M1 - Modeling and Numerical Analysis (MANU)

Partial differential equations (PDEs) are nowadays an essential mathematical tool in the study and understanding of physical and biological phenomena. Their great complexity often makes them impossible to solve analytically; hence the need for numerical resolution methods.

This course is dedicated to the introduction of PDEs, then to their resolution using well-known numerical schemes such as finite difference and finite volume methods. A more analytical part, necessary for the introduction of finite volume methods, will be devoted to the analytical resolution of scalar conservation laws. Four programming practical exercises will illustrate the scientific computing tools learnt in class with simple examples.

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Description : Partial differential equations (PDEs) are nowadays an essential mathematical tool for studying and understanding physical and biological phenomena. Their great complexity often makes them impossible to solve analytically; hence the need to use numerical solution methods.

This course is dedicated to the introduction of PDEs, then to their resolution using well-known numerical schemes such as finite difference and finite volume methods. A more analytical part, necessary for the introduction of finite volume methods, will be devoted to the analytical resolution of scalar conservation laws. Four programming practical exercises will illustrate the scientific computing tools seen in class with simple examples.

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The construction of solutions to partial differential equations (PDEs) and the theoretical study of their qualitative behavior is essentially based on the use of analytical results, and in particular functional analysis. This course introduces the first important tools for solving PDEs from analytical or geometrical points of view. These tools will be put to use in the study of a few examples of PDEs representative of major classes of equations.

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This course develops the classical theory of Banach spaces and provides an introduction to spectral analysis.

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This course is a continuation of the optimization course given in the second semester of L3.

After a review of results and numerical methods for first- and second-order optimization problems, unconstrained and constrained by equality and inequality, the course focuses on issues of current interest in industrial optimization, in particular robust, multicriteria optimization in the presence of uncertainty.

The course then illustrates the role of optimization in the main machine learning algorithms. These issues are illustrated by examples of classification and regression problems in supervised learning. These examples provide an opportunity to discuss metrics and procedures for evaluating learning, validation and inference (crossfold, overfitting, etc.).

The course introduces the different classes of learning: unsupervised, supervised, transfer, reinforcement, incremental, etc.

Database management issues are addressed: generation, imputation, visualization, slicing.

The course introduces the links between transfer learning and numerical simulation to address issues such as synthetic database generation, imputation, non-intrusive prediction, rapid inference and more.

The course includes a significant number of ongoing IT projects. All sessions take place in a computerized environment, enabling immediate implementation of theoretical elements.

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This course complements the concepts developed in PDE Analysis 1, and provides an opportunity to study in depth certain linear PDEs posed on an open space of Rn, such as the Dirichlet problem, the heat equation, the Schrödinger equation and the wave equation.

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This 42-hour course introduces the basics of continuum mechanics: we study motions, deformations and stress fields in media that we consider from a macroscopic point of view, as opposed to a corpuscular description. More precisely, we analyze these physical phenomena by describing them mathematically.

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Internship supervised by a researcher or teacher-researcher.

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Introductory course in differential geometry, focusing on the notions of subvarieties of ^Rn, vector fields and flows.

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This course covers the basic aspects of the C++ language as applied to numerical analysis.

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Finite elements are a widely used numerical method. This course will explain the principles of the method, provide useful equations for a variety of problems and give the keys to computer implementation of the method.

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