An Introduction to Computational Stochastic PDEs

By Gabriel J. Lord, Catherine Elizabeth Powell, Tony Shardlow. 2014

641 citations recorded by the source corpus.

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 modeling and materials science.

Subfields: Applied Mathematics, Computational Theory and Mathematics

View Paper: https://www.semanticscholar.org/paper/fec0671343359bdd4691559a4f57c386126487d6

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Citations in the corpus

Complexity Analysis of stochastic gradient methods for PDE-constrained optimal Control Problems with uncertain parameters (2021, background): “, [4, 23, 35]), the analysis of PDE constrained control problems under uncertainty is much more recent and incomplete, although the topic has received increasing attention in the last few years.”
Model Reduction and Neural Networks for Parametric PDEs (2020, methodology): “V d;N w.r.t. V d is related to the approximation quality of ˚ j by ˚ j;N for j= 1;:::;Nand therefore to the approximation quality of Cby C N. Another perspective is via the KarhunenLoeve Theorem (KL) [36]. For simplicity, assume that is mean zero, then u˘ admits an expansion of the form u= P 1 j=1 p”
Structured Bayesian Gaussian process latent variable model: applications to data-driven dimensionality reduction and high-dimensional inversion (2018, methodology): “One notable example of this is to use a Karhunèn-Loeve expansion (KLE) [13] to parameterize the prior with relatively few latent variables; this has been explored in the context of Bayesian inversion in [9].”
Low-rank solvers for unsteady Stokes–Brinkman optimal control problem with random data (2016, positive): “This allows for a simultaneous generalized polynomial chaos (PCE) approximation of these random functions [16, 17, 35, 44, 57] in the stochastic Galerkin finite element method (SGFEM) discretization of the model.”
Weak convergence rates of splitting schemes for the stochastic Allen–Cahn equation (2018, background): “Numerical schemes for SPDEs have been extensively studied in the last two decades, see for instance the monographs [16, 21, 24].”
Convergence Types and Rates in Generic Karhunen-Loève Expansions with Applications to Sample Path Properties (2014, background): “d of convergence,and their relation to path properties have recently been considered in the area of numerics for stochastic partial differential equations, too. Without going into details we refer to [62, 28, 45, 37] and the references therein. Moveover,[42, 41, 79] consider Karhunen-Loe`ve expansio”