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

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This course is a continuation of the optimization course from the second semester of the third year of mathematics and the course on optimization and machine learning from the first year of the MANU master's program. The course builds on the content covered in other modules of the MANU master's program in PDE analysis and numerical simulation.

After reviewing the results and numerical methods for numerical simulation of PDEs on adaptive meshes, a posteriori error estimation results, and supervised learning methods from the M1, the course focuses on the generation of high-quality databases and their completion and certification through certified numerical simulation using error control.

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

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

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

Next, the properties of the Fourier transform and the convolution operation are reviewed 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, these concepts are applied to the study and inversion of Radon transforms, 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 the use of polyhedral meshes. The first part of the course is devoted to general analysis tools. The second part focuses on the design and analysis of Hybrid High-Order methods, which are an example of the latest generation of numerical methods. The third part develops applications of these methods in connection with the research activities carried out at IMAG: fluid mechanics, solid mechanics, and flows in porous media.

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Minimum 4-month internship in a company, EPIC (public industrial and commercial establishment), or research laboratory, supervised by a researcher, teacher-researcher, or research/development engineer.

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This course covers advanced aspects of the C++ language applied to scientific computing, supplemented by a presentation of pre/post-processing tools, modern collaborative work tools (version managers), and non-regression tools (test managers).

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

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