AMCS 301 Random PDEs - Modern Numerical Methods

Many processes across science and engineering can be modelled by partial differential equations (PDEs). However, these PDE models are often affected by uncertainties due to a lack of knowledge, intrinsic variability in the system, or an imprecise manufacturing process. These uncertainties could appear for instance in material properties, source terms, or boundary conditions. The goal of this course is to provide basic knowledge of the random PDEs as well as various efficient numerical solution techniques for this class of problems. The course will cover modern computational approaches as well as their mathematical foundations. After a brief recap of fundamentals of probability theory and statistics, as well as numerical analysis for PDEs, the focus of this course will be on the following topics: • Sampling methods for random PDEs: o Monte Carlo and Quasi-Monte Carlo methods • Hierarchical Sampling methods for random PDEs: o Multilevel Monte Carlo methods o Multi-Index Monte Carlo methods • Approximation methods for random PDEs o Stochastic Galerkin methods and (generalised) polynomial chaos methods o Low rank approximation o Regression and multilevel regression o Physics-informed neural networks • Inverse Uncertainty quantification techniques for random PDEs: o Bayesian inversion problems