Random Partial Differential Equation Research Articles

Statistical laws of a one-dimensional model of turbulent flows subject to an external random force

This paper provides mathematical and numerical analysis of a one-dimensional model of turbulent flow generating the anomalous cascade of the inviscid conserved quantity. The model is based on the generalized Constantin–Lax–Majda–DeGregorio (gCLMG) equation with viscous dissipation under a large-scale forcing. Suppose that the forcing function and the initial data are random variables defined on a certain probability space. Then, the equation is regarded as a random partial differential equation. We prove the global existence of a unique solution to the gCLMG equation, from which a stochastic process is defined. In addition, by approximating the solutions numerically by Galerkin approximation of random variables with generalized polynomial chaos, we confirm the existence of a steady distribution. We find that the steady distribution reproduces qualitatively the same cascades of the energy and the enstrophy spectra as those of a turbulent flow generated by randomly moving pulse (Matsumoto and Sakajo 2016 Phys. Rev. E 93 053101). We also investigate the structure functions, showing intermittency.

Robust a posteriori estimates for the stochastic Cahn-Hilliard equation

We derive a posteriori error estimates for a fully discrete finite element approximation of the stochastic Cahn-Hilliard equation. The a posteriori bound is obtained by a splitting of the equation into a linear stochastic partial differential equation and a nonlinear random partial differential equation. The resulting estimate is robust with respect to the interfacial width parameter and is computable since it involves the discrete principal eigenvalue of a linearized (stochastic) Cahn-Hilliard operator. Furthermore, the estimate is robust with respect to topological changes as well as the intensity of the stochastic noise. We provide numerical simulations to demonstrate the practicability of the proposed adaptive algorithm.

On Solutions Of Random Partial Differential Equations With Laplace Adomian Decomposition Method

In this study, random partial differential equations obtained by randomly choosing the coefficients or initial conditions of partial differential equations will be analyzed. With the help of Laplace Adomian Decomposition Method and Homotopy Analysis Method, approximate analytical solutions of random partial differential equations were obtained. Initial conditions and parameters are made into random variables with normal distribution and gamma distribution. Probability characteristics such as expected value, variance and confidence intervals of the obtained random partial differential equation are calculated. Obtained results will be plotted with the help of MATLAB (2013a), package program and random results will be interpreted.

Large time behaviour of semilinear stochastic partial differential equations perturbed by a mixture of Brownian and fractional Brownian motions

We study the trajectorywise blowup behaviour of a semilinear partial differential equation that is driven by a mixture of multiplicative Brownian and fractional Brownian motion, modelling different types of random perturbations. The linear operator is supposed to have an eigenfunction of constant sign, and we show its influence, as well as the influence of its eigenvalue and of the other parameters of the equation, on the occurrence of a blowup in finite time of the solution. We give estimates for the probability of finite time blowup and of blowup before a given fixed time. Essential tools are the mild and weak form of an associated random partial differential equation.

Numerical analysis of two-dimensional Navier–Stokes equations with additive stochastic forcing

Abstract We propose and study a temporal and a spatio-temporal discretisation of the two-dimensional stochastic Navier–Stokes equations in bounded domains supplemented with no-slip boundary conditions. Considering additive noise, we base its construction on the related nonlinear random partial differential equation, which is solved by a transform of the solution of the stochastic Navier–Stokes equations. We show a strong rate (up to) $1$ in probability for a corresponding discretisation in space and time (and space-time).

Liouville’s equations for random systems

Given a random system, a Liouville’s equation is an exact partial differential equation that describes the evolution of the probability density function of the solution. In this article, we derive Liouville’s equations for the first-order homogeneous semilinear random partial differential equation. This is done for all finite-dimensional distributions of the random field solution, starting with dimension one, then dimension two, and finally generalizing to any dimension. Several examples, including the linear advection equation with random coefficients, are treated. As a corollary, we deduce Liouville’s equations for path-wise stochastic integrals and nonlinear random ordinary differential equations.

A Spatiotemporal Prediction Model for Regional Scheduling of Shared Bicycles Based on the INLA Method

Dock-less bicycle-sharing programs have been widely accepted as an efficient mode to benefit health and reduce congestions. And modeling and prediction has always been a core proposition in the field of transportation. Most of the existing demand prediction models for shared bikes take regions as research objects; therefore, a POI-based method can be a beneficial complement to existing research, including zone-level, OD-level, and station-level techniques. Point of interest (POI) is the location description of spatial entities, which can reflect the cycling route characteristics for both commuting and noncommuting trips to a certain extent, and is also the main generating point and attraction point of shared-bike travel flow. In this study, we make an effort to model a POI-level cycling demand with a Bayesian hierarchical method. The proposed model combines the integrated nested Laplace approximation (INLA) and random partial differential equation (SPDE) to cope with the huge computation in the modeling process. In particular, we have adopted the dock-less bicycle-sharing rental records of Mobike as a case study to validate our method; the study area was one of the fastest growing urban districts in Shanghai in August 2016. The operation results show that the method can help better understand, measure, and characterize spatiotemporal patterns of bike-share ridership at the POI level and quantify the impact of the spatiotemporal effect on bicycle-sharing use.

Approximation of Probability Density Functions for PDEs with Random Parameters Using Truncated Series Expansions

The probability density function (PDF) of a random variable associated with the solution of a partial differential equation (PDE) with random parameters is approximated using a truncated series expansion. The random PDE is solved using two stochastic finite element methods, Monte Carlo sampling and the stochastic Galerkin method with global polynomials. The random variable is a functional of the solution of the random PDE, such as the average over the physical domain. The truncated series are obtained considering a finite number of terms in the Gram-Charlier or Edgeworth series expansions. These expansions approximate the PDF of a random variable in terms of another PDF, and involve coefficients that are functions of the known cumulants of the random variable. To the best of our knowledge, their use in the framework of PDEs with random parameters has not yet been explored.

A stochastic corporate claim value model with variable delay

Stochastic functional differential equation (SFDE) with constant delay has been used for modelling the price of a firm and European option. However, we can find some weak points such as bounded memory and availability of past event, when using delayed model with constant delay in some applications. Stochastic pantograph differential equation (SPDE) is very special SFDE with variable delay and unbounded memory. In this work, in order to overcome the weaknesses of delayed model with constant delay, we develop a nonlinear delayed model with variable delay for pricing corporate when claim value depends on firm value and time follows a nonlinear SPDE. We proved that, the proposed model is feasible. In addition, a random partial differential equation (RPDE) is derived which satisfied with the corporate claim of value depends on firm value and time. The solution of RPDE is evaluate the debt and equity of a firm. The proposed model with variable delay is more sufficiently flexible to fit real market data, particularly if practitioners are considering the future volatility depends on the past volatility with take into account the variable delay, which can reflect a historical image of volatility more randomly with unbounded memory.

Existence and uniqueness results for mild solutions of random impulsive abstract neutral partial differential equation over real axis

In this paper, we discuss the existence and uniqueness of mild solutions of random impulsive abstract neutral partial differential equations in a real separable Hilbert space. The results are obtained by using Leray-Schauder Alternative and Banach Contraction Principle. Finally an example is given to illustrate our problem.

A Probabilistic Analysis of Volume Transmission in the Brain

Volume transmission is a fundamental neural communication mechanism in which neurons in one brain nucleus modulate the neurotransmitter concentration in the extracellular space of a second nucleus. In this paper, we formulate and analyze a mathematical model of volume transmission to calculate the neurotransmitter concentration in the extracellular space. Our model consists of the diffusion equation in a bounded two- or three-dimensional domain that contains a set of interior holes that randomly switch between being either sources or sinks. The interior holes represent nerve varicosities that are sources of neurotransmitter when firing an action potential and are sinks otherwise. To analyze this random partial differential equation, we show that each realization of its solution can be represented as a certain expected local time of a Brownian particle in a corresponding realization of a random environment. Using this representation, we prove two surprising results. First, the expected neurotransmitter con...

Stochastic consistency and stochastic stability in mean square sense for Cauchy advection problem

This work deals with the construction of finite difference solutions of random advection Cauchy type partial differential equation containing uncertainty through the coefficient of the velocity. Under appropriate hypothesis on the velocity random variable, we establish that the constructed random finite difference solution is mean square consistent and mean square stable over the whole real line. In addition, the main statistical functions, such as the mean, of the approximate solution stochastic process generated by truncation of the exact finite difference solution are given. Finally, we apply the proposed technique to several illustrative examples which show our discussing for the mean square stability.

Intrusive uncertainty quantification for hyperbolic-elliptic systems governing two-phase flow in heterogeneous porous media

We model random locations of spatial interfaces of heterogeneities in porous media by means of the hybrid stochastic Galerkin (HSG) approach. This approach extends the concept of the generalized polynomial chaos (PC) expansion for a multi-element decomposition of the multidimensional stochastic space. In this way, the physically two-dimensional hyperbolic-elliptic fractional flow formulation for two-phase flow in heterogeneous porous media is transformed from a random partial differential equation into a deterministic system for the coefficients of the PC expansion of the primary unknown saturation, total velocity, and global pressure. The hyperbolic part is discretized based on a central-upwind finite volume scheme along with a mixed finite element method for the elliptic part. The latter partly uses the tensor product structure of the saddle point system together with appropriate preconditioning to speed up the computations. Since we use the sequential implicit pressure explicit saturation (IMPES) approach, the elliptic part is solved in every time step. The proposed method is particularly well-suited for parallel computations and allows for the consideration of a huge variety of complex flow problems. We illustrate the power of the method by means of several striking numerical examples of different complexities including their numerical convergence analysis.

Multi-index Stochastic Collocation Convergence Rates for Random PDEs with Parametric Regularity

We analyze the recent Multi-index Stochastic Collocation (MISC) method for computing statistics of the solution of a partial differential equation (PDE) with random data, where the random coefficient is parametrized by means of a countable sequence of terms in a suitable expansion. MISC is a combination technique based on mixed differences of spatial approximations and quadratures over the space of random data, and naturally, the error analysis uses the joint regularity of the solution with respect to both the variables in the physical domain and parametric variables. In MISC, the number of problem solutions performed at each discretization level is not determined by balancing the spatial and stochastic components of the error, but rather by suitably extending the knapsack-problem approach employed in the construction of the quasi-optimal sparse-grids and Multi-index Monte Carlo methods, i.e., we use a greedy optimization procedure to select the most effective mixed differences to include in the MISC estimator. We apply our theoretical estimates to a linear elliptic PDE in which the log-diffusion coefficient is modeled as a random field, with a covariance similar to a Matern model, whose realizations have spatial regularity determined by a scalar parameter. We conduct a complexity analysis based on a summability argument showing algebraic rates of convergence with respect to the overall computational work. The rate of convergence depends on the smoothness parameter, the physical dimensionality and the efficiency of the linear solver. Numerical experiments show the effectiveness of MISC in this infinite dimensional setting compared with the Multi-index Monte Carlo method and compare the convergence rate against the rates predicted in our theoretical analysis.

Multi-step methods for random ODEs driven by Itô diffusions

Linear multi-step methods are derived for random ordinary differential equations (RODEs) driven by the solutions of Itô stochastic differential equations (SODEs) via strong Itô–Taylor schemes for SODEs. Due to the special structure of the RODE–SODE pair it is not necessary to restrict the intensity of the noise. Pathwise convergence is established as well as the B-stability of implicit multi-step methods. Numerical comparisons are provided for explicit schemes applied to a low dimensional RODE and implicit schemes applied to a high dimensional RODE obtained with the method of lines by spatially discretizing a random partial differential equation with finite difference quotients.

A stochastic delay model for pricing debt and equity: Numerical techniques and applications

Delayed nonlinear models for pricing corporate liabilities and European options were recently developed. Using self-financed strategy and duplication we were able to derive a Random Partial Differential Equation (RPDE) whose solutions describe the evolution of debt and equity values of a corporate in the last delay period interval in the accompanied paper (Kemajou et al., 2012) [14]. In this paper, we provide robust numerical techniques to solve the delayed nonlinear model for the corporate value, along with the corresponding RPDEs modeling the debt and equity values of the corporate.Using financial data from some firms, we forecast and compare numerical solutions from both the nonlinear delayed model and classical Merton model with the real corporate data. From this comparison, it comes up that in corporate finance the past dependence of the firm value process may be an important feature and therefore should not be ignored.

Properties of the Parabolic Anderson Model and the Anderson Polymer Model

In this article we examine some properties of the solutions of the parabolic Anderson model. In particular we discuss intermittency of the field of solutions of this random partial differential equation, when it occurs and what the field looks like when intermittency doesn't hold. We also explore the behavior of a polymer model created by a Gibbs measure based on solutions to the parabolic Anderson equation.

DISCRETIZATION OF STATIONARY SOLUTIONS OF SPDE'S BY EXTERNAL APPROXIMATION IN SPACE AND TIME

We consider a stochastic partial differential equation with additive noise satisfying a strong dissipativity condition for the nonlinear term such that this equation has a random fixed point. The goal of this article is to approximate this fixed point by space and space-time discretizations of a stochastic differential equation or more precisely, a conjugate random partial differential equation. For these discretizations external schemes are used. We show the convergence of the random fixed points of the space and space-time discretizations to the random fixed point of the original partial differential equation.

Stationary Conjugation of Flows for Parabolic SPDEs with Multiplicative Noise and Some Applications

The purpose of this paper is to transform a nonlinear stochastic partial differential equation of parabolic type with multiplicative noise into a random partial differential equation by using a bijective random process. A stationary conjugation is constructed, which is of interest for asymptotic problems. The conjugation is used here to prove the existence of the stochastic flow, the perfect cocycle property and the existence of the random attractor, all nontrivial properties in the case of multiplicative noise.

Scaling laws and convergence for the advection-diffusion equation

In this paper we study the convergence of stochastic processes related to a random partial differential equation (PDE with random coefficients) of heat equation propagation type in a Kolmogorov's random velocity field. Then we are able to improve the results of Avellanda and Majda in the case of "shear-flow" advection-diffusion because we prove a convergence in law of the solution of the RPDE instead of just convergence of the moments.