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.

*Planned contents:*

- 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)

- Introductory slides
**Lecture notes**, version of Feb 5,

note in particular: corrected statement of Prop 1.3.8; corrected derivation of (2.17); more detailed cost breakdown after (2.19); additional hints for Exercise 3.1.1; statement of Thm 3.2.4(ii)

(thanks to Friedrich Klaus, Jakiw Pidstrigach, and Katharina Schuh for pointing out typos and inconsistencies)- Slides on MCMC

- Overview of Julia [View] [Download]
- Random Sampling and a basic example of the Monte Carlo method [View] [Download]
- Monte Carlo and conditioning for a basic 1D random boundary value problem [View] [Download]
- Circulant embedding and multilevel Monte Carlo [View] [Download]
- Spectral expansions for dependence on one random variable [View] [Download]
- Stochastic Galerkin for dependence on one random variable [View] [Download]
- Metropolis-Hastings for Bayesian inverse problems [View] [Download]

**Mo 14 (c.t.) - 16** and **Wed 8 - 10**, Wegelerstr 6, SemR We 5.002

**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