Random Partial Differential Equation Research Articles

A Deterministic Approach To Stochastic Optimal Control With Application To Anticipative Control

Using the decomposition of solution of SDE, we consider the stochastic optimal control problem with anticipative controls as a family of deterministic control problems parametrized by the paths of the driving Wiener process and of a newly introduced Lagrange multiplier stochastic process (nonanticipativity equality constraint). It is shown that the value function of these problems is the unique global solution of a robust equation (random partial differential equation) associated to a linear backward Hamilton-Jacobi-Bellman stochastic partial differential equation (HJB SPDE). This appears as limiting SPDE for a sequence of random HJB PDE's when linear interpolation approximation of the Wiener process is used. Our approach extends the Wong-Zakai type results [20] from SDE to the stochastic dynamic programming equation by showing how this arises as average of the limit of a sequence of deterministic dynamic programming equations. The stochastic characteristics method of Kunita [13] is used to represent the ...

A Neumann expansion approach to flow through heterogeneous formations

A stochastic approach is used for the study of flow through highly heterogeneous aquifers. The mathematical model is represented by a random partial differential equation in which the permeability and the porosity are considered to be random functions of position, defined by the average value, constant standard deviation and autocorrelation function characterized by the integral scale. The Laplace transform of the solution of the random partial differential equation is first written as a solution of a stochastic integral equation. This integral equation is solved using a Neumann series expansion. Conditions of convergence of this series are investigated and compared with the convergence of the perturbation series. For mean square convergence, the Neumann expansion method may converge for a larger range of variability in permeability and porosity than the classic perturbation method. Formal expressions for the average and for the correlation moments of the pressure are obtained. The influence of the variability of the permeability and porosity on pressure is analyzed for radial flow. The solutions presented for the pressure at the well, as function of the permeability coefficient of variation, may be of practical interest for evaluating the efficiency of well stimulation operations, such as hydraulic fracturing or acidizing methods, aimed at increasing the permeability around the well.

Analysis of stochastic systems in continuum physics: Some developments of the stochastic adaptive interpolation method

This paper deals with the analysis of stochastic systems in continuum physics which are modelled by a random partial differential equation using the so called stochastic adaptive interpolation method. This paper proposes some developments of the method, mainly oriented towards the solution of stochastic inverse problems.

Analysis of polarization fluctuation in single-mode optical fibers with continuous random coupling

A new analysis method is developed to study the polarization fluctuation in conventional single-mode optical fibers. A random partial differential equation is derived to describe the continuous random coupling between two eigenpolarization modes caused by the perturbation of permittivity along fibers. The calculation formulas for the polarization fluctuation are given. By these formulas, it is proved that the average noise power of the fluctuation at the fiber output depends on the fiber length, the correlation length of perturbation, and the variance of disturbances.