Backward Fokker-planck Equation for Determination Of
نویسندگان
چکیده
1. INTRODUCTION It is widely recognized that uncertainty in atmospheric and oceanic models can be traced back to two factors (Lorenz 1984a, 1987). First, in defining the state of atmosphere (or ocean), a number of errors are involved arising from the finite resolution of measurement or from discretization in a numerical experiment, as a result of which small-scale " subgrid " processes are either discarded or parameterized. Second, once present, small errors of the kind mentioned above trigger a complex response leading to their subsequent amplification. The model predictability versus boundary condition error was discussed by Chu (1999) using the Lorenz system. The model predictability can be measured by two parameters: instantaneous error (IE) and predictability time (PT). The IE and PT are used for models with and without given initial condition errors, respectively. The IE measure is widely used for model evaluation. The predictability is regarded as the model error growth due to the initial condition error. This implies that the initial condition error should be given. The evaluation process becomes to study the stability of the dynamical system with a given initial condition error and to determine either the leading (largest) Lyapunov exponent (e.g., Lorenz 1969) or the amplification factors calculated from the leading singular vectors (e.g., Farrell and Ioannou 1996 a, b). It is well known that the stability analysis using the Lyapunov exponents and the singular vectors is not unique (Has'minskii, 1980). Probabilistic stability analysis becomes available in practical application (Ehrendorfer 1994 a, b; Nicolis 1992). The statistical properties of the prediction error are described through the probability density function (PDF) satisfying the Liouville equation or the Fokker-Plank equation. Solving this
منابع مشابه
Pseudo-spectral Matrix and Normalized Grunwald Approximation for Numerical Solution of Time Fractional Fokker-Planck Equation
This paper presents a new numerical method to solve time fractional Fokker-Planck equation. The space dimension is discretized to the Gauss-Lobatto points, then we apply pseudo-spectral successive integration matrix for this dimension. This approach shows that with less number of points, we can approximate the solution with more accuracy. The numerical results of the examples are displayed.
متن کاملNumerical Studies and Simulation of the Lower Hybrid Waves Current Drive by using Fokker – Planck Equation in NSST and HT-7 Tokamaks
Recent experiments on the spherical tokamak have discovered the conditions to create a powerful plasma and ensure easy shaping and amplification of stability, high bootstrap current and confinement energy. The spherical tours (ST) fusion energy development path is complementary to the tokamak burning plasma experiment such as NSTX and higher toroidal beta regimes and improves the design of a po...
متن کاملFokker-planck Equations: Uncertainty in Network Security Games and Information
We have significant accomplishments on uncertainty quantification in inverse problems for dynamical systems, generalized sensitivity and optimal design of experiments, elasticity and viscoelasticity modeling for buried target detection, and general inverse problem methodology for proliferating populations. Our efforts have continued on development of efficient accurate integration of Fokker Pla...
متن کاملStatistical Analysis of Neural Data: the Integrate-and-fire Neuron and Other Continuous-time State-space Models *
3 The “Fokker-Planck” equation is a partial differential equation that controls the evolution of the forward (and backward) probabilities 9 3.1 Deriving the “free” Fokker-Planck equation (no spike observations) . . . . . . 10 3.1.1 Conductance-based model . . . . . . . . . . . . . . . . . . . . . . . . . 12 3.1.2 Computing mean firing rates in a network of GLM neurons . . . . . . 13 3.2 Incorpo...
متن کاملThe Quadrature Discretization Method (QDM) in comparison with other numerical methods of solution of the Fokker–Planck equation for electron thermalization
The determination of the relaxation of electrons in atomic gases continues to be an important physical problem. The main interest is the determination of the time scale for the thermalization of electrons in different moderators and the nature of the time-dependent electron energy distribution. The theoretical basis for the study of electron thermalization is the determination of the electron d...
متن کامل