a large body of approximation theory. One such approach successfully applied to the solution of these problems in the past is the finite element method, though for systems of dimension three or less. In this paper, a multi-scale finite element method is applied to the Fokker–Planck equation in an effort to develop a formulation that can yield higher accuracy on cruder spatial discretizations, thus reducing the computational overhead associated with large scale problems that arise in higher dimensions.     

Analysis of nonlinear stochastic systems by means of the Fokker–Planck equation†

Nonlinear systems disturbed by Gaussian white noises (or by signals obtained from Gaussian white noises) can sometimes be analysed by setting up and solving the Fokker–Planck equation for the probability density in state space. In the present paper a simplified derivation of the Fokker–Planck equation is given. The uniqueness of the steady-state solution is discussed. Steady-state solutions are obtained for certain classes of system. These solutions correspond to or slightly generalize the Maxwell–Boltzmann distribution which is well known in classical statistical mechanics     

Hermite approximation of a hyperbolic Fokker–Planck optimality system to control a piecewise-deterministic process

The Hermite spectral approximation of a hyperbolic Fokker–Planck (FP) optimality system arising in the control of an unbounded piecewise-deterministic process (PDP) is discussed. To control the probability density function (PDF) corresponding to the PDP process, an optimal control based on an FP strategy is considered. The resulting optimality system consists of a hyperbolic system with opposite-time orientation and an integral optimality condition equation. A Hermite spectral discretisation is investigated to approximate solutions to the optimality system in unbounded domains. It is proven that the proposed scheme satisfies the conservativity requirement of the PDFs. The spectral convergence rate of the discretisation scheme is proved and validated by numerical experiments.     

A Semi-Lagrangian Method for 3-D Fokker Planck Equations for Stochastic Dynamical Systems on the Sphere

In this paper, we consider stochastic dynamical systems on the sphere and the associated Fokker–Planck equations. A semi-Lagrangian method combined with a Finite Volume discretization of the sphere is presented to solve the Fokker–Planck equation. The method is applied to a typical problem in fiber dynamics and textile production. The numerical results are compared to explicit solutions and Monte-Carlo solutions.     

Finite element method analysis of Fokker–Plank equation in stationary and evolutionary versions

The problems that often arise in stochastic dynamics can be investigated using the Fokker–Planck (FP) equation. The response of a such systems being subjected to additive and/or multiplicative random noise is represented by probability density function (PDF) that gives the full information about a response random character. Various analytic and semi-analytic solution methods have been developed for various systems to obtain results requested. However numerical approaches offer a powerful alternative. In particular the Finite Element Method (FEM) seems to be very effective. A couple of single dynamic linear/non-linear (Duffing and Van Der Pol type) systems under additive and multiplicative random excitations are discussed using FEM as a solution tool of the FP equation. The resulting PDFs are analyzed and if the analytic results exist mutually compared.     

Stochastic elastic–plastic finite elements

A computational framework has been developed for simulations of the behavior of solids and structures made of stochastic elastic–plastic materials. Uncertain elastic–plastic material properties are modeled as random fields, which appear as the coefficient term in the governing partial differential equation of mechanics. A spectral stochastic elastic–plastic finite element method with Fokker–Planck–Kolmogorov equation based probabilistic constitutive integrator is proposed for solution of this non-linear (elastic–plastic) partial differential equation with stochastic coefficient. To this end, the random field material properties are discretized, in both spatial and stochastic dimension, into finite numbers of independent basic random variables, using Karhunen–Loève expansion. Those random variables are then propagated through the elastic–plastic constitutive rate equation using Fokker–Planck–Kolmogorov equation approach, to obtain the evolutionary material properties, as the material plastifies. The unknown displacement (solution) random field is then assembled - using polynomial chaos - as a function of known basic random variables and unknown deterministic coefficients, which are obtained by minimizing the error of finite representation, by Galerkin technique.     

On the accuracy of the direct discrete simulation of the Landau collision integral by the Boltzmann integral

The Landau (Fokker–Planck) integral of Coulomb collisions is an integral component of the physical and mathematical models of both laboratory and space plasma, in which the intermediate collisionality regime is important. This paper discusses the method for the direct statistical simulation of the Monte Carlo type for a kinetic equation with a nonlinear operator of the Landau–Fokker–Planck (LFP) Coulomb collisions. This method is based on the approximation of the Landau collision integral by the Boltzmann collision integral. The paper has two main objectives: firstly, to obtain numerical estimates of the order of approximation of the Landau collision integral by the Boltzmann integral and, secondly, to explore the possibilities of optimizing the algorithm for multiply charged ions. The results are illustrated by the calculations of the problem on the relaxation of the initial distribution to equilibrium for one and two components.     

An Explicit Solution to the Fokker-Planck Equation for an Ordinary Differential Equation†

An explicit, steady–state solution is found to the Fokker–Planck equation for an nth order, ordinary linear differential equation with constant coefficients excited by a single white noise source by making use of a Liapunov function. Liapunov functions are also used to derive a more general result of Wang and Uhlenbeck.     

Stochastic dynamics simulation with generalized interval probability

Two types of uncertainties are generally recognized in modelling and simulation, including variability caused by inherent randomness and incertitude due to the lack of perfect knowledge. In this paper, a generalized interval-probability theory is used to model both uncertainty components simultaneously, where epistemic uncertainty is quantified by generalized interval in addition to probability measure. Conditioning, independence, and Markovian probabilities are uniquely defined in generalized interval probability such that its probabilistic calculus resembles that in the classical probability theory. A path-integral approach can be taken to solve the interval Fokker–Planck equation for diffusion processes. A Krylov subspace projection method is proposed to solve the interval master equation for jump processes. Thus, the time evolution of both uncertainty components can be simulated simultaneously, which provides the lower and upper bound information of evolving probability distributions as an alternative to the traditional sensitivity analysis.     

Two alternative methods for solving the Vlasov Fokker–Planck equation

The expansion of moments technique for generating short time expansions for the moments with the distribution function is used to solve the reduced Vlasov Fokker–Planck equation. The obtained results are compared with those found by other theories such as the operator technique. The results obtained by expansion of moments confirm the correctness of those obtained by the operator method. The method is straightforward and concise, and its applications are promising and can be applied to other moment equations arising in physics.     

Biologically Plausible Associative Memory: Continuous Unit Response + Stochastic Dynamics

A neural network model of associative memory is presented which unifies the two historically more relevant enhancements to the basic Little-Hopfield discrete model: the graded response units approach and the stochastic, Glauber-inspired model with a random field representing thermal fluctuations. This is done by casting the retrieval process of the model with graded response neurons, into the framework of a diffusive process governed by the Fokker-Plank equation, which leads to a Langevin system describing the process at a microscopic level, while the time evolution of the probability density function is governed by a multivariate Fokker Planck equation operating over the space of all possible activation patterns. The present unified approach has two notable features: (i) greater biological plausibility and (ii) ability to escape local minima of energy (associated with spurious memories), which makes it a potential tool for those complex optimization problems for which the previous models failed.     

Application of Markov chains on image enhancement

Stochastic image processing tools have been widely used in digital image processing in order to improve the quality of the images. Markov process is one of the well-known mathematical modeling tools in stochastic theory. In this study, a Markov chain model has been developed and applied to image denoising. The transition probabilities were obtained from Fokker–Planck diffusion equation. According to the results, the proposed Markov chain model supplies very good peak signal to noise ratio values along with low computational cost.     

A Fokker–Planck equation method predicting Buffer occupancy in a single queue

The VIrtual Predictor BuffER (VIPER) algorithm is a novel algorithm for performing online prediction of the buffer space requirement of each competing traffic stream. It accomplishes its task through the employment of a theoretical, infinite capacity, virtual buffer. Information acquired from the virtual buffer is used to construct a probability distribution function that is based on the Fokker–Planck equation. This distribution function is central to the VIPER algorithm and is used to compute the queue length predictions. The predictions are shown to be promising.     

The PDF shape control of the state variable for a class of stochastic systems

The shape control of probability density function (PDF) is an important subject in stochastic systems. The PDF-shaping control study has ranged from linear systems to non-linear systems. In this paper we present a PDF-shaping control technique which is useful for a class of non-linear stochastic systems. Controlling the PDF shape requires designing a controller to make the state PDF follow the goal PDF; it is actually to determine the gains of the controller. As we know, the stationary PDF of the state variable is equivalent to the solution of the Fokker–Planck–Kolmogorov (FPK) equation arising from the stochastic system driven with Gaussian white noise. After designing the controller, we derive the solution with some parameters to the corresponding FPK equation, and then solve out the parameters in the solution by the linear-least-squares method, therefore obtaining the gains of the controller. Finally, simulation experiments have been carried out to verify the effectiveness of the approach.     

Probabilistic relaxation labelling using the Fokker–Planck equation

In this paper we develop a new formulation of probabilistic relaxation labelling using the theory of diffusion processes on graphs. Our aim is to tackle the problem of labelling objects consistently and unambiguously using information concerning label consistency and initial label probabilities. We abstract this problem using a support graph with each graph node an object-label assignment. Initial object-label probabilities then evolve across the graph under the governance of the Fokker–Planck equation in terms of an infinitesimal generator matrix computed from the edge weights of the support graph. In this way we effectively kernelise probabilistic relaxation. Encouraging results are obtained in applying the new relaxation process in the applications of scene labelling, data classification, and feature correspondence matching.     

A fully deterministic micro–macro simulation of complex flows involving reversible network fluid models

Micro–macro models associate the coarse-grained molecular scale of the kinetic theory to the macroscopic scale of continuum mechanics. The conservation equations are solved along with the microscopic equation or the so-called Fokker–Planck equation. In this paper, a micro–macro approach based on the separated representation introduced in 2 and 3 with the Stream-Tube method 10, 11, 12, 21 and 22 is implemented to study the main features of fiber and polymer networks solutions in complex flows. The Fokker–Planck equation, that defines the fluid microstructure, is solved using a separated representation strategy and is coupled to the macroscopic equations through the macroscopic extra-stress tensor evaluated at the microscopic level. Then, the flow kinematics is solved by applying the Stream-Tube method.     

Control of quasi non-integrable Hamiltonian systems for targeting a specified stationary probability density

An innovative design procedure for the feedback control of quasi non-integrable Hamiltonian systems to target a specified stationary probability density function (SPDF) is proposed based on the averaged Fokker–Planck–Kolmogorov equation. The control force is split into a conservative part and a dissipative part. The conservative part is designed to make the controlled system and the target SPDF having the same Hamiltonian structure. The dissipative part is then determined to make the target SPDF to be the stationary solution of the controlled system. Then, a Lyapunov function method is adopted for proving that the transient solution of the controlled system approaches to the target SPDF as time t → ∞. The simulation result for an example shows the effectiveness of the proposed control strategy.