M1 - Modeling and Numerical Analysis (MANU)

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

This course is dedicated to introducing EDPs and then solving them using well-known numerical methods such as finite difference and finite volume methods. A more analytical section, necessary for introducing finite volume methods, will be devoted to the analytical solution of scalar conservation laws. Four programming labs will illustrate the scientific computing tools covered in class using simple examples.

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

This course is dedicated to introducing EDPs and then solving them using well-known numerical methods such as finite difference and finite volume methods. A more analytical section, necessary for introducing finite volume methods, will be devoted to the analytical solution of scalar conservation laws. Four programming labs will illustrate the scientific computing tools covered in class using 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, particularly functional analysis. This course presents important initial tools for solving PDEs from analytical or geometric perspectives. These tools will be applied in the study of several examples of PDEs representative of large classes of equations.

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

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

After reviewing the results and numerical methods for first- and second-order optimization problems, without constraints and under equality and inequality constraints, the course focuses on issues of current interest in industrial optimization, in particular robust, multi-criteria optimization in the presence of uncertainty.

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

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

Issues surrounding database management are addressed: generation, allocation, visualization, and segmentation.

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

The course includes a significant amount of ongoing IT projects. All sessions take place in a computerized environment and allow for immediate implementation of theoretical concepts.

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This course complements the concepts developed in the EDP Analysis 1 course. In particular, it provides an opportunity to study in depth certain linear EDPs posed on an open set 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 provides the basics of continuum mechanics: we study movements, deformations, and stress fields within media that are considered from a macroscopic point of view, as opposed to a corpuscular description. More specifically, 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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Introductory course in differential geometry, focusing on the concepts of submanifolds of^Rn, vector fields, and flow.

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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 various problems, and give the keys to implementing the method in computer science.

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