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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.
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karin.johansson@liu.se
Last updated: 2022-11-14
