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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In this course, we will present analytical methods for solving partial differential equations (PDEs)—including nonlinear ones—and for studying the qualitative behavior of solutions. The class of PDEs—and thus the methods studied—may vary depending on the instructor. They will be related to the applications developed at IMAG: fluid mechanics, solid mechanics, and mathematical biology.

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This course is a continuation of the optimization course offered in the second semester of the L3 Mathematics program and the course on optimization and machine learning in the M1 MANU program. The course builds on the concepts covered in other modules of the MANU master’s program in PDE analysis and numerical simulation.

Following a review of the results and numerical methods for the numerical simulation of PDEs on adaptive meshes, results on a posteriori error estimation, and supervised learning methods covered in the M1 program, the course focuses on the generation of high-quality databases and their completion and certification through certified numerical simulation based on error control.

This issue is fundamental to the certified use of machine learning in industry. Indeed, the accuracy of mathematical learning during inference depends heavily on the quality of the database.

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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We study inverse problems, emphasizing the concept of a “well-posed problem.”

First, inverse problems in finite dimensions are introduced (the concepts of conditioning, singular values, etc.), followed by those in infinite dimensions (problem stability, regularization, pseudo-inverse, etc.).

Next, we review the properties of the Fourier transform and the convolution operation in Lp spaces and their usual subspaces. We examine how these two operations lead to well-posed or ill-posed problems. For convolution, in particular, the concepts of “filter” and “deconvolution” are discussed in the context of signal and image analysis.

Finally, we apply these concepts to the study and inversion of the Radon transform—or related transforms—derived from medical imaging techniques.

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This course focuses on the study of advanced numerical methods for partial differential equations that allow for the use of polyhedral meshes. The first part of the course is devoted to analytical tools of general interest. In the second part, we focus on the design and analysis of hybrid high-order methods, which are an example of state-of-the-art numerical methods. In the third part, we develop applications of these methods related to the research activities currently underway at IMAG: fluid mechanics, solid mechanics, and flow in porous media.

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A minimum 4-month internship at a company, a public industrial and commercial establishment (EPIC), or a research laboratory, supervised by a researcher, a faculty researcher, or a research engineer.

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This course covers advanced aspects of the C++ language as applied to scientific computing, supplemented by an introduction to pre- and post-processing tools, modern collaborative work tools (version control systems), and non-regression testing tools (test management systems).

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This course allows students to become familiar with real-world problems involving the theoretical or numerical solution of partial differential equations.

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