V5E5 Advanced Topics in Numerical Analysis
Numerical Methods in Uncertainty Quantification
In practice, input data for mathematical models are never precisely known, but subject
to various types of uncertainties. These can arise, for instance, from measurement errors
or from limited available information. The subject of this lecture are methods for
quantifying the resulting uncertainties in model outputs.
The course focuses on recent advances in approaches based on probabilistic models of uncertainty.
An important class of applications are PDE models with uncertain coefficients, where one
aims to extract information on the probability distributions of solutions.
In a Bayesian framework, one can also treat corresponding inverse problems, where distributions
of coefficients are reconstructed from noisy partial measurements of solutions.
Besides Monte Carlo-type methods based on random sampling, one can also consider purely deterministic
approximations of probability distributions, which leads to high-dimensional approximation problems.
- Probability measures on Banach spaces
- Monte Carlo and related methods
- Spectral approximations
- Uncertainty propagation in PDE models
- The Bayesian approach to inverse problems
Prerequisites: The course assumes basic knowledge on probability theory and on partial differential equations.
Oral exam, by individual appointment.
Please choose from the following dates: February 5, 6, 26, 27. (Dates for a second attempt will be offered in the last week of March, i.e. 26th to 30th)
Subject matter of the exam: chapters 1, 2, and 3 of the lecture notes (i.e., chapter 4 is excluded)
Coding examples in Julia
(To run the code in these notebooks yourself, you can either log in to JuliaBox
and use them remotely after uploading them there, or install Julia and IJulia on your own machine by first downloading Julia from here
and following the installation instructions here
When & where:
Mo 14 (c.t.) - 16 and Wed 8 - 10, Wegelerstr 6, SemR We 5.002
For further literature, see also the lecture notes. In each category below, the items that are most relevant for the course come first:
- Uncertainty quantification and stochastic simulation
G. J. Lord, C. E. Powell, T. Shardlow, An Introduction to Computational Stochastic PDEs, Cambridge University Press, 2014
T. J. Sullivan, Introduction to Uncertainty Quantification, Springer, 2015
S. Asmussen, P. W. Glynn, Stochastic Simulation, Springer, 2007
R. C. Smith, Uncertainty Quantification: Theory, Implementation, and Applications, SIAM, 2014
- Basic probability theory
R. M. Dudley, Real Analysis and Probability, Cambridge University Press, 2002
O. Kallenberg, Foundations of Modern Probability, Springer, 1997
S. Saeki, A Proof of the Existence of Infinite Product Probability Measures, The American Mathematical Monthly 103, pp 682-683, 1996
- Random fields
V. I. Bogachev, Gaussian Measures, AMS, 1998
M. Hairer, An introduction to stochastic PDEs, arXiv:0907.4178
G. Da Prato, J. Zabczyk, Stochastic Equations in Infinite Dimensions, 1992
G. Da Prato, An Introduction to Infinite-Dimensional Analysis, 2006
M. Ledoux, M. Talagrand, Probability in Banach Spaces, Springer, 1991
- Polynomial approximations
Ch. Schwab, C. J. Gittelson, Sparse tensor discretizations of high-dimensional parametric and stochastic PDEs, Acta Numerica 20, pp 291-467, 2011
A. Cohen, R. DeVore, Approximation of high-dimensional parametric PDEs, Acta Numerica 24, pp 1-154, 2015
- Bayesian inverse problems
M. Dashti, A. M. Stuart, The Bayesian approach to inverse problems, arXiv:1302.6989
A. M. Stuart, Inverse problems: A Bayesian perspective, Acta Numerica 19, pp 451-559, 2010
- Elliptic PDEs and Finite Element Method
W. Hackbusch, Theorie und Numerik elliptischer partieller Differentialgleichungen, lecture notes (see also German Springer print version and English translation)
D. Braess, Finite Elemente, Springer; english translation: Finite Elements, Cambridge University Press
A. Ern, J.-L. Guermond, Theory and Practice of Finite Elements, Springer
K. Atkinson, W. Han, Theoretical Numerical Analysis, Springer