Mathematical and numerical tools for uncertainty analysis

The aim of this course is to give students initial experience in uncertainty analysis for decision support. To achieve this, they will be required to work with machine learning tools (random variables, sampling, linear or non-linear regression models, i.e. neural networks, Bayesian inference, etc.). However, the course does not require any pre-requisites on these topics, and will only provide a reminder (targeted to needs).
In this course, we will answer "What is uncertainty analysis?", "Why do we need it (as a complement to numerical analysis and modeling)?", "What are its objectives?", "Why is it very important in an engineering curriculum and in the industrial world in general?".
At the end of this course, students will be familiar with the difficulties encountered in the industrial world in terms of uncertainties. They will be able to formalize the problem mathematically, identify the type of uncertainty analysis problems they are faced with and propose simple solutions (references will supplement the course when simple solutions are not applicable/possible). Some simplified problems representative of industrial uncertainty analysis/decision-aid issues will be addressed. At the end of the course, students will have written codes enabling the tools to be applied to any physics model/code/experimental data.
Among the main types of uncertainty analysis studies covered in this course are
- uncertainty propagation: transformation of random variables through a model or simulation code (difficulties and issues).
- sensitivity analysis: given a model (or code), determine which input variables explain most of the fluctuations in a model/code output, or which variables are negligible.
- Calibration under uncertainties: given physical experimental results fraught with uncertainties, how can we trace model parameters and their uncertainties, and how can we rigorously identify a model error?
- Probability of failure: using simulation to guarantee in an uncertain context.
- Metamodeling: when should we use reduced models to achieve our objectives? How can we control their errors and ensure that they are usable? Which models should be used (linear regression, kriging, neural networks, deep or not)?

Uncertainty analysis for decision support. What is uncertainty analysis? Why is it needed (as a complement to numerical analysis and modeling)? What are its objectives? Why is it so important in an engineering curriculum and in the industrial world in general?