# MMath Mathematics with Financial Mathematics

Year of entry: 2023

## Course unit details:Introduction to Uncertainty Quantification

Unit code MATH44071 15 Level 4 Semester 1 Department of Mathematics No

### Overview

This unit introduces the core concepts necessary to quantify, analyse, and understand the effects of randomness and uncertainty in models of real-world systems. We begin by introducing some basic concepts from probability theory, before building on this to develop a range of practical methods for analysing real-world models affected by uncertainty.

Though the concepts introduced in this module will require a theoretical grounding, the primary intention is to focus on the application of the methods to models derived from environmental, industrial, and/or biological applications. Computational exercises will reinforce understanding of the methods introduced here.

### Pre/co-requisites

Students are not permitted to take, for credit, MATH44071 in an undergraduate programme and then MATH64071 in a postgraduate programme at the University of Manchester, as the courses are identical.

### Aims

To provide an introduction to the core ideas of uncertainty quantification (UQ), using a range of sophisticated sampling methods and other techniques, focusing on the forward propagation of uncertainty (from uncertain model inputs to outputs), the dynamics of stochastic processes (specifically Ito stochastic differential equations), and inference in inverse problems (estimating uncertain model inputs from data). The course includes an introduction to basic concepts in probability, stochastic processes and measure theory. The focus will be on UQ for models consisting of differential equations, of the type frequently encountered in environmental, industrial, and biological applications.

### Learning outcomes

On successful completion of the course unit students will be able to:

• classify types of uncertainty, and describe its possible effects;
• apply probabilistic and statistical properties and theoretical results, to models in applied mathematics which are subject to uncertainties;
• describe and implement computational methods for the generation of random variables from simple or classical probability distributions;
• implement and analyse the error of appropriate methods for the propagation of uncertainty in deterministic dynamical systems;
• identify and explain the meaning of terms within an Ito stochastic differential equation (SDE) model in the context of applications;
• implement Euler-Maruyama approximations of Ito SDEs and analyse their error;
• apply the Bayesian framework to inverse problems arising in applied mathematics;
• construct and simulate Markov chains in order to sample from probability distributions arising from Bayesian inverse problems in applied mathematics;
• apply the tools and methods from the course to quantify uncertainty in dynamical systems arising in environmental, biological and industrial applications.

### Syllabus

1. Modelling of uncertainty. 

General introduction to the concept of incorporating uncertainty into mathematical models, classification of uncertainty and problems in UQ.

2. Introduction to probability theory 

Probability spaces, univariate random variables, multivariate random variables, statistical theorems, Brownian motion.

3. Sampling-based methods for uncertainty in ODEs. 

Finite difference approximation of ODEs, Pseudo-random number generation, inverse CDF method, Box-Muller transform, Monte Carlo for propagating uncertainty in deterministic dynamical systems, Monte Carlo error analysis.

4. Stochastic differential equations (SDEs). 

Introduction to stochastic calculus, Fokker-Planck equation, the Ornstein-Uhlenbeck process, Euler–Maruyama method, strong and weak convergence, examples of modelling using SDEs.

5. Bayesian inverse problems. 

Bayes’ theorem,  application to inference problems for ODEs, explicit solutions for linear inverse problems, error analysis for approximated forward models.

6. Sampling methods for Bayesian inverse problems 

Introduction to Markov chain theory, Markov chain Monte Carlo (MCMC) methods, Metropolis-Hastings, Random Walk Metropolis-Hastings, error analysis, application to examples of inference for ODEs.

### Assessment methods

Method Weight
Other 20%
Written exam 80%
• Near end-of-semester courseworks weighting 20%
• End of semester examination: weighting 80%

### Feedback methods

Feedback tutorials will provide an opportunity for students' work to be discussed and provide feedback on their understanding. Coursework or in-class tests (where applicable) also provide an opportunity for students to receive feedback. Students can also get feedback on their understanding directly from the lecturer, for example during the lecturer's office hour.

• J. Voss, An Introduction to Statistical Computing: A Simulation-based Approach, Wiley, 2013. (Recommended)
• Ralph Smith, Uncertainty Quantification, SIAM, 2014. (Recommended)
• J. Kaipio, E. Somersalo, Statistical and Computational Inverse Problems, Springer, 2005. (Recommended)
• T.J. Sullivan, Introduction to Uncertainty Quantification, Springer, 2015. (Further reading)
• G.J. Lord, C.E. Powell, T. Shardlow. An introduction to computational stochastic PDEs. Cambridge University Press, 2014. (Further reading)
• P. Kloeden, E. Platen, Numerical Solution of Stochastic Differential Equations, Springer, 1992. (Further reading)

### Study hours

Scheduled activity hours
Lectures 11
Tutorials 11
Independent study hours
Independent study 128

### Teaching staff

Staff member Role
Simon Cotter Unit coordinator

The independent study hours will normally comprise the following. During each week of the taught part of the semester:

·         You will normally have approximately 75-120 minutes of video content. Normally you would spend approximately 2.5-4 hrs per week studying this content independently

·         You will normally have exercise or problem sheets, on which you might spend approximately 2-2.5hrs per week

·         There may be other tasks assigned to you on Blackboard, for example short quizzes, short-answer formative exercises or directed reading

·         In some weeks you may be preparing coursework or revising for mid-semester tests

Together with the timetabled classes, you should be spending approximately 9 hours per week on this course unit.

The remaining independent study time comprises revision for and taking the end-of-semester assessment.

The above times are indicative only and may vary depending on the week and the course unit. More information can be found on the course unit’s Blackboard page.