Modeling and Numerical Analysis (MANU)

Partial Differential Equations (PDE) are nowadays an essential mathematical object for the study and understanding of physical or 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 solution 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 exercises will illustrate in simple examples the tools of scientific calculation seen in the course.

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Description : Partial Differential Equations (PDE) are nowadays an essential mathematical object for the study and understanding of physical or 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 solution 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 exercises will illustrate the scientific computing tools seen in the course with simple examples.

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The construction of solutions of partial differential equations (PDEs) and the theoretical study of their qualitative behavior is essentially based on the use of analytical results and in particular of functional analysis. This course presents the first important tools for the solution of PDEs via analytical or geometrical points of view. These tools will be implemented in the study of some examples of PDEs representative of large classes of equations.

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

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

After a reminder of the 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, and in particular, robust, multi-criteria optimization, in the presence of uncertainties.

The course then illustrates the place of optimization in the main machine learning algorithms. These issues are illustrated by examples of classification and regression problems in supervised learning. These examples are used to discuss issues of metrics and procedures for learning evaluation, validation and inference (crossfold, overfitting, etc).

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

The issues surrounding database management are addressed: generation, imputation, visualization, slicing.

The course presents the links between transfer learning and numerical simulation to address issues of synthetic database generation, imputation, non-intrusive prediction, rapid inference, etc.

The course includes an important part of computer projects along the way. All the sessions take place in a computerized environment and allow an immediate implementation of the theoretical elements.

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This course completes the notions developed in the course Analysis of PDEs 1. In particular, it is an opportunity to study in depth some linear PDEs posed on an open of Rn, such as the Dirichlet problem, the heat equation, the Schrödinger equation or the wave equation.

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This 42-hour course gives the basic elements of continuum mechanics: we study the motions, deformations and stress fields within 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 from a mathematical point of view.

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Internship supervised by a researcher or teacher-researcher.

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An 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, give the useful equations on various problems and give the keys for the computer implementation of the method.

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

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

After a reminder of the results and numerical methods for the numerical simulation of PDEs on adaptive mesh, of the results of a posteriori error estimation, and of the supervised learning methods of the M1, the course focuses on the generation of quality databases and their completion and certification thanks to the numerical simulation certified by an error control.

This question is fundamental for a 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 an important part of computer projects along the way. All the sessions take place in a computerized environment and allow an immediate implementation of the theoretical elements.

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Inverse problems are studied, emphasizing the notion of a "well-posed problem".

A presentation is first made of inverse problems in finite dimension (notion of conditioning, singular values, ...), then in infinite dimension (stability of the problem, regularization, pseudo-inverse...).

In a second step, the properties of the Fourier transform and the convolution operation are recalled in Lp spaces and their usual subspaces. We examine how these two operations lead to well or badly posed problems. For convolution, in particular, the notions of "filter" and "deconvolution" are discussed in the context of signal and image analysis.

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

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This course deals with the study of advanced numerical methods for partial differential equations allowing the use of polyhedral meshes. The first part of the course is devoted to analysis 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 the latest generation of numerical methods. In the third part, we develop applications of these methods in connection with the research activities present at IMAG: 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 study/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), non-regression tools (test managers).

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

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