# LMS short course: Introduction to the Bayesian approach to inverse problems

### When

11 May, 2017

### Where

University of Edinburgh

### Organisers

### Description

Uncertainty Quantification (UQ) is an interesting, fast growing research area aimed at developing methods to address the impact of parameter, data and model uncertainty in complex systems. In this course we will focus on the identification of parameters through observations of the response of the system - the inverse problem. The uncertainty in the solution of the inverse problem will be described via the Bayesian approach. We will derive Bayes' theorem in the setting of finite dimensional parameter spaces, and discuss properties such as well-posedness, statistical estimates and connections to classical regularization methods. We will briefly examine the extension of the Bayesian approach to the infinite dimensional setting. The remainder of this course will be devoted to algorithms for the efficient approximation of the solution of the Bayes inverse problem. We will give a detailed discussion on Monte Carlo methods in this context, and review alternative approaches such as multilevel Monte Carlo and Markov-chain Monte Carlo.

### Expected Audience

PhD and MSc students, and other researchers with a maths-based degree (e.g. mathematics, statistics with maths component, computer science, physics, engineering etc), as the focus of the course is on theory and algorithms.

### Lecturers

- Dr Aretha Teckentrup (University of Edinburgh and the Alan Turing Institute)
- Dr Claudia Schillings (Mannheim, Germany)

### Schedule

Time | Title | Lecturer |

9:15 - 10:30 | Mathematical Foundations of Bayesian Inverse Problems | Dr Claudia Schillings |

11:00 - 12:00 | Well-posedness and Statistical Estimates in Bayesian Inverse Problems | Dr Claudia Schillings |

12:00 - 13:00 | Lunch Break | |

13:00 - 14:15 | Convergence Properties of Ratio Estimators for Posterior Expectations | Dr Aretha Teckentrup |

14:45 - 15:45 | Convergence Properties of Markov-chain Monte Carlo Estimators for Posterior Expectations | Dr Aretha Teckentrup |

16:00 - 16:45 | Discussion on Current Research Directions | Led by Dr Claudia Schillings and Dr Aretha Teckentrup |