Ingénieur 2A

This course has three objectives:

The first one is to introduce mathematical statistics tools and statistical learning ("machine learning"). We will describe everything from the choice of a statistical model, to parameter estimation, inference and model selection. We will learn how to build estimators, tests and classification rules, and how to evaluate the performance of these rules. We will introduce a number of theoretical tools - decision theory, empirical process.

The second objective is to describe, in the course and in small classes, concrete examples of modeling in various fields (signal and image processing, econometrics, environmental sciences, shape classification, etc.).

The third objective is to develop a well-founded practical savoir-faire enabling students to understand how theoretical tools can be implemented in concrete applications (use of R or Python).

The last two courses will be devoted to an introduction to statistical learning.

Evaluation: Written exam, two take-home assignments, two quizzes.

Partial differential equations play a fundamental role in modeling complex phenomenon in fields as varied as mechanics, physics, and biology. Since the 1950s and the advent of computers, the development and use of numerical methods for the approximate machine calculation of solutions to partial differential equations have become routine in engineering. In automotive engineering, for example, not only deformations of the passenger compartment in the event of an impact but also air conditioning, ambient noise and electromagnetic compatibility are nowadays calculated by computer.

The course aims to highlight the link between classical mechanical and physical models based on partial differential equations, the underlying mathematical analysis, and the development of the finite element method. The common thread will be the variational point of view, which enables problems to be rewritten as minimization problems, making the link with optimization. A significant part of the course will be devoted to machine implementation of the finite element method and to the explicit approximate solution of certain partial differential equations using FreeFem++ software. In particular, students will be asked to carry out a mini-project using this software.

In addition, some of the PCs will illustrate on computer the concepts seen in class.

Evaluation: A written exam, quiz and mandatory mini-project.

Introduction to Applied Mathematics through an experimental approach

The aim of this project is to propose methods for solving the resulting optimization issues, implement these methods and test them.

Topic 1(Frédéric Meunier): the organization of the production line in "Bucket Brigades" ensures great flexibility, while often having a production rate that is close to optimal. For order-picking lines, this type of organization presents a number of difficulties, and Pyung-Hoi Koo (Operations Research Spectrum, 2009) proposed ways to offset them while leaving several aspects open to further optimization.

Topic 2 (Frédéric Meunier): it is considered a workshop production issue, in which employees have a variety of qualifications. Products arrive according to a Poisson process, and each must visit a subset of the machines in a specific order. The arrival rate being quite high, queues can be formed in front of machines. As the employees are less numerous than the machines, the challenge is to pick which machine will be sent to an employee who has just completed a task. The aim of the project is to find the assignment rules that will keep the average stay time as low as possible.

Topic 3 (Eric Gourdin): We're interested in routing in Internet networks, which we'll first simulate and then optimize. The network is modeled by a graph whose links (edges/arcs) are equipped with capacities. We also have a set of requests, each request is defined by a source node, a destination node, and a volume of traffic to be handled. It is assumed that the network uses a shortest-path routing protocol: each link is assigned a (presumably given) administrative metric value, and traffic between the source and destination nodes is routed along the shortest paths (as defined by these administrative metrics). If there is more than one shortest path, traffic is distributed over the paths in accordance with the ECMP protocol. Link load is the ratio of traffic flow and capacity. Routing "quality" is measured by calculating the load of the fullest link. We'll take an interest in routing protocol evolution called  "Segment Routing" where a request can be routed on a continuation of segments, the routing within each segment being by itself managed by the shortest paths.

Topic 4 (Eric Gourdin): in an Internet network, we're seeking to calculate a "robust" routing called "oblivious routing". The general context is similar to the previous topic: a graph with links provided with capacities, a set of requests to be handled. Unlike the previous topic, the routing to be calculated is totally free (traffic can be split over as many paths as you like, as long as you satisfy the flow conservation on each intermediate vertex). It is assumed that request volumes are unknown. We are looking for a routing that is valid for all possible requests and such the maximum overall request sets of the ratio between the max load with this single routing and the ratio obtained with a routing that would be optimized for this request set be as small as possible (oblivious routing). The modal approach consists in offering ways of calculating (or approximating) this "oblivious" routing.