Fiche de cours

Numerical methods for random PDEs and uncertainty

MATH-616

Fiche de cours

Enseignant(s) :

Invited lecturers (see below)
Nobile Fabio
Vanzan Tommaso

Langue:

English

Frequency

Only this year

Remark

Fall semester

Summary

The course focuses on mathematical models based on PDEs with random parameters, and presents numerical techniques for forward uncertainty propagation, inverse uncertainty analysis in a Bayesian framework and optimal control under uncertainty.

Content

When building a mathematical model to describe the behavior of a physical system, one has often to face a certain level of uncertainty in the proper characterization of the model parameters and input data.  The increasing computer power and the need for reliable predictions have pushed researchers to include uncertainty models, often in a probabilistic setting, for the input parameters of otherwise deterministic mathematical models.
The course will  focus on mathematical models based on Partial Differential Equations with random parameters, and presents numerical techniques for forward uncertainty propagation, including Monte Carlo, Multilevel Monte Carlo,  polynomial chaos and rational approximation techniques; inverse uncertainty analysis in a Bayesian framework and Markov Chain Monte Carlo methods; optimal control under uncertainty.
Particular attention is devoted to addressing the case of a large (even infinite) number of input parameters thus leasing to High Dimensional Approximation problems and presenting recent results such as the "Cohen-Devore" theory on polynomial approximation in infinite dimensions.

Keywords

Random PDEs, Forward Uncertainty Propagation, Bayesian Inverse Problems, Optimization Under uncertainty; Monte Carlo, Multi Level Monte Carlo, Polynomial Chaos, Sparse grids, rational approximations

Learning Prerequisites

Required courses

The students are expected to have basic knowledge on probability theory, approximation theory, Partial Differential Equations, numerical analysis in general and finite element analysis in particular.

Resources

Bibliography

A. Cohen, R. DeVore “Approximation of high-dimensional parametric PDEs”. Acta Numer. 24 (2015).
A. Stuart, “Inverse problems: a Bayesian perspective”. Acta Numer, 19 (2010).
D. Kouri, A. Shapiro, “Optimization of PDEs with uncertain inputs.” in Frontiers in PDE-constrained optimization, 41–81, IMA Vol. Mat. Appl., 163, Springer, 2018.

Ressources en bibliothèque

• Approximation of high-dimensional parametric / Cohen & DeVore
• Inverse problems: a Bayesian perspective / Stuart
• Optimization of PDEs with uncertain inputs / Kouri & Shapiro

Moodle Link

• https://go.epfl.ch/MATH-616