Exact solutions for the probability density function of turbulent scalar fields

Summary The basic property of a turbulent scalar field is its probability density function p( x, t). Here, for the first time, some exact solutions for p( x, t) are derived and discussed. These apply to the case of a finite mass of passive scalar — called a cloud for short — dispersing in simple, but conceptually important, turbulent flows, namely those associated with constant rates of strain. Extensions of the solutions to cases where the cloud is meandering, and where there are several clouds, are obtained. Applications of the results are discussed, with particular emphasis on their potential value for testing and validating approximate closure schemes applied to the evolution equation for p( x, t).     

New super-kurtic probability density function for use in computer simulation

The philosophy of computer simulation and its application to hydrological processes is described in this paper. The structure of natural hydrologic time processes is indicated and the techniques to filter out white noise is explained. The limitations of the well-known probability density functions (PDF) such as the Gaussian, Pearson’s and Johnson’s etc. in hydrologic applications are set forth. A new super-kurtic PDF developed by the author specially for hydrological processes is introduced and a numerical example is given.     

A probability density closure model for turbulence

Summary Starting from the Navier-Stokes equation, the evolution equation of the first order probability density functions of turbulence is derived. Closure assumptions for the pressure terms are introduced and the closed pdf equation is obtained. The resulting evolution equation takes the form of the Fokker-Planck equation and contains the existing Langevin based models as special cases. It is shown that the model predicts the known behaviors of turbulence in the limit of extremely high Reynolds number as well as a decaying turbulence. The moment equations are also examined. It is shown that the model is consistent with the recent second and third order closure models of turbulence.     

Radial basis function neural networks solution for stationary probability density function of nonlinear stochastic systems

A radial basis function neural networks (RBF-NN) solution of the reduced Fokker–Planck-Kolmogorov (FPK) equation is proposed in this paper. The activation functions consist of normalized Gaussian probability density functions (PDFs). The use of normalized Gaussian PDFs leads to a simple constraint on the coefficients for normalization of the RBF-NN solution, which as a constraint is imposed with the help of the method of Lagrange multiplier. The relationship between the proposed RBF-NN PDF solution and the generalized cell mapping with short-time Gaussian approximation is discussed, which provides a justification for Gaussian PDFs with varying means and variances in the state space. The optimal number of neurons or activation functions, which leads to the smallest error, is investigated. Four examples are presented to show the effectiveness of the proposed solution method. The results indicate that the proposed solution method is a very efficient and accurate way to compute the stationary PDF of nonlinear stochastic systems. It is also found that the distribution of the optimal coefficients as a function of the mean of Gaussian activation functions is similar to the steady-state PDF solution. Finally, we should point out that an important advantage of the RBF-NN method over methods such as finite element and finite difference is its ability to obtain solutions of the FPK equation for multi-degree-of-freedom stochastic systems.     

A radial basis function approach to compute the first-passage probability density function in two-dimensional jump-diffusion models for financial and other applications

We consider the problem of computing the survival (first-passage) probability density function of jump-diffusion models with two stochastic factors. In particular the Fokker–Planck partial integro-differential equation associated to these models is solved using a meshless collocation approach based on radial basis functions (RBF). To enhance the computational efficiency of the method, the calculation of the jump integrals is performed using a suitable Chebyshev interpolation procedure. In addition, the RBF discretization is carried out in conjunction with an ad hoc change of variables, which allows to use radial basis functions with equally spaced centers and at the same time yields an accurate resolution of the gradients of the survival probability density function near the barrier. Numerical experiments are presented showing that the RBF approach is extremely accurate and fast, and performs significantly better than the conventional finite difference method.     

On joint stationary probability density function of nonlinear dynamic systems

Summary In this paper a general procedure is developed to differentiate the independent problem of nonlinear dynamic systems from the state of statistical stationary conditions. We give four examples to illustrate the application of the differential method. One example demonstrates that the joint probability density function of coordinate and velocity possesses the form of the separated product for both nonlinear damping and nonlinear restoring force systems.     

Explicit solutions for the response probability density function of nonlinear transformations of static random inputs

This work is the second paper of two companion ones. Both of them show the use of a new version of the Probabilistic Transformation Method (PTM) for finding the probability density function (pdf) of a limited number of response quantities in the transformations of static random inputs. This is made without performing multi-dimensional integrals of the response total joint pdf for saturating the non-interested variables. While in the first paper the linear transformations have been considered, in the present one some nonlinear systems are taken into account. In particular, first the case when the loads on a linear structural system are a nonlinear combination of static random inputs is studied. Then the attention is placed on the case of nonlinear structural systems, for which the new version of the PTM allows to determine approximated, but accurate, results.     

A method for the evaluation of the response probability density function of some linear dynamic systems subjected to non-Gaussian random load

This paper deals with the characterization of the random response of linear systems subjected to stochastic load. It proposes a new method based on the new version of the Probabilistic Transformation Method (PTM) that allows obtaining, with a very low computational effort, the probability density function of the response. An important aspect of the proposed approach is the ability to join directly the pdfs of the input load with those of the response. Based on the step-by-step integration method, explicit solutions will be proposed for the random response of systems loaded by seismic and windy sampled inputs.     

Explicit solutions for the response probability density function of linear systems subjected to random static loads

In the present work a new version of the Probabilistic Transformation Method (PTM) has been reported for the study of linear systems subjected to static random loads. Even if this application could appear trivial, it allows to find some exact results, difficulty obtainable by other approaches. In particular, some interesting results have been obtained in the case of uniformly distributed random loads. For a generic vector of random loads this version of the PTM has allowed to obtain the characteristic function (cf) of any response elements in a very simple and effective way.     

A closed equation for the joint probability density function of the magnitudes of fluctuations of a turbulent scalar reacting field and its gradient

We derived a closed system of equations for calculating the single-point joint probability density function (JPDF) of the magnitudes of fluctuations of a scalar reacting field and its gradient. The system of equations includes an equation for the JPDF and two equations for functions that describe the distribution of turbulent energy and of the reacting-scalar intensity over various length scales. The latter functions are necessary for calculation of the time-dependent coefficients in the equation for the JPDF. Translated from Inzhenerno-Fizicheskii Zhurnal, Vol. 71, No. 5, pp. 827–849, September–October, 1998.     

Computing the survival probability density function in jump-diffusion models: A new approach based on radial basis functions

We propose a numerical method to compute the survival (first-passage) probability density function in jump-diffusion models. This function is obtained by numerical approximation of the associated Fokker–Planck partial integro-differential equation, with suitable boundary conditions and delta initial condition. In order to obtain an accurate numerical solution, the singularity of the Dirac delta function is removed using a change of variables based on the fundamental solution of the pure diffusion model. This approach allows to transform the original problem to a regular problem, which is solved using a radial basis functions (RBFs) meshless collocation method. In particular the RBFs approximation is carried out in conjunction with a suitable change of variables, which allows to use radial basis functions with equally spaced centers and at the same time to obtain a sharp resolution of the gradients of the survival probability density function near the barrier. Numerical experiments are presented in which several different kinds of radial basis functions are employed. The results obtained reveal that the numerical method proposed is extremely accurate and fast, and performs significantly better than a conventional finite difference approach.     

A scatter search heuristic for maximising the net present value of a resource-constrained project with fixed activity cash flows

In this paper, we present a meta-heuristic algorithm for the resource-constrained project scheduling problem with discounted cash flows. We assume fixed payments associated with the execution of project activities and develop a heuristic optimisation procedure to maximise the net present value of a project subject to the precedence and renewable resource constraints. We investigate the use of a bi-directional generation scheme and a recursive forward/backward improvement method from literature and embed them in a meta-heuristic scatter search framework. We generate a large dataset of project instances under a controlled design and report detailed computational results. The solutions and project instances can be downloaded from a website in order to facilitate comparison with future research attempts.     

Maximizing the net present value of a project with linear time-dependent cash flows

The paper studies the unconstrained project-scheduling problem with discounted cash flows where the cash flow functions are assumed to be linear-dependent on the completion times of the corresponding activities. Each activity of this unconstrained project-scheduling problem has a known deterministic cash flow function that is linear and non-increasing in time. Progress payments and cash outflows occur at the completion times of activities. The objective is to schedule the activities in order to maximize the net present value (npv) subject to the precedence constraints and a fixed deadline. Despite the growing amount of research concerning the financial aspects in project scheduling, little research has been done on the problem with time-dependent cash flow functions. Nevertheless, this problem gives an incentive to solve more realistic versions of project-scheduling problems with financial objectives. We introduce an extension of an exact recursive algorithm that has been used in solving the max-npv problem with time-independent cash flow functions and which is embedded in an enumeration procedure. The recursive search algorithm schedules the activities as soon as possible and searches for sets of activities to shift towards the deadline in order to increase the npv. The enumeration procedure enumerates all sets of activities for which such a shift has not been made but could, eventually, have been advantageous. The procedure has been coded in Visual C++ v.4.0 under Windows NT and has been validated on a randomly generated problem set.