MMath&Phys Mathematics and Physics

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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.

Please note

Students are not permitted to take, for credit, MATH44082 in an undergraduate programme and then MATH64082 in a postgraduate programme at the University of Manchester, as the courses are identical.

• 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.

1. Representation of Random Inputs [3]

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 [3]

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

3. Sampling-based methods for uncertainty in ODEs [4]

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

4. Numerical Integration [5]

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

5. Galerkin approximation [3]

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

6. Stochastic Spectral Methods [4]

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

7. Gaussian Process Regression. [5]

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

• Mid-semester coursework: 20%  
• Written exam : 80% 

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.

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.