An Introduction to Computational Stochastic PDEs

Abstract

This book gives a comprehensive introduction to numerical methods and analysis of stochastic processes, random fields and stochastic differential equations, and offers graduate students and researchers powerful tools for understanding uncertainty quantification for risk analysis. Coverage includes traditional stochastic ODEs with white noise forcing, strong and weak approximation, and the multi-level Monte Carlo method. Later chapters apply the theory of random fields to the numerical solution of elliptic PDEs with correlated random data, discuss the Monte Carlo method, and introduce stochastic Galerkin finite-element methods. Finally, stochastic parabolic PDEs are developed. Assuming little previous exposure to probability and statistics, theory is developed in tandem with state-of-the-art computational methods through worked examples, exercises, theorems and proofs. The set of MATLAB codes included (and downloadable) allows readers to perform computations themselves and solve the test problems discussed. Practical examples are drawn from finance, mathematical biology, neuroscience, fluid flow modelling and materials science.

Gaussian process modelling for uncertainty quantification in convectively-enhanced dissolution processes in porous media

“…Thus, for a given RF, Z, we can set K = exp(Z) to obtain the desired discrete permeability field [33]. Furthermore, if Z is chosen to be normally distributed then K is log normal.…”

“…Even for a few hundred sampling points, however, the round-o↵ error in this method cannot be neglected due to the fact that the associated covariance matrix is likely to become extremely ill-conditioned [19]. An alternative method for simulating a Gaussian RF is the circulant embedding algorithm [20] described, e.g., in [33]. This method provides an exact simulation of a Gaussian RF, although its implementation is not straightforward.…”

“…where m j is the predicted expected value at a given design point,⇠ j , given by (32), s 2 j its corresponding variance, given by (33), and y j the corresponding observed value at⇠ j . In the following section, we will describe how we select appropriate parameters and functions for the GP emulator.…”

“…where # i ⇠ N (0, 1), and m D (·) and k D (·, ·) are the predictive mean and predictive variance given, respectively, by (32) and (33). Finally, we approximate F (i) (·) by using the empirical cumulative distribution function:…”

On the statistical dependence : from nonlinearity to spatial structural estimation

“…Thus, the block bandable matrices can be used to model the dependency structure of two-dimensional data that decays according to its spatial distance. It is worth noting that matrices that exhibit block structure plays a significant role in spatial modeling, such as the circulate embedding of random fields Lord et al [2014].…”