Mathematical modelling of complex systems using stochastic partial differential equations: Review and development of mathematical concepts

Abstract

Modelling real life stochastic phenomena is difficult due to heterogeneity in associated parameters or insufficient information, or both; hence it is important to develop stochastic models to understand the behaviours of complex Systems. Stochastic Partial Differential Equations (SPDEs) (Holden et al. 2010; Nicholas 2017), are used to model physical, engineering, and biological systems in which small scale effects and related uncertainties are modelled as stochastic processes. In this chapter, we discuss the modelling of complex systems using SPDEs, giving reasons for such developments and focusing on numerical solutions. We primarily discuss Polynomial Chaos Expansion (PCE) approach (Wan et al. 2004). In recent years, several extensions of PCE, such as Generalised PCE using the Askey scheme (Xiu and Em Karniadakis 2003), have been developed to overcome the slow computational speed of the method for non-Gaussian random variables. When there are viable restrictions on model predictability due to interdependencies between the physical system and the associated parameters, PCE provides a robust mathematical structure and acceptable probabilistic solutions. We also discuss the use of Wick products (Venturi et al. 2013) in stochastic analysis. Using the Wick product instead of the usual pointwise multiplication makes it possible to study anticipating processes. The Wick product is innate in stochastic analysis (with respect to Wiener chaos space), as it is implicit in the Ito integral. For anticipating processes, the Skorohod integral is similar to the Ito integral. Thus, Wick calculus works equally well for adapted and non-adapted processes.