AMCS 301 Numerical methods for random partial differential equations: hierarchical approximation and machine learning approaches

Random PDEs stochastic algorithms Monte carlo methods Quasi-Monte Carlo Hierarchical regression Multilevel Monte Carlo Stochastic collocation Multi-index Low-rank approximation hierarchical and sparse approximation Bayesian Inversion Bayesian optimal experimental design

A course on modern numerical methods for random partial differential equations

COURSE OBJECTIVES: 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.

COURSE OUTLINE: 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:
Random fields and Gaussian fields
Sampling methods for random PDEs:
• Monte Carlo
• Quasi-Monte Carlo methods
Hierarchical Sampling methods for random PDEs:
• Multilevel Monte Carlo methods
• Multi-Index Monte Carlo methods
Approximation methods for random PDEs:
• Stochastic Galerkin, Stochastic collocation methods and (generalised) polynomial
chaos methods
• Low rank approximation
• Regression and multilevel regression
• Physics-informed neural networks

Inverse Uncertainty quantification techniques for random PDEs:
• Bayesian inversion problem and Bayesian optimal experimental design (optimal data acquisition)