Introductory description

This module provides students with fundamental knowledge for predictive modelling and uncertainty quantification. It gives an overview of the essential elements of the mathematical, statistical, and computational techniques needed to provide well-calibrated predictions for the behaviour of physical systems.

Module aims

Outline syllabus

This is an indicative module outline only to give an indication of the sort of topics that may be covered. Actual sessions held may differ.

• Univariate probability pre-reading – discrete and continuous distributions, the Law of Large Numbers, expectations and variance
• Multivariate probability
• Bayesian probability
• Multivariate Gaussians and their conditioning
• State estimation via Kalman filtering
• Forward and inverse UQ
• Quadrature and sampling (MC, MCMC, QMC, LHS, …)
• Finite-dimensional linear algebra
• Basic functional analysis (infinite-dimensional linear algebra)
• Perspectives on random functions
• Interpolation, regression, orthogonal projection
• Motivation for regularisation
• Tikhonov regularisation / ridge regression
• Sparse approximation (\ell^1 regularisation)
• Optimisation approach to inverse problems, linear and nonlinear case, connect to Tikhonov regularisation
• Bayesian approach to inverse problems, well-posedness of BIPs
• High- or infinite-dimensional aspects
• Approximate Bayesian inference e.g. Laplace approximation,

Learning outcomes

By the end of the module, students should be able to:

• Understand and apply univariate and multivariate probability to complex scientific modelling problems
• Understand and interpret forward and inverse uncertainty quantification
• Apply and evaluate quadrature and sampling schemes to solve predictive modelling tasks
• Appreciate, synthesise, and apply modelling strategies for uncertain quantities of interest in high dimension
• Contextualise the relations among regression, approximation, and orthogonal projection
• Recognise, formulate, analyse and solve Bayesian inverse problems

Indicative reading list

McClarren, Uncertainty Quantification and Predictive Computational Science, available from SpringerLink on campus at Warwick - aimed at a general Physical Sciences / Engineering audience

Sullivan, Introduction to Uncertainty Quantification, available from SpringerLink on campus at Warwick

Boyd and Vandenberghe, Introduction to Applied Linear Algebra - Vectors, Matrices and Least Squares, freely available online. Companion exercises that implement the material in Python and Julia are available from the same webpage.

View reading list on Talis Aspire

Interdisciplinary

The MSc programme will recruit students with backgrounds across the physical and mathematical sciences, including engineering, and will provide an interdisciplinary perspective on predictive modelling.

The fundamentals of uncertainty quantification and predictive modelling are drawn from the mathematical, statistical and computational sciences, and rely on disciplinary fusion with application domains such as physical/chemical science and engineering.

Subject specific skills

• Computational statistics
• Mathematical modelling
• Predictive modelling
• Reliability assessment of descriptive and predictive scientific models

Transferable skills

• Data analysis and modelling
• Oral presentation skills
• Scientific computing