Numerical analysis of stochastic PDEs in traffic flow: Investigating density-flow relations

A Limit Theorem for the Solutions of Differential Equations with Random Right-Hand Sides

Understanding phenomena ranging from biological processes to financial markets involves uncertainty. Stochastic Differential Equations (SDEs) and Stochastic Partial Differential Equations (SPDEs) serve as robust mathematical frameworks for modelling such systems. Given the stochastic influences within these models, comprehending the dynamics of complex systems becomes pivotal for accurately predicting system behaviour. However, traditional numerical techniques frequently encounter challenges in effectively addressing the intricacies and stochastic properties inherent in these equations. This chapter explores several numerical methods that offer streamlined and dependable solutions capable of handling the complexities inherent in stochastic differential and partial differential equations. Also, numerical challenges associated with these methods are discussed and the solution strategies are also suggested.

Macroscopic traffic flow modeling is essential for describing and forecasting the characteristics of traffic flow. However, the classic Lighthill–Whitham–Richards (LWR) model only provides equilibrium values for steady‐state conditions and fails to capture common stochastic variabilities, which are a necessary component of accurate modeling of real‐time traffic management and control. In this paper, a stochastic LWR (SLWR) model that randomizes free‐flow speed is developed to account for the stochasticity incurred by the heterogeneity of drivers, while holding individual drivers’ behavior constant. The SLWR model follows a conservation law of stochastic traffic density and flow and is formulated as a time‐dependent stochastic partial differential equation. The model is solved using a dynamically bi‐orthogonal (DyBO) method based on a spatial basis and stochastic basis. Various scenarios are simulated and compared with the Monte Carlo (MC) method, and the results show that the SLWR model can effectively describe dynamic traffic evolutions and reproduce some commonly observed traffic phenomena. Furthermore, the DyBO method shows significant computational advantages over the MC method.

Recent years have seen an explosion of interest in stochastic partial differential equations where the driving noise is discontinuous. In this comprehensive monograph, two leading experts detail the evolution equation approach to their solution. Most of the results appeared here for the first time in book form. The authors start with a detailed analysis of Lévy processes in infinite dimensions and their reproducing kernel Hilbert spaces; cylindrical Lévy processes are constructed in terms of Poisson random measures; stochastic integrals are introduced. Stochastic parabolic and hyperbolic equations on domains of arbitrary dimensions are studied, and applications to statistical and fluid mechanics and to finance are also investigated. Ideal for researchers and graduate students in stochastic processes and partial differential equations, this self-contained text will also interest those working on stochastic modeling in finance, statistical physics and environmental science.

Stochastic factors and string stability of traffic flow: Analytical investigation and numerical study based on car-following models

Stochastic partial differential equations are derived for the reaction-diffusion process in one, two, and three dimensions. Specifically, stochastic partial differential equations are derived for the random dynamics of particles that are reacting and diffusing in a medium. In the derivation, a discrete stochastic reaction-diffusion equation is first constructed from basic principles, that is, from the changes that occur in a small time interval. As the time interval goes to zero, the discrete stochastic model leads to a system of It stochastic differential equation. As the spatial intervals approach zero, a stochastic partial differential equation is derived for the reaction-diffusion process. The stochastic reaction-diffusion equation can be solved computationally using numerical methods for systems of It stochastic differential equations. Comparisons between numerical solutions of stochastic reaction-diffusion equations and independently formulated Monte Carlo calculations support the accuracy of the derivations.

Exponential stability of energy solutions to stochastic partial differential equations with variable delays and jumps

In the present work, we consider an optimal control for a three-factor stochastic factor model. We assume that one of the factors is not observed and use classical filtering technique to transform the partial observation control problem for stochastic differential equation (SDE) to a full observation control problem for stochastic partial differential equation (SPDE). We then give a sufficient maximum principle for a system of controlled SDEs and degenerate SPDE. We also derive an equivalent stochastic maximum principle. We apply the obtained results to study a pricing and hedging problem of a commodity derivative at a given location, when the convenience yield is not observable.

As the name suggests, Stochastic Partial Differential Equations is an interdisciplinary area at the crossroads of stochastic processes and partial differential equations (SPDEs). The beginnings of SPDEs as a new discipline can be traced to the late sixties and early seventies of the previous century. It is safe to say that in the last four decades SPDEs have been one of the most dynamic areas of probability theory and stochastic processes. Generally speaking, any partial differential equation is an SPDE if its coefficients, forcing terms, initial and boundary conditions, or some of the above are random/uncertain. Needles to say, this constitutes an extremely diverse area. The accelerating progress in research on stochastic partial differential equations has stimulated involvement of many experts from other fields in the research on stochastic PDEs. As of now, the subject of SPDEs with its numerous important applications is an exciting mosaic of interconnected topics revolving around stochastics and partial differential equations. Interacting particle systems, fluid dynamics, statistical physics, financial modeling, nonlinear filtering, super-processes, continuum physics and, recently, uncertainty quantification are important contributors to and heavy users of the theory and practice of SPDEs. In the last two decades numerical methods and large scale computations have become a very important and popular part of SPDEs. In fact, this development has made the applied side of SPDEs truly relevant to a wide range of high-tech areas, e.g. computer-based prediction and design, risk assessment and decision making in financial markets, energy, environment etc.

Since Pardoux and Peng firstly studied the following nonlinear backward stochastic differential equations in 1990. The theory of BSDE has been widely studied and applied, especially in the stochastic control, stochastic differential games, financial mathematics and partial differential equations. In 1994, Pardoux and Peng came up with backward doubly stochastic differential equations to give the probabilistic interpretation for stochastic partial differential equations. Backward doubly stochastic differential equations theory has been widely studied because of its importance in stochastic partial differential equations and stochastic control problems. In this article, we will study the theory of doubly stochastic systems and related topics further.

In this paper, we introduce a stochastic partial differential equation model for the spatial dynamic of tumor–immune interactions. We perform numerical simulations in order to investigate the propagation of traveling waves in model system under the influence of random space-time fluctuations. One of methods is to solve a stochastic partial differential equation system for tumor–immune cell densities. The second method is to solve a stochastic partial differential algebraic equation system in order to assess the wave behavior of the solution in comparison with the deterministic approach. Finally, we discuss the implications of the model results.

The present review discusses recent developments in numerical techniques for the solution of systems with stochastic uncertainties. Such systems are modelled by stochastic partial differential equations (SPDEs), and techniques for their discretisation by stochastic finite elements (SFEM) are reviewed. Also, short overviews of related fields are given, e.g. of mathematical properties of random fields and SPDEs and of techniques for high-dimensional integration. After a summary of aspects of stochastic analysis, models and representations of random variables are presented. Then mathematical theories for SPDEs with stochastic operator are reviewed. Discretisation-techniques for random fields and for SPDEs are summarised and solvers for the resulting discretisations are reviewed, where the main focus lies on series expansions in the stochastic dimensions with an emphasis on Galerkin-schemes.

We develop a perturbation theory for stochastic differential equations (SDEs) by which we mean both stochastic ordinary differential equations (SODEs) and stochastic partial differential equations (SPDEs). In particular, we estimate the $L^{p}$-distance between the solution process of an SDE and an arbitrary Itô process, which we view as a perturbation of the solution process of the SDE, by the $L^{q}$-distances of the differences of the local characteristics for suitable $p,q>0$. As one application of the developed perturbation theory, we establish strong convergence rates for numerical approximations of a class of SODEs with nonglobally monotone coefficients. As another application of the developed perturbation theory, we prove strong convergence rates for spatial spectral Galerkin approximations of solutions of semilinear SPDEs with nonglobally monotone nonlinearities including Cahn–Hilliard–Cook-type equations and stochastic Burgers equations. Further applications of the developed perturbation theory include regularity analyses of solutions of SDEs with respect to their initial values as well as small-noise analyses for ordinary and partial differential equations.

On Stochastic Processes Defined by Differential Equations with a Small Parameter

Stochastic partial differential equations (SPDEs) are important tools in modeling complex phenomena, and they arise in many physics and engineering applications. Developing efficient numerical methods for simulating SPDEs is a very important while challenging research topic. In this thesis, we study a numerical method based on the Wiener chaos expansion (WCE) for solving SPDEs driven by Brownian motion forcing. WCE represents a stochastic solution as a spectral expansion with respect to a set of random basis. By deriving a governing equation for the expansion coefficients, we can reduce a stochastic PDE into a system of deterministic PDEs and separate the randomness from the computation. All the statistical information of the solution can be recovered from the deterministic coefficients using very simple formulae. We apply the WCE-based method to solve stochastic Burgers equations, Navier-Stokes equations and nonlinear reaction-diffusion equations with either additive or multiplicative random forcing. Our numerical results demonstrate convincingly that the new method is much more efficient and accurate than MC simulations for solutions in short to moderate time. For a class of model equations, we prove the convergence rate of the WCE method. The analysis also reveals precisely how the convergence constants depend on the size of the time intervals and the variability of the random forcing. Based on the error analysis, we design a sparse truncation strategy for the Wiener chaos expansion. The sparse truncation can reduce the dimension of the resulting PDE system substantially while retaining the same asymptotic convergence rates. For long time solutions, we propose a new computational strategy where MC simulations are used to correct the unresolved small scales in the sparse Wiener chaos solutions. Numerical experiments demonstrate that the WCE-MC hybrid method can handle SPDEs in much longer time intervals than the direct WCE method can. The new method is shown to be much more efficient than the WCE method or the MC simulation alone in relatively long time intervals. However, the limitation of this method is also pointed out. Using the sparse WCE truncation, we can resolve the probability distributions of a stochastic Burgers equation numerically and provide direct evidence for the existence of a unique stationary measure. Using the WCE-MC hybrid method, we can simulate the long time front propagation for a reaction-diffusion equation in random shear flows. Our numerical results confirm the conjecture by Jack Xin that the front propagation speed obeys a quadratic enhancing law. Using the machinery we have developed for the Wiener chaos method, we resolve a few technical difficulties in solving stochastic elliptic equations by Karhunen-Loeve-based polynomial chaos method. We further derive an upscaling formulation for the elliptic system of the Wiener chaos coefficients. Eventually, we apply the upscaled Wiener chaos method for uncertainty quantification in subsurface modeling, combined with a two-stage Markov chain Monte Carlo sampling method we have developed recently.