Modeling and Numerical Analysis (MANU)

Partial differential equations (PDEs) are nowadays an essential mathematical tool in the study and understanding of physical and biological phenomena. Their great complexity often makes them impossible to solve analytically; hence the need for numerical resolution methods.

This course is dedicated to the introduction of PDEs, then to their resolution using well-known numerical schemes such as finite difference and finite volume methods. A more analytical part, necessary for the introduction of finite volume methods, will be devoted to the analytical resolution of scalar conservation laws. Four programming practical exercises will illustrate the scientific computing tools learnt in class with simple examples.

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

This course is dedicated to the introduction of PDEs, then to their resolution using well-known numerical schemes such as finite difference and finite volume methods. A more analytical part, necessary for the introduction of finite volume methods, will be devoted to the analytical resolution of scalar conservation laws. Four programming practical exercises will illustrate the scientific computing tools seen in class with 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, and in particular functional analysis. This course introduces the first important tools for solving PDEs from analytical or geometrical points of view. These tools will be put to use in the study of a few examples of PDEs representative of major classes of equations.

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

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

After a review of results and numerical methods for first- and second-order optimization problems, unconstrained and constrained by equality and inequality, the course focuses on issues of current interest in industrial optimization, in particular robust, multicriteria optimization in the presence of uncertainty.

The course then illustrates the role of optimization in the main machine learning algorithms. These issues are illustrated by 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 (crossfold, overfitting, etc.).

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

Database management issues are addressed: generation, imputation, visualization, slicing.

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

The course includes a significant number of ongoing IT projects. All sessions take place in a computerized environment, enabling immediate implementation of theoretical elements.

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This course complements the concepts developed in PDE Analysis 1, and provides an opportunity to study in depth certain linear PDEs posed on an open space 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 introduces the basics of continuum mechanics: we study motions, deformations and stress fields in media that we consider from a macroscopic point of view, as opposed to a corpuscular description. More precisely, we analyze these physical phenomena by describing them mathematically.

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Internship supervised by a researcher or teacher-researcher.

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Introductory course in differential geometry, focusing on the notions of subvarieties of ^Rn, vector fields and flows.

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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 a variety of problems and give the keys to computer implementation of the method.

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This course will present analytical methods for solving partial differential equations (PDEs) -- possibly nonlinear -- and studying the qualitative behavior of solutions. The class of PDEs and therefore the methods studied may depend on the speaker. They will be related to the applications developed within IMAG: fluid mechanics, solid mechanics, maths-bio.

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This course is a continuation of the optimization course in the second semester of L3 Mathematics and the optimization and machine learning course in M1 MANU. The course builds on the ingredients given in the other MANU master's modules in PDE analysis and numerical simulation.

After a reminder of the results and numerical methods for numerical simulation of PDEs on adaptive meshes, a posteriori error estimation results, and supervised learning methods from M1, the course looks at the generation of quality databases and their completion and certification thanks to numerical simulation certified by error checking.

This question is fundamental to the certified use of machine learning in industry. Indeed, the accuracy of mathematical learning during inference is strongly conditioned by the quality of the database.

The course includes a significant number of ongoing IT projects. All sessions take place in a computerized environment, enabling immediate implementation of theoretical elements.

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

Inverse problems are first presented in finite dimension (notion of conditioning, singular values, etc.), then in infinite dimension (problem stability, regularization, pseudo-inverse, etc.).

Secondly, 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 problems that are well or badly posed. For convolution, in particular, the notions 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 covers advanced numerical methods for partial differential equations using polyhedral meshes. The first part of the course is devoted to analysis tools of general interest. The second part focuses on the design and analysis of Hybrid High-Order methods, an example of the latest generation of numerical methods. In the third part, applications of these methods are developed in relation to IMAG's research activities in fluid mechanics, solid mechanics and flows in porous media.

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Minimum 4-month internship in a company, EPIC or research laboratory, supervised by a researcher, teacher-researcher or design/research engineer.

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This course covers advanced aspects of the C++ language applied to scientific computing, complemented 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 introduces students to real-life problems involving the theoretical or numerical solution of partial differential equations.

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