Undergraduate-level mathematics (Years 1-3), with advanced Year-3 coursework in probability and statistics (linear algebra, optimization, distribution functions, parametric estimation, limit theorems, and confidence intervals).

Uncertainty quantification is used in analysis and modeling (e.g. sources of uncertainty in a physical system), optimization and operations research (e.g. uncertainties on supply and demand in the modeling of a supply chain), vision and machine learning (processing noisy data...), or quantitative finance (financial risk management...).

Numerical simulation is an essential tool for modeling and analyzing complex systems in many scientific fields (physics, biology, chemistry, finance, economics, etc.). The expectations associated with it are multiple: better identification and control of risks, limitation of the costs of real experiments (car crash tests for example), search for cost-performance trade-offs, etc. The presence of many sources of uncertainty (on the models, their parameters, the reference measurements) nevertheless raises the question of the confidence that can be associated with the decisions and predictions provided by the simulation.

The objective of this course is to present a set of mathematical methods to model, propagate and analyze these uncertainties in numerical simulations. At the end of the module, students will :

- know the theoretical foundations of the usual methods of uncertainty quantification;

- be able to implement these methods (using R) on concrete case studies

The course is composed of 10 sessions of 3 hours each. Each session starts with a 1h30 lecture, and ends with a 1h30 numerical practical works. During the last session, the practical work will be replaced by an exam.

Topics which will be dealt with are :

- Modeling of uncertainties

- Uncertainty propagation (quadrature and Monte Carlo methods)

- Sensitivity analysis - Risk analysis

- Solving inverse problems in the presence of uncertainties

- Data filtering and assimilation

Written exam during the last session.