# MSc Applied Mathematics / Course details

Year of entry: 2020

## Course unit details:Advanced Uncertainty Quantification

Unit code MATH64082 15 FHEQ level 7 – master's degree or fourth year of an integrated master's degree Semester 2 Department of Mathematics No

### Overview

This unit introduces theoretical tools and numerical methods for incorporating random inputs into models consisting of differential equations.  We begin by introducing stochastic processes and random fields and numerical methods for simulating them. We then introduce the multilevel Monte Carlo method for propagating uncertainty in ODE models with random inputs and sparse grid techniques for estimating intergrals in high dimensions. Finally, we investigate intrusive and non-intrusive surrogate modelling techniques in the form of stochastic Galerkin approximation and Gaussian process regression.

Although the concepts and tools introduced in this module will require a theoretical grounding, the primary intention is to focus on the application of the methods to models consisting of ordinary and partial differential equations, derived from environmental, industrial and biological applications. Computational exercises will reinforce understanding of the methods introduced and their theoretical properties.

### Pre/co-requisites

Unit title Unit code Requirement type Description
Introduction to Uncertainty Quantification MATH64071 Pre-Requisite Compulsory
Pre-Requsites for MATH64082

### Aims

To further develop the ideas introduced in the first semester course An Introduction to Uncertainty Quantification (MATH64071), focusing on the representation of random inputs, multilevel Monte Carlo sampling, numerical integration in more than one dimension and intrusive and non-intrusive surrogate modelling techniques. Again, the focus will be on UQ for models consisting of differential equations, of the type frequently encountered by applied mathematicians working on environmental, industrial and biological applications.

### Learning outcomes

• Represent second order random fields as series expansions and explain key theoretical results.
• Describe and implement numerical methods for generating realisations of second order random fields on one and two-dimensional domains.
• Apply multilevel Monte Carlo sampling to ODEs with random inputs in combination with standard time-stepping methods, and analyse the associated error.
• Construct and implement standard tensor product quadrature rules in multiple dimensions and explain their disadvantages.
• Derive sparse grid approximation rules, apply them to the computation of expectations and other statistical quantities of interest, and state key approximation theory results.
• Explain the concept of a surrogate model for differential equations with random inputs and state common intrusive and non-intrusive approaches.
• Define the concept of a weak solution for test problems consisting of differential equations and derive the finite-dimensional problems associated with Galerkin approximation.
• Recognise families of orthogonal polynomials associated with common probability distributions and explain how to construct appropriate spaces of multivariate polynomials for stochastic Galerkin approximation.
• Describe and implement stochastic Galerkin approximation schemes for test problems consisting of differential equations with random inputs, and perform error analysis.
• Explain how to apply Gaussian process regression to approximate a function whose value is known only at a finite set of points and derive the predictive distribution from the prior.
• Implement Gaussian process regression for selected test problems consisting of differential equations with random inputs and analyse the properties of the predictive mean.

### Syllabus

1. Representation of Random Inputs 

Stochastic processes/random fields. Stationary and isotropic cases. Covariance functions and regularity results. Mercer's theorem. Hilbert–Schmidt theorem. Karhunen-Loeve expansions. Examples of ODEs and PDEs with random inputs.

2. Numerical Methods for Generating Random Fields 

Cholesky factorisation, singular value decomposition, circulant embedding in one dimension.

3. Sampling-based methods for uncertainty in ODEs 

Multilevel Monte Carlo sampling. Telescoping sums. Error analysis and comparison to standard Monte Carlo sampling.

4. Numerical Integration 

Review of Newton-Cotes and Gauss rules in one dimension. Tensor product rules. Sparse grid integration and interpolation in higher dimensions.

5. Galerkin approximation 

Hilbert spaces. Riesz representation theorem. Lax-Milgram Lemma. Weak solution of differential equations. Galerkin approximation.

6. Stochastic Spectral Methods 

Univariate orthogonal polynomials. Legendre and Hermite polynomials. Multivariate orthogonal polynomomials. Stochastic Galerkin approximation.

7. Gaussian Process Regression. 

Statistical models. Linear regression. Gaussian processes and conditioning. Choice of prior. Approximation theory and link to radial basis functions.

### Assessment methods

Method Weight
Other 20%
Written exam 80%
• Mid-semester coursework: 20%

• Written exam : 80% (3 Hours)

### 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.

Ralph Smith, Uncertainty Quantification, SIAM, 2014.

C. E. Rasmussen & C. K. I. Williams, Gaussian Processes for Machine Learning, the MIT Press, 2006

J. Voss, An Introduction to Statistical Computing: A Simulation-based Approach, Wiley, 2013.

T.J. Sullivan, Introduction to Uncertainty Quantification, Springer, 2015.

G.J. Lord, C.E. Powell, T. Shardlow. An introduction to computational stochastic PDEs. Cambridge University Press, 2014.

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### Study hours

Scheduled activity hours
Lectures 27
Tutorials 6
Independent study hours
Independent study 117

### Teaching staff

Staff member Role
Catherine Powell Unit coordinator