Differential Calculus, Wavelets and Applications
M2 MSIAM (DS, MSCI)
Sylvain Meignen and Kevin Polisano
Numerical simulation is ubiquitous in today’s world. Initially confined to well-mastered physical problems, it has spread to all fields (oceanography, biology, ecology, etc.), the aim being to make forecasts of the systems under study. This has been possible thanks to the combination of numerical models and access to a considerable amount of data. However, there are many sources of uncertainty in these modelling systems. They can come from poorly known processes, approximations in the model equations and/or in their discretization, partial and uncertain data, … The objective of this course is to explore in depth the mathematical methods that have allowed these two worlds to meet. Firstly, we will focus on sensitivity analysis approaches that allow us to study the behavior of the system and its response to perturbations. In particular, this permits to study the way in which uncertainties are propagated. Next, we will look at data assimilation methods that aim at reducing said uncertainties by combining numerical models and observation data. Finally, the notions of model reduction will be discussed, which allow the implementation of the previous methods on high dimensional problems.
This course is intended for DS and MSCI students and will start with a differentiated refresher course on the necessary basic mathematical notions.
Part 1: Differential calculus in finite and infinite dimensions (S. Meignen), 1D and 2D wavelet transforms in continuous and discrete paradigms (K. Polisano)
Part 2: Image processing (S. Meignen & K. Polisano):
- Edge detection: Canny method and multiscale detector
- Compression, denoising and deblurring}: with linear approaches (Wiener filtering, heat equation, wavelet approximation, …) and non linear approaches (anisotropic diffusion e.g Perona-Malick model, minimization approaches, wavelet thresholding, …).
Part 3: Other applications (K. Polisano): the scattering transform and wavelets on graphs;
- N1 = 1 exam
- N2 = 3/4 project + 1/4 lab sessions
Final mark = (N1+N2)/2