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.

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

- Introductory slides
**Lecture notes**, draft of Oct 18

**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

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