# MA4K0 Introduction to Uncertainty Quantification

**Lecturer: **Dr Tim Sullivan

**Term(s): **Term 2

**Status for Mathematics students: **List A

**Commitment: **30 hours of lectures

**Assessment: **Three hour exam in summer (50%), term-time assessments (50%)

Assessment in term-time will be a mix of written assessed work and a min-project.

**Prerequisites:**

Essential: ST112 Probability B, MA3G7 Functional Analysis I and either MA359 Measure Theory or ST318 Probability Theory.

Useful or related:

MA4A2 Advanced PDEs, ST407 Monte Carlo Methods.

Some programming background in e.g. C, Mathematica, Matlab, Python, or R.

**Leads to:**

Graduate study in a range of problems at the interface of differential equations and probability, including UQ theory, data assimilation, inverse problems and filtering. These subjects may be studied within mathematics departments, or in applications departments throughout the sciences and engineering.

**Content: **This is a list of *possible* topics, not all of which will necessarily be covered in the module. In particular, sections marked *** are likely to be omitted.

**Introduction and Course Outline**- Typical UQ problems and motivating examples: uncertainty propagation, inverse problems, certification, prediction
- Epistemic and aleatoric uncertainty. Bayesian and frequentist interpretations of probability

**Preliminaries**- Recap of Hilbert space theory [e.g. from MA3G7 Functional Analysis I]: direct sums; orthogonal decompositions and orthogonal projection; compact operators
- Recap of measure/probability theory [e.g. from MA359 Measure Theory or ST318 Probability Theory]: basic axioms for measure/probability spaces, Lebesgue integration of real-valued functions
- More Hilbert space theory: tensor products; trace-class and Hilbert–Schmidt operators; scales of Hilbert spaces
- More probability theory (on function spaces): Bochner integration of vector-valued functions; probability measures (especially Gaussian measures) on function spaces; various representations of random functions, e.g. Karhunen–Loève expansions, random series, gPC expansions, Gaussian processes and Gaussian mixtures

**Inverse Problems and Bayesian Perspectives**- Examples of inverse problems (linear and nonlinear, static and dynamic) and their ill-posedness
- Deterministic solution of linear inverse problems in Hilbert spaces; the Moore–Penrose pseudo-inverse
- Regularisation of inverse problems: regularisation of operators; convergence of regularisation schemes; variational regularisation; perspectives on nonlinear inverse problems
- Bayesian inverse problems: formulation and well-posedness; special treatment of linear Gaussian problems and the Kálmán filter

**Computational Methods**- Deterministic numerical evaluation of integrals: univariate and multivariate quadrature rules; sparse quadratures
- Random quadratures: Monte Carlo; Markov chain Monte Carlo; dimension-independent proposals.
- Particle and sequential Monte Carlo methods for sampling: ensemble Kálmán filtering and inversion
- Pseudo-random methods***: quasi-Monte Carlo, low-discrepancy sequences, Koksma–Hlawka inequality
- Intrusive and non-intrusive calculation of gPC expansions: stochastic Galerkin, non-intrusive spectral projection, stochastic collocation
- Visualisation of uncertainty***
- Computing with Gaussian processes***

**Sensitivity Analysis*****- Estimation of derivatives
- “
*L*^{∞}” sensitivity indices, e.g. McDiarmid subdiameters; associated concentration-of-measure inequalities - ANOVA and “
*L*^{2}” sensitivity indices, e.g. Sobol' indices - Active subspaces and model reduction

**Second-Order ("Knightian") Uncertainty*****- Mixed epistemic/aleatoric uncertainty; the robust Bayesian paradigm
- Finite-dimensional parametric studies; convex programs
- Optimal UQ / distributionally-robust optimization: formulation, reduction, computation

**Aims:**

Uncertainty Quantification (UQ) is a research area of theoretical and practical importance at the intersection of applied mathematics, probability, statistics, computational science and engineering (CSE) and many application areas. UQ can be seen as the theory and numerical application of probability/statistics to problems and models with a strong “real-world” (especially physics- or engineering-based) setting.

This course will provide an introduction to the basic problems and methods of UQ from a mostly mathematical point of view, with numerical exercises so that the methods can be seen to work in (small) practical settings. More generally, the aim is to provide an introduction to some relatively diverse methods of applied mathematics and applied probability as they are used in practice, through the particular unifying theme of UQ.

**Objectives:**

By the end of the module students should be able to understand both the basic theory of, and in example settings perform:

- deterministic and Bayesian solution of inverse problems
- forward propagation of uncertainty
- orthogonal systems of polynomials and their diverse applications
- data assimilation and filtering
- finite- and infinite-dimensional optimization methods
- sensitivity and variance analysis

**Literature:**

The following books may be of interest:

- T. J. Sullivan.
*Introduction to Uncertainty Quantification*, Texts in Applied Mathematics, Springer, 2015. doi:10.1007/978-3-319-23395-6 - R. C. Smith.
*Uncertainly Quantification: Theory, Implementation, and Applications*, SIAM, 2013. ISBN:978-1-611973-21-1 - K. J. H. Law, A. M. Stuart, and J. Voss.
*Data Assimilation: A Mathematical Introduction*, Texts in Applied Mathematics, Springer, 2015. doi:10.1007/978-3-319-20325-6 - O. P. Le Maître and O. M. Knio.
*Spectral Methods for Uncertainty Quantification. With Applications to Computational Fluid Dynamics*. Scientific Computation, Springer, 2010. doi:10.1007/978-90-481-3520-2 - D. Xiu.
*Numerical Methods for Stochastic Computations. A Spectral Method Approach*. Princeton University Press, 2010. ISBN:978-0-691-14212-8