M2 - Modeling and Numerical Analysis (MANU)

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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