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 methods.

This course is devoted to an introduction to EDPs, followed by their solution using well-known numerical methods such as the finite difference and finite volume methods. A section focused more on analysis—necessary for the introduction to the finite volume method—will be devoted to the analytical solution of scalar conservation laws. Four programming lab sessions will use simple examples to illustrate the scientific computing tools covered in class.

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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 devoted to an introduction to EDPs, followed by their solution using well-known numerical methods such as the finite difference and finite volume methods. A section focused more on analysis—necessary for the introduction to the finite volume method—will be devoted to the analytical solution of scalar conservation laws. Four programming lab sessions will use simple examples to illustrate the scientific computing tools covered in class.

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The construction of solutions to partial differential equations (PDEs) and the theoretical study of their qualitative behavior are based primarily on the use of results from analysis, particularly functional analysis. This course introduces key tools for solving PDEs from analytical or geometric perspectives. These tools will be applied in the study of several examples of PDEs representative of major classes of equations.

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This course unit covers the classical theory of Banach spaces and also serves as an introduction to spectral analysis.

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This course is a continuation of the optimization course offered in the second semester of the third year of the bachelor’s program.

After reviewing the results and numerical methods for first- and second-order optimization problems—both unconstrained and subject to equality and inequality constraints—the course focuses on topics currently of interest in industrial optimization, particularly robust, multi-criteria optimization under uncertainty.

The course then illustrates the role of optimization in the main machine learning algorithms. These topics are illustrated with 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 (cross-validation, overfitting, etc.).

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

Topics related to database management are covered: generation, imputation, visualization, and segmentation.

The course explores the connections between transfer learning in mathematics and numerical simulation to address issues such as the generation of synthetic datasets, imputation, non-intrusive prediction, fast inference, and more.

The course includes a significant number of ongoing computer projects. All sessions take place in a computer-based environment and allow students to immediately apply the theoretical concepts they have learned.

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This course builds on the concepts covered in the course “Analysis of PDEs 1.” In particular, it provides an opportunity to study in depth certain linear PDEs defined 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 covers the fundamentals of continuum mechanics: it examines motion, deformation, and stress fields within media considered from a macroscopic perspective, as opposed to a particle-based description. More specifically, these physical phenomena are analyzed by describing them from a mathematical perspective.

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Internship supervised by a female or male researcher or a female or male faculty member who is also a researcher.

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An introductory course in differential geometry, focusing on the concepts of subvarieties of^Rn, vector fields, and flows.

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

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The finite element method is a widely used numerical method. This course will explain the principles of the method, present the equations relevant to a variety of problems, and provide the keys to implementing the method computationally.

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