Content:

Partial differential equations (PDEs) appear in the mathematical modelling of a great variety of processes. Most of these equations contain various parameters that describe some physical properties, for example permeability or thermal conductivity. Often there is a degree of uncertainty regarding the given data. Or, sometimes one introduces noise to a given model. Clearly, one would like to quantify the effect of uncertain parameters and noise. On this background, uncertainty quantification (UQ) has developed into a very active mathematical field. Since these problems naturally appear in applications and modelling, it is important to derive analysis based computations.

This course is at the interface of PDEs with random coefficients, stochastic differential equations and numerical analysis.

The course will cover a selection from the following topics:

• Karhunen–Loève expansion of random fields
• Galerkin methods for semilinear PDEs (in particular FEM)
• Elliptic PDEs with random coefficients
• FEM for semilinear stochastic PDEs
• Spectral Galerkin method for semilinear stochastic PDEs
• Monte-Carlo and Multilevel Monte-Carlo sampling methods

Target audience: 

M.Sc. Mathematik/Physik, BMS course

Requirements:

Stochastic I. Basic knowledge from measure theory, functional analysis and numerical analysis.

Suggested reading:

[1] T. J. Sullivan. Introduction to Uncertainty Quantification, volume 63 of Texts in Applied Mathematics. Springer, 2015.

[2] Lord, Gabriel J., Catherine E. Powell, and Tony Shardlow. An Introduction to computational stochastic PDEs. Vol. 50. Cambridge University Press, 2014.

[3] Le Maître, Olivier, and Omar M. Knio. Spectral methods for uncertainty quantification: with applications to computational fluid dynamics. Springer Science & Business Media, 2010.

[4] Zhang, Zhongqiang, and George Karniadakis. Numerical methods for stochastic partial differential equations with white noise. Springer International Publishing, 2017.