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MAI0129
Stochastic Galerkin Methods for Partial Differential Equations

Number of credits: 5 hp

Examiner: Jan Nordström

Course literature:

GX08: Gottlieb, Xiu, Galerkin Method for Wave Equations with Uncertain Coefficients, Commun. Comput. Phys., Vol. 3, No. 2, pp. 505-518, 2008.

PIN15: Pettersson, Iaccarino, Nordström, Polynomial Chaos Methods for Hyperbolic Partial Differential Equations, Springer, 2015.

TPME11: Tuminaro, Phipps, Miller, Elman, Assessment of Collocation and Galerkin Approaches to Linear Diffusion Equations with Random Data, International Journal for Uncertainty Quantification, Vol. 1, No. 1, pp. 19-33, 2011.

XK02: Xiu, Karniadakis, Modeling uncertainty in steady state diffusion problems via generalized polynomial chaos, CMAME, Vol. 191, pp. 49274948, 2002.

Course contents:

Basic Concepts

Introduction

Representation of random fields via spectral expansions:

PDE Theory

Reading Material: TPME11.

 

Linear Problems

Hyperbolic Problems

Parabolic Problems

Elliptic Problems

Reading Material: XK02, GX08

 

Non-intrusive Methods

Summary of the material from day 1-2

Non-intrusive methods

Reading material: TPME11

 

Nonlinear Problems (Burgers' equation)

Nonlinear analysis for stochastic problem guided by deterministic analysis

Analysis of the exact solution of the stochastic Burgers' equation 

Introduction to project work/assignments

Reading material: PIN15 Ch. 6

 

Advanced topics

Sensitivity of different PDEs

Multiple stochastic dimensions

Alternative gPC basis functions: wavelets, spatially adaptive gPC

 

Organisation: Lectures and exercises, mixed.

Examination: Work in small groups (or individually) on a mini project. Project topics will be provided by the lecturers. The deliverables include a report (5 pages) and mandatory homework.

Prerequisites: Basic knowledge in computational mathematics and mathematical statistics.

 

Course web page

 

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Last updated: 2022-11-14