# 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

There will be weekly exercises in this module, which will involve a mixture of theoretical and computer-based questions.

**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.

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

**Preliminaries**- Hilbert space theory: direct sums; orthogonal decompositions and approximations; tensor products; Riesz representation and Lax–Milgram theorems. [Mostly recap of MA3G7 Functional Analysis I.]
- Probability theory: axioms, integration, sampling, key inequalities and limit theorems. [Mostly recap of MA359 Measure Theory / ST318 Probability Theory.]
- Optimization: least squares; linear/quadratic/convex programming; extreme points.

**Inverse Problems and Bayesian Perspectives**- Ill-posedness of inverse problems, regularization.
- Bayesian inversion in Banach spaces.
- State estimation and data assimilation, e.g. Kálmán filter.

**Orthogonal Polynomials**- Basic definitions and properties.
- Polynomial interpolation and approximation.

**Numerical Evaluation of Integrals**- Deterministic methods: uniform sampling, Newton–Cotes formulae, Gaussian quadrature, Clenshaw–Curtis quadrature, sparse quadrature.
- Random methods: Monte Carlo and variants.
- Pseudo-random methods: low-discrepancy sequences, Koksma–Hlawka inequality.

**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. - Model reduction.

**Spectral Methods**- Polynomial chaos: Wiener–Hermite expansions, generalized PC expansions, changes of PC basis.
- Intrusive (Galerkin) methods: deterministic and stochastic Galerkin projection.
- Non-intrusive spectral projection, stochastic collocation methods.

**Optimization Methods**- 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 growing 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:

- sensitivity and variance analysis
- orthogonal systems of polynomials and their applications
- spectral decomposition methods
- finite- and infinite-dimensional optimization methods
- data assimilation and filtering
- Bayesian perspectives on inverse problems.

**Literature:**

The course will be based on two sets of lectures notes; details of how to access these are given under Additional Resources.

The following books may also be of interest:

Berger, James O. "An overview of robust Bayesian analysis." *Test* **3**(1):5–124, 1994.

Le Maître, O. P.; Knio, O. M. *Spectral methods for uncertainty quantification. With applications to computational fluid dynamics*. Scientific Computation. Springer, New York, 2010. xvi+536 pp. ISBN: 978-90-481-3519-6

Xiu, Dongbin. *Numerical methods for stochastic computations. A spectral method approach*. Princeton University Press, Princeton, NJ, 2010. xiv+127 pp. ISBN: 978-0-691-14212-8