M1 - Modeling and Numerical Analysis (MANU)

Partial Differential Equations (PDE) are nowadays an essential mathematical object for the study and understanding of physical or 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 solution 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 exercises will illustrate in simple examples the tools of scientific calculation seen in the course.

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Description : Partial Differential Equations (PDE) are nowadays an essential mathematical object for the study and understanding of physical or 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 solution 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 exercises will illustrate the scientific computing tools seen in the course with simple examples.

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The construction of solutions of partial differential equations (PDEs) and the theoretical study of their qualitative behavior is essentially based on the use of analytical results and in particular of functional analysis. This course presents the first important tools for the solution of PDEs via analytical or geometrical points of view. These tools will be implemented in the study of some examples of PDEs representative of large classes of equations.

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

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

After a reminder of the 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, and in particular, robust, multi-criteria optimization, in the presence of uncertainties.

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

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

The issues surrounding database management are addressed: generation, imputation, visualization, slicing.

The course presents the links between transfer learning and numerical simulation to address issues of synthetic database generation, imputation, non-intrusive prediction, rapid inference, etc.

The course includes an important part of computer projects along the way. All the sessions take place in a computerized environment and allow an immediate implementation of the theoretical elements.

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This course completes the notions developed in the course Analysis of PDEs 1. In particular, it is an opportunity to study in depth some linear PDEs posed on an open of Rn, such as the Dirichlet problem, the heat equation, the Schrödinger equation or the wave equation.

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This 42-hour course gives the basic elements of continuum mechanics: we study the motions, deformations and stress fields within 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 from a mathematical point of view.

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Internship supervised by a researcher or teacher-researcher.

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An 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, give the useful equations on various problems and give the keys for the computer implementation of the method.

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