• Course title: Computational Science
• Number of ECTS: 4
• Course code: F1_BAINFOR-47
• Module(s): Mathematical Modelling and Computational Sciences 1
• Language: EN
• Mandatory: No

Numerous problems in engineering, physical and economical industries and application domains essentially boil down to minimising a single function: the objective function in optimization terminology. Minimisation is not only the basis of many simulation tools, but also the basis of many parameter identification approaches. Unfortunately, there is not one minimisation method that outperforms the others. In this module, the student will therefore become familiar with three numerical minimisation techniques, each with its own advantages and disadvantages. Furthermore, three ways to deal with constraints in minimisation problems will be considered, also each with its own advantages and disadvantages. The student will implement the minimisation techniques herself/himself in a programming language of her/his choice, hereby effectively implementing her/his own simulations. In this way, the student will truly be exposed to the methods’ advantages and disadvantages, and she/he will capture the relevant technical complexities of the methods. Thus, the aim of the module is to teach the student a variety of unconstrained and constrained minimisation approaches and understand their beneficial and disadvantageous differences.

At the end of the course, the student will be able to:•Understand and be able to work with descent methods. •Understand and be able to work with Newton’s method in optimization.  •Understand and be able to work with quasi-Newton methods in optimization. •Understand and be able to incorporate constraints in objective functions using substitution. •Understand and be able to incorporate constraints in objective functions using the penalty method. •Understand and be able to incorporate constraints using the method of Lagrange multipliers. 

1. Descent methods: steepest descent method, line search using the Armijo rule, conjugate gradient method, compute multivariate derivatives, implement the methods.2. Newton’s method: Compute multivariate second-order derivatives, solve linear systems, implement the method.3. Quasi-Newton methods: line search using the Wolfe conditions, implement the BFGS method and the L-BFGS method.4. Implement constraints in previous unconstrained objective functions using substitution.5. Implement constraints in previous unconstrained objective functions using the penalty method.6. Implement constraints in previous unconstrained objective functions using the method of Lagrange multipliers.

Assessment modality:   Combined assessmentAssessment tasksTask 1: Written exam (20%)Grading scheme: 20 points (0-20)Objectives: Assess the student’s understanding of (1) descent methods and (2) Newton’s method.Assessment rules: The lecture notes and even the internet may be used. However, any means of communication is forbidden.Assessment criteria: The student must use its own implementations, made during the semester, to calculate some minimization problems. Open questions may also be posed, which require a textual response.Task 2: Written exam (20%)Grading scheme: 20 points (0-20)Objectives: Assess the student’s understanding of (3) quasi-Newton methods.Assessment rules: The lecture notes and even the internet may be used. However, any means of communication is forbidden.Assessment criteria: The student must use its own implementations, made during the semester, to calculate some minimization problems. Open questions may also be posed, which require a textual response.Task 3: Written exam (60%)Grading scheme: 20 points (0-20)Objectives: Assess the student’s understanding of (1) descent methods, (2) Newton’s method, (3) quasi-Newton methods, (4) constraint incorporation using substitution, (5) constraint incorporation using the penalty method, (6) constraint incorporation using the method of Lagrange multipliers.Assessment rules: The lecture notes and even the internet may be used. However, any means of communication is forbidden.Assessment criteria: The student must use its own implementations, made during the semester, to calculate some minimization problems. Open questions may also be posed, which require a textual response.Task 4: Written exam – RETAKE (100%)Grading scheme: 20 points (0-20)Objectives: Assess the student’s understanding of (1) descent methods, (2) Newton’s method, (3) quasi-Newton methods, (4) constraint incorporation using substitution, (5) constraint incorporation using the penalty method, (6) constraint incorporation using the method of Lagrange multipliers.Assessment rules: The lecture notes and even the internet may be used. However, any means of communication is forbidden. In principle, again the resit exams consist of 2 midterm exams and one final exam – with the same weights for the final grade as for the standard exams.Assessment criteria: The student must use its own implementations, made during the semester, to calculate some minimization problems. Open questions may also be posed, which require a textual response.

Course materialsSyllabus☒Yes☐NoRemarks:Available on the Moodle page.Literature list☒Yes☐NoRemarks:Lecture notes are provided by the instructor.Moodle page☒Yes☐NoRemarks:https://moodle.uni.lu/course/view.php?id=4194 

• Course title: Probability Theory and Mathematical Statistics I
• Number of ECTS: 5
• Course code: MA_DS-31
• Module(s): Mathematical Modelling and Computational Sciences 1
• Language: EN
• Mandatory: No

At the end of the course, a student should be familiar with the basic concepts of probability theory (event, random variables, distributions) and master the tools that allow him or her to make calculations (calculations of expectations, variances, distributions etc). The student should also be able to master the law of the large numbers as well as the central limit theorem. In particular, he or she must be able to calculate limits of random variables. Given an integral, the student should also be able to provide a probabilistic approximation of it and assess the probability of error.  

What is an event, the probability of an event, a random variable?What are the main notations associated to probability theory?What is a law or a distribution? The main probability distributions: Bernoulli, binomial, Poisson, uniform, exponential, Cauchy, Gaussian. What random phenomena do they model? What is a density with respect to the Lebesgue measure? What is a density with respect to the counting measure? What is the distribution of a random variable?What is the expectation (or mean) of a random variable? How can we interpret it? What is the variance of a random variable?How to calculate the probability that a random variable belongs to a set from its distribution? How to calculate the expectation of a function of a given random variable from its distribution? What is a distribution function? What are its properties? What link densities and distribution functions? What are the quantiles of a real-valued random variable? What are the median and quartiles of a random variable?The main inequalities: the Markov inequality, the Bienaymé-Tchebychev inequality, the Jensen inequality.What does it mean that two random variables are independent? What is the density of a pair of independent random variables? How to generalize to n independent random variables? What is a random vector? What are the marginal distributions? How to calculate them?How to calculate the distributions of random variables (from distribution functions, change of variables)What does it mean that a sequence of random variables converges almost surely? In probability? In distribution? What are the connections between these convergence modes?What is Law of large numbers? What is the Monte Carlo method for calculating integrals?What is the central limit theorem? How can it be used to evaluate the error in the Monte Carlo method.

First sessionCombined assessment (end of course assessment + continuous assessment)Retake examOral exam

• Course title: Functional Analysis
• Number of ECTS: 6
• Course code: F1_MA_MAT_MMCS2-2
• Module(s): Mathematical Modelling and Computational Sciences 1
• Language: EN
• Mandatory: Yes

The aim of the course is to provide students with efficient tools to study linear operators between infinite dimensional vector spaces. In particular, this class will deal with normed vector spaces, Banach and Hilbert spaces, with bounded linear operators on normed vector spaces, fundamental principles such as Fourier analysis, Lebesgue integral, Han-Banach Theorem, Uniform Boundedness Principle or Closed Graph Theorem, and with spectral theory of compact (self-adjoint) linear operators. 

Students will acquire a solid understanding of functional analysis, its fundamental results and basic techniques. In particular, students will understand applications to measure theory, Fourier theory, and the spectral theorem for (unbounded) operators. Students will know the relevance of a theorem, its underlying motivation and a precise idea of its proof. Hopefully, students will demonstrate capacity for mathematical reasoning through analyzing, proving and explaining concepts from functional analysis. 

Functional analysis aims to study infinite-dimensional spaces of functions and features of linear operators on these spaces. Though, from the point of view of Mathematics, functional analysis has its own interest, it plays a crucial role in many related areas, in particular in Physics, Engineering or Finance. Roughly speaking, most “real-life problems” involve non-linear partial differential equations with infinite-dimensional solution spaces of functions (or distributions). This course provides Master students with the basic tools and the fundamental results to develop skills to solve such problems.

Exam modality for the first sessionWritten test on week of October 28, duration 1 hour, 30% of the final grade,Final written exam (scheduled by the administration), 60% of the final grade,Written homework (problems to be determined), 10% of the final grade.Exam modality for the retake examRetake exam (scheduled by the administration) will be in written form.AbsenceA make-up will be possible only for an unavoidable reason supported by a proof.

Note / Literature / Bibliography Introductory Functional Analysis with Applications, by Erwin Kreyszig.Functional Analysis, Spectral Theory and Applications, by Manfred Einsiedler and Thomas Ward.An Introduction to Fourier Analysis, by Russell L. Herman

• Course title: Partial Differential Equations I
• Number of ECTS: 7
• Course code: F1_MA_MAT_GM-3
• Module(s): Mathematical Modelling and Computational Sciences 1
• Language: EN
• Mandatory: Yes

The goal of the course it to get acquainted with Partial differential equations (PDE) as a powerful tool for modeling problems in science, providing functional analytic techniques in order to deal with PDE.

On successful completion of the course the student should be able to: Apply methods of Fourier Analysis to the discussion of constant coefficient differential equationsWork freely with the classical formulas in dealing with boundary value problems for the Laplace equationProve acquaintance with the basic properties of harmonic functions (maximum principle, mean value property) and solutions of the wave equation (Huygens property)Solve Cauchy problems for the heat and the wave equationsGive a pedagogic talk for peers on a related topic 

Fourier transform, the classical equations, spectral theory of unbounded operators, distributions, fundamental solutions.

Written exam

LiteraturRudin: Functional analysisJost: Postmodern analysisFolland: Introduction to partial differential equations.Reed-Simon: Methods of mathematical physics I-IV

• Course title: Scientific Python
• Number of ECTS: 1
• Course code: F1_MA_MAT_MMCS2-4
• Module(s): Mathematical Modelling and Computational Sciences 1
• Language: EN
• Mandatory: Yes

IntroductionGetting startedBasics of PythonArray computations with numpyArray computations with numpy (cont.)Plotting with matplotlibTabular data manipulation with pandasTabular data manipulation with pandas (cont.)Writing good quality and robust Python code

First sessionYou should feel comfortable writing basic scientific programs in Python, and be able to participate fully in future courses that require an element of programming.Retake examRetake exam not possible, course must be retaken.

This course covers the basics of scientific programming with Python. It is aimed at people who have done some programming before, perhaps on an undergraduate course, but need a refresher before starting their Masters or Doctoral degrees at the University.https://jhale.github.io/scientific-python/ 

A coursework will be distributed at the end of the class. To pass the course and receive the ECTS credits you must complete the coursework.

• Course title: Introduction to Graph Theory
• Number of ECTS: 3
• Course code: MA_DS-8
• Module(s): Mathematical Modelling and Computational Sciences 1
• Language: EN
• Mandatory: No

On successful completion of the course the students should be able tounderstand the relevance of the topics covered in the course for their applications,master the proofs of the main results of the course,solve problems using the toolkit developed in the coursebe autonomous in learning in the field of Graph Theory.

Through a presentation of selected topics, the course aims to be an introduction to graph theory, its applications and its algorithmic aspects. It is designed as a self-contained course and focused on problems pertaining to Data Science. Possible topics for the course include, but are not limited toGraphs and digraphs, degree and the degree sequence algorithmConnectedness, distance, shortest paths and connected components algorithmsGraph matching problems and algorithmsElements of algebraic graph theory and PageRank algorithmGraph traversal algorithmsTrees and applicationsMinimum spanning tree algorithmsNetwork flow, min cut – max flow theorem and Ford–Fulkerson algorithmCentrality and betweness measuresCluster analysisRandom Graphs

First sessionWritten exam and homework, and possibly algorithm implementation project during the semester.  Retake exam Writen exam 

Note / Literature / BibliographyR. Diestel, Graph Theory, Springer, 2017D. Jungnickel, Graphs, Networks and algorithms, Springer 2017

• Course title: Mathematical Modelling
• Number of ECTS: 5
• Course code: F1_MA_MAT_MMCS2-3
• Module(s): Mathematical Modelling and Computational Sciences 1
• Language: EN
• Mandatory: Yes

By the end of the course, students are expected to:

Understand the principles of mathematical modeling and its applications, including agent-based methods, in diverse fields.

Demonstrate proficiency in selecting appropriate mathematical techniques, including agent-based modeling, for specific problems.

Develop the ability to translate real-world scenarios into mathematical equations, agent-based models, and simulations.

Gain proficiency in using computational tools and programming languages, including agent-based modeling platforms, for simulations and analysis.

Apply critical thinking to interpret and validate results from mathematical models and agent-based simulations in practical contexts.

Collaborate effectively in group projects involving mathematical modeling and agent-based simulations.

Communicate complex mathematical concepts, agent-based modeling principles, and findings to both technical and non-technical audiences.

This graduate-level course offers an in-depth exploration of mathematical modeling and computational techniques, including agent-based methods, applied to real-world problems across various disciplines. Through a blend of theoretical foundations and practical applications, students will develop the skills necessary to formulate, analyze, and solve complex problems using mathematical approaches, agent-based modeling, and computational tools.Course Topics1. Introduction to Mathematical Modeling and Agent-Based MethodsUnderstanding the modeling process and the role of agent-based methods.Types of models: deterministic, probabilistic, agent-based.2. Differential Equations, Systems, and Agent-Based SimulationsOrdinary differential equations (ODEs), partial differential equations (PDEs), and their role in agent-based models.Implementing agent-based simulations using Python, NetLogo, or other platforms.3. Numerical Methods, Simulations, and Agent-Based ModelingFinite difference methods, finite element methods for agent-based simulations.Hands-on experience with agent-based modeling software.4 Optimization Techniques for Agent-Based ModelsIncorporating optimization into agent-based models.Applications in social sciences, economics, and ecology.5. Statistical Models, Data Analysis, and Agent-Based ApproachesProbability distributions, statistical inference for agent-based models.Analyzing agent-based simulation results and validating against data.6. Agent-Based Models in Complex SystemsExploring emergent behaviors and self-organization in complex systems.Applications in urban planning, traffic flow, and network dynamics.7. Case Studies and Applications of Agent-Based ModelingModeling epidemics, social dynamics, market behavior.Simulating ecological interactions, ecosystem dynamics.8. Sensitivity Analysis, Validation, and InterpretationAssessing agent-based model sensitivity and robustness.Validating agent-based model outcomes against real-world data.9. Project-Based Learning with Agent-Based ModelingCollaborative projects involving real-world scenarios and agent-based simulations.Developing, implementing, and analyzing agent-based models.10. Presentation and Communication of Agent-Based ResultsEffectively communicating complex mathematical concepts and agent-based findings.Presenting agent-based simulation results to technical and non-technical audiences.

Assessment will be based on a combination of individual and group projects, quizzes, assignments, presentations, and a final examination. Emphasis will be placed on practical application, agent-based modeling, and the ability to integrate theoretical knowledge into real-world contexts.

Will be given to the students in class.

• Course title: Numerical Analysis
• Number of ECTS: 6
• Course code: F1_MA_MAT_MMCS2-5
• Module(s): Mathematical Modelling and Computational Sciences 1
• Language: EN
• Mandatory: Yes

 The objective of the course is to provide the main tools in numerical analysis to manage rigorously the computer solution of practical problems in physics and engineering. 

At the end of the course, a student will be able

to understand the main methods, the rigorous mathematical analysis and algorithms developed in numerical analysis

to master their concrete implementation on a computer

to formulate some basic physics or engineering applications in view of their treatment by a numerical method and its computer solution.

The main results about the theory of numerical methods will be explained. During the course, the student will implement algorithms in Python to propose some concrete, robust and efficient scientific computing solutions to concrete problems. The course content is the following:Notion of error in numerical analysisPolynomial interpolation and approximationNumerical derivation and integrationNumerical solution of Ordinary Differential EquationsDirect and iterative methods for solving linear systemsNonlinear equations in one and several variablesProject presentation + Complementary topics (optimization, eigenvalues problems)

The evaluation will be based on continuous assessment (weekly programming exercises – 50 % of the grade) and a project presentation with a report (50 % of the grade).

X. Antoine, Numerical Analysis, course at the University of Luxembourg

Ascher, Uri M and Greif, Chen, A First Course in Numerical Methods, SIAM, 2011.

Heath, Michael T, Scientific computing: an introductory survey, revised second edition SIAM, 2018.

• Course title: Student project 1
• Number of ECTS: 4
• Course code: F1_MA_MAT_GM-7
• Module(s): Mathematical Modelling and Computational Sciences 1
• Language: EN
• Mandatory: No

On successful completion of the project the student should be able to:

analyse complex tasks

propose solution strategies,

break up a longer project into subsequent steps,

apply a variety of methods in one project,

present a task and its solution in a scientific way.

The student project consists of project work that is carried out under the supervision of a professor or a postdoc. The work is either individual or group work. Group work needs the explicit approval of the Study Director.At the beginning of the project, supervisor and student(s) define tasks to be carried out by the student(s), corresponding to the volume of 100 working hours (4 ECTS). The student(s) need to notify the Study Director of the project and the tasks at the latest on 15 October.The project outcome is a pdf document written by the student. Additional outcomes (such as computer code, images, videos) can be asked for. The required outcome has to be handed in on Moodle at the latest on 31 December.

The students are marked for their project work.In the case of a retake exam, a new project has to be done. The retake can be done with other supervisors.

Note / Literature / Bibliography

Depends on the project.

• Course title: Student seminar 1
• Number of ECTS: 1
• Course code: F1_MA_MAT_GM-8
• Module(s): Mathematical Modelling and Computational Sciences 1
• Language: EN
• Mandatory: No

On successful completion of the Student Seminar 1, the students should be able to:Acquire good insight into a field by means of individual workDeliver a mathematical lecture on a topic of their choiceShare their knowledge with others

Every participant chooses a supervisor among the academic staff of the Department of Mathematics or among the instructors of the Master in Mathematics and, jointly with the supervisor, a topic for a talk.The audience of the talk consists at least of the participants of the Student Seminar 1, the supervisor and one other academic staff member of the Department of Mathematics (e.g. the supervisor of another talk). The duration of the talk is 40 minutes (time for questions included).

The evaluation is based on the quality of the talk delivered by the student

Note / Literature / Bibliography Depends on the subject and is communicated by the chosen supervisor

• Course title: Introduction to Machine Learning Methods and Data Mining
• Number of ECTS: 5
• Course code: MA_DS-13
• Module(s): Mathematical Modelling and Computational Sciences 2
• Language: EN
• Mandatory: Yes

After successfully finishing the course, a student will become familiar with the basics of supervised and unsupervised ML methods, understand their theoretical background, advantages, limitations, and get practical experience in solving real problems employing ML techniques. Particular attention will be focused on the interdisciplinary aspect of ML applications and advantages, which provides an understanding of the nature of the data. Within the course, the students will learn how to implement the basic elements of ML models by themselves, as well as how to use state-of-the-art ML software packages.

The main chapters arePreprocessing of collected data, understanding their structure, visualization. (1 hour)Introduction into Scikit-Learn and TensorFlow. (7 hours)Unsupervised methods: clustering, nearest neighbor task, association rules mining; rule- and tree-based classifications. (12 hours)(Kernel) ridge regression. (4 hours)Support vector machines. (4 hours)Artificial neural networks. (12 hours)Advanced topics: model evaluation and selection, anomaly detection, conformal learning (prediction with guarantees of accuracy), causal inference (identification of causal relationships). (4 hours)Combining different machine learning methods for solving actual problems in natural sciences. (4 hours)Presentation of personal projects. (8 hours)The course will be split into series of lectures with following practical exercises. The ideal schedule will be one day per week in a computer class, where the lecture is directly followed by practical exercises. At the end of the course each student will have to present his individual project.

The evaluation will be based on the presence on the lectures and practical exercises (25%), the individual project (50%), and the answers on the questions following the presentation (25%).

BibliographyGéron, Aurélien. Hands-on machine learning with Scikit-Learn, Keras, and TensorFlow: Concepts, tools, and techniques to build intelligent systems. O’Reilly Media, 2019.Andriy Burkov. The Hundred-Page Machine Learning Book, 2019

• Course title: Advanced Stochastic Models and Financial Applications
• Number of ECTS: 5
• Course code: F1_MA_MAT_FM-4
• Module(s): Mathematical Modelling and Computational Sciences 2
• Language: EN
• Mandatory: No

Learning the concept of stable convergenceReviewing the theory of semimartingalesProving the law of large numbers for power variation of semimartingalesShowing the stable central limit theorem for power variation of semimartingalesDiscuss statistical applications

This course gives an introduction to statistics of high frequency data. The main focus is on power variation statistics of semimartingales, and in particular on the law of large numbers and a stable limit theorem.

Written exam at the end of the course

LiteraturJean Jacod and Philip Protter “Discretization of Processes”, Springer 2012.

• Course title: Algebraic Topology
• Number of ECTS: 8
• Course code: F1_MA_MAT_GM-12
• Module(s): Mathematical Modelling and Computational Sciences 2
• Language: EN
• Mandatory: No

On successful completion of the course, the student should be able to:Explain the main definitions and results of Algebraic Topology;Comment on new concepts;Apply the new techniques and solve problems;Structure the acquired abilities and summarize essential aspects adopting a higher standpoint;Give a talk for peers or students on a related topic and write scientific texts or lecture notes, observing modern standards in scientific writing, in Didactics and in Pedagogy;Provide evidence for the mastery of the Mathematical Method. 

Topology is a field of mathematics that deals with the fundamental properties of spaces. In the topological world, two spaces are considered identical if they can be continuously transformed into one another or more generally, if they are related by what topologists call homotopy equivalence. For example, a sphere and a pyramid are identical, as are a donut and a coffee cup… A main goal is therefore to classify spaces up to homotopy equivalence.Algebraic topology uses algebraic methods to solve this and other topological problems. An important method is the search for algebraic homotopy invariants. In particular, we try to assign an algebraic object such as a group to each topological space and to prove that this group is invariant if we replace the space with a homotopy equivalent space. One of the proofs of the fundamental theorem of algebra uses an algebraic invariant.The main homotopy invariants we discuss are the ‘singular homology functor’ and the ‘homotopy functor’. An amazing result called the Hurewicz isomorphism, shows that the first singular homology group and the (abelianized) first homotopy group of a space are two variants of the same information. Van Kampen’s theorem allows us to practice our understanding of category theory and to compute homotopy groups of larger spaces from those of smaller spaces from which they are constructed. The final chapter will highlight the relationship between the coverings of a space and subgroups of its first homotopy group.The objective is to allow the student to familiarize himself with a very active field of mathematics, with broad applications throughout science. Beyond this goal, special emphasizes will be put on the Mathematical Method, i.e., the optimal technique to learn and apply mathematics. This method is the most important of the objectives of any study program in mathematics.Fundamental concepts of homological algebraChain complexes, chain maps, chain homotopies, homology functor, connecting homomorphism, Künneth isomorphism, simplicial and singular homology, homotopy invariance, examples and applications.Homotopy groupsAlgebra and topology (revision), first homotopy group, Hurewicz isomorphism, homotopy functor, homotopy invariance, typical examples and applications.Van Kampen theoremFree product with amalgamation, Van Kampen-Seifert theorem (basic, general and groupoid version), examples and applications (in particular: first homotopy group of a product and of a coproduct).Covering space theoryCoverings of spaces, lifting property, coverings of groupoids, orbit categories, classifications and constructions of coverings, connections with the preceding chapters, examples and applications. 

Written exam 

LiteraturAllen Hatcher, Algebraic Topology, Cambridge University Press, 2002 – Mathematics – 544 pages, ISBN-13: 978-0521795401 Peter May, A concise course in Algebraic Topology, University of Chicago Press, 1999 – Mathematics – 243 pages, ISBN-13: 978-0226511832Charles Weibel, An introduction to homological algebra, Cambridge studies in advanced mathematics 38, 1997, ISBN: 0-521-55987-1Saunders Mac Lane, Categories for the Working Mathematician, Graduate Texts in Mathematics, 5(2), Springer, 1978, ISBN: 978-0-387-98403-2

• Course title: HPC Software Environment
• Number of ECTS: 6
• Course code: F1_MA_HPC_2-3
• Module(s): Mathematical Modelling and Computational Sciences 2
• Language: EN
• Mandatory: No

 

This course aims to provide students with a comprehensive understanding of the software environment used in High Performance Computing (HPC) from the perspective of application developers and users. It covers aspects related to application development, software deployment and execution of HPC workflows.

 

 

 

By the end of the course, students will be able to:

Effectively use the HPC platform

Execute applications and complex workflows reliably

Compile, debug and optimise applications

Install, deploy and test software on HPC

Understand the integration of HPC with cloud computing

 

 

 This comprehensive course provides you with the knowledge and skills to effectively use HPC systems. It delves into the intricacies of the HPC software environment, to master the art of developing applications and deploying software focusing on the specifics of HPC platforms. It describes the key techniques for efficiently designing and executing complex HPC workflows and robust job campaigns. It explores the use of HPC containers and examines the convergence of HPC workload with cloud computing platforms. It is designed to provide a broad overview, but also an advanced practical insight into the techniques for creating the next generation of HPC power users. – HPC platform and command line environment- Compilation, debugging and profiling on HPC- Software installation on HPC- Testing, CD/CI and testing for HPC- Containers on HPC- HPC & Cloud Computing- Complex HPC jobs- Job campaign, workflow and reproducibility 

Final exam

during the exam period (30%)

Continuous evaluation (assignments, project and presentation)

(70%)

Students’ laptop required

• Course title: Numerical Linear Algebra
• Number of ECTS: 5
• Course code: F1_MA_MAT_MMCS2-6
• Module(s): Mathematical Modelling and Computational Sciences 2
• Language: EN
• Mandatory: Yes

Topics : matrices, linear systems, eigenvalues/eigenvectors. The course is devided into three parts. Part one is devoted to “reminders” about matrix theory: matrix norms, triangularization, diagonalization, spectral decomposition, perturbation theory, complexity of some matrix operations Part two is concerned with the numerical analysis of several algorithms Solving linear systemsGaussian elimination, LU and Cholesky methods,… The conjugate gradient method The least squares problem Normal equations, QR and Householder algorithms,… The eignevalues problem Power methods, Jacobi, Givens–Householde, Lanczos methods,… Part three is devide to computer sessions to program some of the previous numerical methods. Programming language : Python.

written final exam (50%) project (50%) carried out by groups of 2 to more students 

NoteCoursePart 1 (draft version)ExercisesExercices 1 (Preliminaries (linear algebra))Exercices 2 (Norms, condition number)Computer SessionsExercices 1 Gram-Scmidt process

• Course title: Mathematical Statistics II
• Number of ECTS: 5
• Course code: MA_DS-35
• Module(s): Mathematical Modelling and Computational Sciences 2
• Language: EN
• Mandatory: No

In parametric models, a successful student should be is able to establish the main properties (consistency, asymptotic normality, etc) of the most classical estimators, provide (possibly asymptotic) confidence regions and tests between two hypotheses.  

The main problems of mathematical statistics: estimation, testing, the notion of risk. Construction of confidence intervals and tests. False discovery rate. The empirical measure and its applications: the moment method, estimation by empirical quantiles, the Kolmokorov-Smirnov Test. The maximum likelihood estimator. Exponential families in one dimension. Introduction to the Bayes paradigm and elements of decision theory.

First session

A partial written exam and a final written exam.

Retake exam

Oral exam

Literatur

For probability theory: Real Analysis and Probability, R.M. Dudley

For statistics: Mathematical Statistics, P. Bickel and K. Doksum

• Course title: Numerical solution of partial differential equations and applications
• Number of ECTS: 6
• Course code: F1_MA_MAT_MMCS2-7
• Module(s): Mathematical Modelling and Computational Sciences 2
• Language: EN
• Mandatory: Yes

This course is an introduction to the numerical solution of partial differential equations (PDEs). It contains a theoretical part setting the mathematical foundations necessary for some important numerical methods used to obtain solutions to some classical PDEs, in particular the finite element method and the finite difference method. The theoretical part of the course is supported by the development of a one-dimensional Galerkin finite element code for the Poisson problem and a one-dimensional finite difference code for a scalar hyperbolic transport problem.We largely follow the reference [Qua09] which is available via the National Library’s Website as well as on some less official websites. In particular, we cover the contents of [Qua09, Chapters 1, 2, 3, 4, and 14].

A brief survey of partial differential equationsThis chapter briefly introduces the notion of linear PDEs and their classification into elliptic, parabolic and hyperbolic equations. We mention some classical examples, mainly issued from physics and engineering, such as the transport equation, the Laplace equation, the heat equation and the wave equation.Elements of functional analysisWe introduce the main notions and theoretical results of functional analysis that are extensively used in the numerical analysis of partial differential equations. We consider the Riesz representation theorem regarding representation of continuous linear forms on a Hilbert space, and we survey the notions of bilinear form, of continuous injection of a Hilbert space into another, the notion of derivative in the sense of Fréchet, some elements of the theory of distributions, the basic properties of the Lebesgue and Sobolev spaces, and the notion of adjoint operator.Elliptic equationsWe illustrate boundary-value problems for elliptic equations (in one and several dimensions), present their variational reformulations, treat the boundary conditions and analyze their well-posedness. Several examples of physical interest are introduced, in particular the Poisson equation, starting with the one-dimensional case, for various boundary conditions. We consider some variational formulations of these problems, and then turn to the boundary-value problems associated to the Poisson equation in the two-dimensional case. We establish that under some regularity condition the weak formulation is equivalent to the strong one. For general elliptic problems, the Lax-Milgram theorem ensures that the weak formulation is well-posed.The Galerkin finite element method for elliptic problemsWe formulate Galerkin’s method for the numerical discretization of elliptic boundaryvalue problems and analyze its existence, uniqueness, stability and convergence features in an abstract functional setting. We then introduce the Galerkin finite elements method, first in one dimension, and then in several dimensions.The Galerkin finite element method – a numerical exampleThis section introduces students to the development of a one-dimensional Galerkin finite element solver for the Poisson problem using Python. We focus on building the solver within a provided Jupyter notebook, providing hands-on experience with the computational and algorithmic aspects of the finite element method.Finite differences for hyperbolic equationsThe aim of this final chapter is to study classical finite differences methods for approximating solutions of first-order hyperbolic equations. We start with exposing some hyperbolic equations starting with the scalar transport problem in one dimension which we analyze by the method of characteristics. We also establish an a priori estimate by the energy method. We then turn to systems of linear hyperbolic equations in one dimension and give the example of the wave equation. Then we introduce the finite difference method, together with its variants: the forward/centered Euler scheme, the Lax-Friedrichs scheme, the Lax-Wendroff scheme, in the case of the scalar transport problem, before to move to more general cases, for which we analyze the consistency, stability, convergence, dissipation and dispersion properties of the finite difference methods.Finite differences – a numerical exampleThis section introduces students to the development of a one-dimensional finite difference solver for a hyperbolic transport problem using Python. We focus on building the solver within a provided Jupyter notebook, providing hands-on experience with the computational and algorithmic aspects of the finite difference method.

First session

Assessment is via coursework (30%) and final examination (70%). The due date for the coursework will be set at a later time.

Retake policy

Failure on the coursework can be compensated through the examination, and vice versa, as long as the final mark for the course is greater than or equal to 10.In the event that the final mark is less than 10:

The student may request a retake the examination once.

The student may not resubmit the coursework.

If the final mark remains less than 10 the student has failed, and can take the entire course again in the next semester that it is offered.

Reference

[Qua09] Alfio Quarteroni. Numerical models for differential problems. Vol. 2. Springer, 2009.

• Course title: Statistical Modelling
• Number of ECTS: 5
• Course code: MA_DS-15
• Module(s): Mathematical Modelling and Computational Sciences 2
• Language: EN
• Mandatory: No

The main goal of the course is to raise awareness of model assumptions and critical thinking, and to provide a guide as to how to make reasonable modelling choices. 

This course will deal with two cultures of statistical modelling: top-down via a parametric model, and bottom-up via data-adaptive methods. More concretely, it will cover classical distributions and their limitations, flexible distributions allowing to model complex modern datasets, interpretable machine learning, and the difference between the two modelling cultures. 

First session

Written exam on both theory and exercises for 75% of the mark, and midterm exam for 25% of the mark.

Retake exam

Written exam on both theory and exercises. The student can choose whether they wish to keep their midterm exam mark or not.

 

 

 

Literatur

Ley, C., Babic, S. and Craens, D. (2021) Flexible models for complex data with applications.

Annual Review of Statistics and Its Application 8, 18.1-18.23.

Genuer, R. and Poggi, J.-M. (2020) Random Forests with R, Springer.

Kleinbaum, D.G. and Klein, M. (2012) Survival Analysis – A Self-learning Text, Springer.

• Course title: Sustainable Scientific Software
• Number of ECTS: 3
• Course code: F1_MA_MAT_MMCS2-8
• Module(s): Mathematical Modelling and Computational Sciences 2
• Language: EN
• Mandatory: Yes

During this course you will learn foundational skills that will enable you todevelop numerical and scientific software that is sustainable and reproducible.

Students should be able to:

Understand the motivation for writing scientific software to a high standard.

Work effectively as a team to write consistent and clean code, and share that code using tools, e.g. version control.

Understand appropriate methodologies for improving code quality, robustness, and performance, e.g. linting, writing unit tests, static analysis tools, debuggers, performance profilers.

Effectively plan a small piece of software, including the choice of and combination of different programming paradigms.

Reflect on the tradeoffs in terms of time to write vs overall quality. and the long term sustainability implications.

Definitions of sustainability in a software contextIntroduction to the Unix shell with bashIntroduction to version control with gitClean code, style guidelines and working in a teamTools for improving code.Writing documentationTest driven developmentContainersContinuous integration

Two courseworks are set during the semester with equal weight towards the final mark.

Note / Literature

https://jhale.github.io/sustainable-scientific-software/ for further links.

• Course title: Partial Differential Equations II
• Number of ECTS: 8
• Course code: F1_MA_MAT_GM-15
• Module(s): Mathematical Modelling and Computational Sciences 2
• Language: EN
• Mandatory: No

Learning tools in order to deal with PDE, understanding the interplay between local and global problems and techniques.

Distributions as generalized functions continued, Sobolev spaces, elliptic regularity, elliptic operators on compact manifolds, some non-linear equations. 

Written exam

LiteraturJost: Postmodern analysisFolland: Introduction to partial differential equations Reed-Simon: Methods of mathematical physics I-IV Aubin: Nonlinear analysis on manifolds

• Course title: Advanced Graph Theory
• Number of ECTS: 6
• Course code: F1_MA_MAT_GM-16
• Module(s): Mathematical Modelling and Computational Sciences 2
• Language: EN
• Mandatory: No

 

On successful completion of the course, the student should be able to 

Illustrate the main results and concepts with well-chosen examples

Master the proofs and techniques of the theory 

Solve exercises related to the topics covered in class 

Give an overview of the course content, focusing on his/her own taste and favorite topics 

Communicate his/her own pleasure in solving mathematical problem 

Graphs are structures that are ubiquitous in Mathematics and its applications. Through a presentation of selected topics, the course aims to be an introduction to certain modern aspects of graph theory. Some basic knowledge of the concept of graphs is a prerequisite for this course.According to time and taste, topics covered will be chosen among (and are not limited to) the following ones:Matching, covering and packing problemsEdge and vertex coloringsGraph polynomialsRamsey theory for graphsInfinite Graph Theory 

Oral exam or written exam according to number of participants

Literatur

Reinhard Diestel. Graph Theory, 5th edition. Graduate texts in mathematics 173, Springer, 2017

• Course title: Introduction to Biology for Data Scientists
• Number of ECTS: 5
• Course code: MA_DS-14
• Module(s): Mathematical Modelling and Computational Sciences 2
• Language: EN
• Mandatory: No

structure and function of the cell

basics in biochemistry

basics of genetics

basics of evolution

introduction to plant biology

introduction to animal biology

introduction to ecology

We will follow the Campbell Biology. The course includes fundamental principles of biochemistry, genetics, molecular biology and cell biology.

Written exam

• Course title: Student Project 2
• Number of ECTS: 4
• Course code: F1_MA_MAT_GM-18
• Module(s): Mathematical Modelling and Computational Sciences 2
• Language: EN
• Mandatory: No

On successful completion of the Student Project 2, the student should be able to:analyse complex tasks,propose solution strategies,break up a longer project into subsequent steps,apply a variety of methods in one project,present a task and its solution in a scientific way.

The student project consists of project work that is carried out under the supervision of a professor or a postdoc. The work is either individual or group work. Group work needs the explicit approval of the Study Director. At the beginning of the project, supervisor and student(s) define tasks to be carried out by the student(s), corresponding to the volume of 100 working hours (4 ECTS). The student(s) need to notify the Study Director of the project and the tasks at the latest on 15 March. The project outcome is a pdf document written by the student. Additional outcomes (such as computer code, images, videos) can be asked for. The required outcome has to be handed in on Moodle at the latest on 31 May. The project file will undergo an automatic plagiarism check.

The students are marked for their project work. In the case of a retake exam, a new project has to be done. The retake can be done with other supervisors.

• Course title: Student seminar 2
• Number of ECTS: 2
• Course code: F1_MA_MAT_GM-19
• Module(s): Mathematical Modelling and Computational Sciences 2
• Language: EN
• Mandatory: No

On successful completion of the Student Seminar 2, the students should be able to:

Fully benefit from seminar talks

Acquire good insight into a field by means of individual work

Give themselves lectures on specific topics

Share their knowledge with others

Every participant chooses a supervisor among the academic staff of the Department of Mathematics and, jointly with the supervisor, a topic for a talk.The audience of the talk consists at least of the participants of the seminar, the supervisor and one other academic staff member of the Department of Mathematics (e.g. the supervisor of another talk). The duration of the talk is 75 minutes (time for questions not included).A typewritten version of this (these) lecture(s) is requested.The students’ seminar talks will take place during April and May. The exact timetable depends on the number of registered students and is made during the semester.The supervisor together with the course responsible marks the students’ performance and the written text. 

Continuous evaluation. The mark is based on the talk and the written text delivered by the student.If a student fails, a new talk and a new text can be delivered in the following semester. A supervisor change is allowed.

• Course title: Lie Algebras and Lie Groups
• Number of ECTS: 6
• Course code: F1_MA_MAT_GM-20
• Module(s): Mathematical Modelling and Computational Sciences 3
• Language: EN
• Mandatory: No

The purpose of this course is to give an introduction into the theory of finite dimensional Lie groups and Lie algebras, assuming some basic knowledge of differentiable manifolds.

On successful completion of the course, the student should be able to:Expound the mathematical foundation behind symmetries of solid bodies, dynamics of mechanical systems, and geometric structures in nature.Explain the deep interrelations between Lie groups and Lie algebras, as well as the technical tools behinds these interrelations.Simplify mathematical problems admitting symmetry Lie groups actions to problems admitting symmetry actions of their Lie algebras.Master applications to the theory of manifolds and representation theory, which in turn have applications in physics, engineering and mechanics.

The Lie algebra of a Lie group, the exponential map, the adjoint representation, actions of Lie groups and Lie algebras on manifolds, the universal enveloping algebra, basics of the representation theory.

First sessionEnd of course assessment

Note / Literature / Bibliography “Lie groups and Lie algebras” by Eckhard Meinrenken, 83 pages (free to download)”Prerequisites from Differential Geometry” by Sergei Merkulov (free to download)

• Course title: Advanced Discretization Methods
• Number of ECTS: 5
• Course code: F1_MA_MAT_MMCS2-10
• Module(s): Mathematical Modelling and Computational Sciences 3
• Language: EN
• Mandatory: Yes

The objectives of the course are to introduce some advanced discretization techniques for the numerical solution of partial differential equations arising in engineering and applied sciences. The schemes will be explained in details as well as their mathematical properties (e.g. order of accuracy, stability). In addition, these methods will be implemented by using Matlab and tested on concrete problems.

On successful completion of the course the student should be able to:

Explain the mathematical foundation of advanced discretization techniques for PDEs.

Master their concrete implementation on nontrivial engineering boundary-value problems.

Adapt them according to the problem under consideration.

Complements the Finite Element MethodFinite difference schemes in spaceFinite difference schemes for the discretization of time-dependent PDEsIntroduction to integral equations

First session

The students will have to provide some reports that will be evaluated. In addition, a final written examination will be organized.

Note / Literature / Bibliography 

Support / Arbeitsunterlagen / Support: Lecture notes (french), exercise sheets (english)

X. Antoine, Numerical solution of PDEs, lecture notes.

X. Antoine, Numerical Analysis, course at the University of Luxembourg.

G. Allaire, Analyse Numérique et Optimisation, Presses de l’Ecole Polytechnique.

P.A. Raviart et J.M. Thomas, Introduction à l’Analyse Numérique des Equations aux Dérivées Partielles, Dunod.

• Course title: Combinatorial Geometry
• Number of ECTS: 6
• Course code: F1_MA_MAT_GM-21
• Module(s): Mathematical Modelling and Computational Sciences 3
• Language: EN
• Mandatory: No

Beyond learning exciting material, the course is designed to explore and experience the process of mathematical research. 

The course requires minimal prerequisites (some linear algebra, Euclidean geometry and basic topology) but aims to explore results that are at the limit of current known understanding. In particular, we’ll discuss some open problems and try to illustrate the process of modern research. The subjects are chosen so that they can be treated with a hands-on approach, and this approach and experience are as important for this course as the actual content. 

The course will take the form of a topics course, presenting a selection of themes from combinatorial aspects of geometry. Topics may include cube complexes, convex sets in R^n, combinatorial ways of exploring surfaces, topics in graph theory…

First session3 short in-class tests, worth together a total of 20%; final written exam worth 80%.Retake exam100% final written exam, no kept grade.Absence plan: the final grade will be worked out as follows:Case 1: Student attends all three short tests. The best two out of the three scores will count for 10% each, and the final exam will count for 80%.Case 2: Student attends two short tests. The two scores will count for 10% each and the final exam will count for 80%.Case 3: Student attends one short test. The test score will count for 10% and the final exam will count for 90%.Case 4: Student attends no short test. The final exam will count for 100%. 

Note / Literature / Bibliography Varied. (A mix of lecture notes and articles.)

• Course title: Selected Topics in MMCS
• Number of ECTS: 5
• Course code: F1_MA_MAT_MMCS2-11
• Module(s): Mathematical Modelling and Computational Sciences 3
• Language: EN
• Mandatory: Yes

Students will learn how to:

Formally and critically assess an academic article.

See a range of important modern and classical research advances.

Place fundamental advances in the academic literature in a wider application context.

Deliver a short seminar on a selected topic.

Summarise an article for an audience of peers.

Summarise an article for a non-expert audience.

Understand the process of publication.

Use modern bibliographic tools to understand links with other articles and fields.

For more information:https://jhale.github.io/selected-topics-in-mmcs/

This course will review a selection of recent and classical topics in Mathematical Modelling and Computational Sciences by a process of student-led discussions around articles in the academic literature. The course will be run in an interactive manner with a strong emphasis on student participation.

First session

Written assignments

In-class participation

Presentation

Retake exam

Retake exam not possible, course must be retaken.

Note / Literature / Bibliography 

The topics will be selected based around the interests of the students and instructor.

• Course title: Gaussian processes and applications
• Number of ECTS: 5
• Course code: F1_MA_MAT_FM-9
• Module(s): Mathematical Modelling and Computational Sciences 3
• Language: EN
• Mandatory: No

The objective is to learn the basic concepts and techniques associated with Gaussian processes, that are omnipresent in modelling random phenomena (finance, physics, machine learning, statistics, etc.)

On successful completion of the course, the student should be able to:

Explain the language, basic concepts and techniques associated with Gaussian variables, vectors, and processes

Identify, analyse, and prove relevant properties of models based on a Gaussian structure

Solve exercises involving a Gaussian structure

Gaussian random variables (characteristic function, CLT, stability properties, Stein’s lemma)Gaussian random vectors (definition, characteristic function, existence, uniqueness in law, multivariate CLT, density, Hermite polynomials)Gaussian random processes (definition, modifications, uniqueness in law, function of positive type, existence, Brownian motion, continuity)Fractional Brownian motion (definition, existence, Hölder regularity)

Exam modalities for the first session

Written exam during the exam period (January/February).

Exam modalities for the retake exam

Written exam during the exam period July).

Note / Literature / Bibliography

Will be discussed in class

• Course title: Intelligent Systems – Problem Solving
• Number of ECTS: 3
• Course code: MICS2-38
• Module(s): Mathematical Modelling and Computational Sciences 3
• Language: EN
• Mandatory: No

The objective of this lecture first consists in providing a structured approach to students in terms of optimization problem modeling. Next various solving techniques based on exact methods (A*, B&B, LP), approximated ones (heuristics, meta-heuristics, problem relaxation) and hybrids are described. Students are also taught how to validate the proposed solution by having a scientific approach

Introduction to optimisation and decision problems Linear programming, graphical interpretation and primal simplex Branch&Bound, A* The scheduling problem List algorithms, greedies, heuristics Meta-Heuristics and Evolutionary computation

Final Exam: 100%

• Course title: Machine Learning – Reading course
• Number of ECTS: 5
• Course code: F1_MA_MAT_MMCS2-12
• Module(s): Mathematical Modelling and Computational Sciences 3
• Language: EN
• Mandatory: Yes

• Course title: Data Science
• Number of ECTS: 5
• Course code: F1_MA_MAT_GM-24
• Module(s): Mathematical Modelling and Computational Sciences 3
• Language: EN
• Mandatory: No

The successful candidate will understand the basic theoretical concepts of data-centric aspects and will be able to work on data-centric problems. The aim is to continue the work done in the project within the framework of a Master’s thesis.

On successful completion of the course the student should be able to:Explain and apply basic theoretical concepts on selected aspects of data processing.Develop appropriate solutions for data-centered problems.Consolidation of the acquired competences in the subject area through a Master’s thesis.

In this course, the term ‘data’ is seen centric and we will look at data from different perspectives. We will discuss selected aspects of Data Preparation and Preprocessing, Data Statistics, Data Security, Data Privacy, Data Management, Big and Small Data, Data Retrieval, Data Visualization, and Data Analytics.  

First session50% written assignment or oral interview + 50% practical. Retake examThose who do not pass the course have the opportunity to retake the exam in the summer term.

Note / Literature / Bibliography Elmasri, Navathe: Fundamentals of Database Systems. Pearson Addison Wesley. 2006.Han, Kamber: Data Mining – Concepts and Techniques. Morgan Kaufmann. 2011.Manning, Raghavan, Schütze: Introduction to Information Retrieval. Cambridge University Press.Ware: Information Visualization. Morgan Kaufmann. 2012.Witten, Kamber: Data Mining: Practical Machine Learning Tools and Techniques. Morgan Kaufmann.Aggarwal, Yu: Privacy-Preserving Data Mining – Models and Algorithms. Springer. 2008.Marz: Big Data: Principles and best practices of scalable realtime data systems. Manning. 2015.as well as different articles, reports, and journals contributions.

• Course title: Student Project 3
• Number of ECTS: 4
• Course code: F1_MA_MAT_GM-25
• Module(s): Mathematical Modelling and Computational Sciences 3
• Language: EN
• Mandatory: No

On successful completion of the Student Project 1 (or 3) the student should be able to:analyse complex tasks,propose solution strategies,break up a longer project into subsequent steps,apply a variety of methods in one project,present a task and its solution in a scientific way.

The ability to work on a project, either individually or as part of a team, is one of the key skills expected of a mathematician, in and outside academia. The Student Project 1 offers the student the opportunity to improve this skill by allowing them to work on a well-defined and well chosen project, either individually or as a team, under the guidance of a researcher.The Student Project 3 consists of project work that is carried out under the supervision of a researcher.The work is either individual or group work. At the beginning of the project, supervisor and student(s) define tasks to be carried out by the student(s), corresponding to the volume of 100 working hours (4 ECTS) for each student. The student(s) need to submit the planned topic to the course responsible for approval at the latest on 15 October. The project outcome is a pdf document written by the student(s).Additional outcomes (such as computer code, images, videos) can be asked for. The required outcome has to be handed in on Moodle at the latest on 31 December. The project file will undergo an automatic plagiarism and AI check.Timeline for the studentBeginning of semesterConnect to the Moodle classroomFind a supervisor and a topicOn or before 15 October, get approval from the course responsible of the project participants, the supervisor and the topicDuring the semesterWork on the projectSeek feedback from your supervisorEnd of the semesterHand in your project on or before 31 December on Moodle (for plagiarism and AI check)  

First sessionThe students are marked for their project work. In the case of a retake exam, a new project on the same or a different topic has to be done. Retake examThe retake can be done with a different supervisor.

• Course title: Internship (in a company)
• Number of ECTS: 5
• Course code: F1_MA_MAT_MMCS2-13
• Module(s): Mathematical Modelling and Computational Sciences 3
• Language: EN
• Mandatory: No

• Course title: Numerical Methods for Variational Problems
• Number of ECTS: 5
• Course code: F1_MA_MAT_MMCS2-9
• Module(s): Mathematical Modelling and Computational Sciences 3
• Language:
• Mandatory: Yes

At the end of the course students should have a thorough understanding of the mathematical underpinnings of the polynomial (p) finite element method and its implementation in Python code. If there is sufficient time students will also be exposed to advanced topics such as mixed finite element methods and preconditioning finite element linear systems.

Please see https://jhale.github.io/finite-elements/ for a full syllabus. 

First session

50% written examination.50% coursework.

Retake examRetake exam possible if coursework component passed, otherwise course must be retaken.

Absence

Absence at final exam treated via procedure in Règlement des études.

Note / Literature / Bibliography

https://jhale.github.io/finite-elements/ The Mathematical Theory of Finite Element Methods, Brenner S.C. and Scott, L. R., Springer 2008. https://doi.org/10.1007/978-0-387-75934-0 (available digitally via the uni.lu library)

• Course title: Master Thesis
• Number of ECTS: 20
• Course code: F1_MA_MAT_MMCS2-14
• Module(s): Master Thesis
• Language: FR
• Mandatory: No

• Course title: Master Thesis with Internship
• Number of ECTS: 30
• Course code: F1_MA_MAT_MMCS2-15
• Module(s): Master Thesis with Internship
• Language: FR
• Mandatory: No

• Course title: Reading Course
• Number of ECTS: 5
• Course code: F1_MA_MAT_GM-30
• Module(s): Mathematical Modelling and Computational Sciences 4
• Language: EN
• Mandatory: No

This is an individual reading course about the background and complementary knowledge for a Master thesis.

The assessment is based on the performance in the meetings with the instructor(s).

• Course title: Extended Reading Course
• Number of ECTS: 10
• Course code: F1_MA_MAT_GM-31
• Module(s): Mathematical Modelling and Computational Sciences 4
• Language: EN
• Mandatory: No

This is an individual extended reading course about the background and complementary knowledge for a Master thesis.

The assessment is based on the performance in the meetings with the instructor(s).

• Course title: Complex Geometry
• Number of ECTS: 5
• Course code: F1_MA_MAT_GM-28
• Module(s): Mathematical Modelling and Computational Sciences 4
• Language: EN
• Mandatory: No

On successful completion of the course the student should be able to:Demonstrate proper understanding of the notion of complex manifold;Identify the most important examples;Master the main techniques of the theory;Prove good knowledge of the key theorems;Apply the new techniques to geometric problems;Give a pedagogic talk for peers on a related topic;Write clear and concise lecture notes, including appropriate exercises and applications

The aim of the lecture course is to give an introduction to complex geometry. In particular the course will treat many examples. After the course the students should understand Hermitian vector bundles and the basics of Kahler manifolds and the Hodge decomposition. The students should be acquainted with the basic theorems and examples.

First sessionOral examination (a presentation) and 1-2 homework sets

LiteratureR.O. Wells: Differen9al Analysis on Complex Manifolds (main text);S.S. Chern: Complex Manifolds without Potential Theory;Griffiths & Harris: Principles of Algebraic Geometry;F. Warner: Foundations of Differentiable Manifolds;D. Huybrechts: Complex Geometry;Lecture notes Martin SCHLICHENMAIER, available on the Moodle for download.

• Course title: Optimisation for Computer Science
• Number of ECTS: 5
• Course code: MICS2-17
• Module(s): Mathematical Modelling and Computational Sciences 4
• Language:
• Mandatory: No

Different problems have different nature. In terms of complexity some problems are called intractable and can not be solved by classical computers. But there are also many other aspects of the nature of optimisation problems such as linearity, convexity, continuity, dynamicity, randomness that may lead the choice of different optimisation techniques

* Characterize problems* Identify the key concepts related to optimisation techniques* Use optimization frameworks* Implement optimization algorithms* Validate optimization algorithms and results* Validate approaches for solving optimization problems

This lecture confront the students to real instances of such problems. They are first asked to model the problem and next proposed solutions include exact methods, relaxations, approximations, heuristics and meta-heuristics. And these practical study cases are supported by the theoretical lectures on Problem Solving (1st semester)

Project: 100%