Applied and Computational Mathematics

Program Description

Research interests of the members cover several closely connected areas which include dynamical systems and delay equations; physics of fluids and continua; material sciences; phase transitions and crystal growth; numerical methods in fluid dynamics and asymptotic analysis; shape and structural optimization; control of partial differential equations.

Two research centers are affiliated with the group:

Program Members

Academic Program

The objective of this program is a training in modern mathematics aimed at applications and in the use of computers as a tool in the analysis, optimization, and control of physical and technological systems. It welcomes strong graduate students with a variety of backgrounds (ranging from the physical sciences and engineering to mathematics) wishing to work in partial differential equations and their applications. The program is sufficiently broad to accomodate software development and physical modelling as well as topics requiring delicate techniques in functional analysis or partial differential equations.

It is intended to offer students the possibility of collaborative contact with several local government and industrial research groups such as the Canadian Space Agency and a variety of other organisations with which members of the group have been involved at various times.

The program covers several closely connected areas which include:

  • Dynamical systems and delay equations.
  • Physics of fluids and continua.
  • Material sciences; phase transitions and crystal growth.
  • Numerical methods in fluid dynamics and asymptotic analysis.
  • Shape and structural optimization.
  • Control of partial differential equations.

There are no formal programmatic requirements beyond the departmental requirements. However the following guidelines should be followed and courses must be selected in consultation with an adviser from the group.

  1. All students should take courses in partial differential equations: appropriate courses are MATH 580 and MATH 581 at McGill and MAT 6110 at U de M.
  2. It is essential that most (and desirable that all) students develop their computational skills by taking appropriate courses in numerical analysis. Beyond the introductory courses, generally at an undergraduate level, the essential courses cover computational mathematics (MATH 578 at McGill and MAT6470 at U de M) numerical differential equations (MATH 579 at McGill) finite difference methods (MAT 6165 at U de M) and finite element methods (MTH 6206/7 at Polytechnique and MAT6450 at U de M).
  3. Students should develop an understanding of neighbouring areas of physics such as fluids and continuum mechanics, thermodynamics, etc. Suitable courses include MATH 555 at McGill and MAT 6150 at U de M; other useful courses can be found in Physics or Engineering departments.
  4. Students involved in fluid mechanics or material sciences should take a course on asymptotic and perturbation methods: MATH 651 at McGill or MTH 6506 at Polytechnique.
  5. Students in shape optimization or control should take at least one course in optimization. The following courses are available: MATH 560 at McGill, MAT 6428, MAT 6439 (Optimisation et contrôle), MAT 6441 (Analyse et optimisation de forme) at U de M; MTH 6403 and MTH 6408 at Polytechnique.
  6. Students who wish to work on shape optimization or the control of distributed parameter systems will need to develop a strong background in real analysis and functional analysis.

We expect that future elaboration and formalization of this program will occur within the framework described above which allows also for the introduction of additional areas under the broad umbrella of the program title.

2017-18 Course Listings

Fall

Fondements de l'optimisation

Ce cours a pour objectif de familiariser les étudiants aux différentes techniques fondamentales de l'optimisation ainsi qu'à des logiciels commerciaux largement répandus afin de les mettre en application. Du côté des méthodes exactes on couvrira les méthodes de base de la programmation linéaire, de la programmation non linéaire ainsi que de la programmation linéaire en nombres entiers, tout en faisant ressortir la difficulté inhérente à ces différentes classes de programmes (linéaire vs non linéaire, convexe vs non convexe, entier vs continu, etc.).

On présentera également les principes à la base des méthodes heuristiques et métaheuristiques les plus utilisées dans la pratique afin de donner une vision la plus complète possible des outils disponibles pour résoudre les problèmes d'optimisation rencontrés dans la pratique.

Prof. Patrick Soriano

6-606-13

Institution: HEC Montréal

Algorithmes pour l'optimisation et l'exploitation de données

Le but du cours est de familiariser l'étudiant à l'aspect algorithmique des méthodes d'exploitation de données. L'étudiant développera des notions d'analyse de complexité et de structures de données. Ces notions seront exposées dans le contexte de l'analyse de données et certaines méthodes classiques seront étudiées. Dans le but de mieux familiariser l'étudiant avec la programmation, les présentations seront systématiquement illustrées par des programmes en C++. Il est conseillé que les étudiants aient des notions de programmation car ils devront effectuer un projet informatique simple. Une base de programme leur sera fournie afin de faciliter l'accomplissement de ce projet. Après avoir suivi ce cours, l'étudiant pourra développer ses propres outils et résoudre des problèmes pour lesquels il n'existe pas de logiciels.

Prof. Gilles Caporossi

6-607-11

Institution: HEC Montréal

Applications of Game Theory

The course is a general introduction to non-cooperative and cooperative static and dynamic game theory. The main concepts are defined rigorously and illustrated by a series of examples, exercises and applications to different problems in management which are of interest to the participants.

Prof. Georges Zaccour

80-685-09

Institution: HEC Montréal

Numerical Analysis 1

Development, analysis and effective use of numerical methods to solve problems arising in applications. Topics include direct and iterative methods for the solution of linear equations (including preconditioning), eigenvalue problems, interpolation, approximation, quadrature, solution of nonlinear systems.

Prof. Jean-Christophe Nave

MATH 578

Institution: McGill University

Partial Differential Equations 1

Classification and wellposedness of linear and nonlinear partial differential equations; energy methods; Dirichlet principle. Brief introduction to distributions; weak derivatives. Fundamental solutions and Green's functions for Poisson equation, regularity, harmonic functions, maximum principle. Representation formulae for solutions of heat and wave equations, Duhamel's principle. Method of Characteristics, scalar conservation laws, shocks.

Prof. Rustum Choksi

MATH 580

Institution: McGill University

Dynamic Optimisation in Management

The objective of this course is to introduce students to optimization models in a dynamic contest; more specifically, the students will be acquainted with the major tools used in dynamic optimization, that is, dynamic programming, Markov decision problems and optimal control. In addition, the course will cover many applications where the use of such tools is called for. At the end of the course, the students should be able to:

  • identify a business management situation presenting a dynamic optimization problem
  • represent this problem under an suitable mathematical model,
  • and solve it using one of the available techniques of dynamic optimization.

Prof. Michèle Breton

80-680-04

Institution: HEC Montréal

Winter

Modèles d'optimisation

Ce cours vise à développer les habiletés de modélisation à partir de diverses applications en gestion des opérations et de la logistique, en gestion des ressources humaines, en finance ou autres domaines de la gestion.  
 
Dans ce cours, l'étudiant apprendra :
i) à reconnaître le type de modèle adapté à une situation particulière en gestion,
ii) à décrire cette situation sous la forme d’un modèle mathématique approprié,
iii) à implanter et résoudre ce modèle grâce au logiciel IBM ILOG CPLEX Optimization Studio,
iv) à valider, analyser et présenter les résultats obtenus. 

Thèmes couverts :

1 : Introduction à la modélisation mathématique
2 : Implantation d'un modèle d'optimisation avec le langage OPL du logiciel IBM ILOG CPLEX Optimization Studio
3 : Modèles linéaires.  
4 : Modèles de réseau
5 : Modèles linéaires en nombres entiers et techniques de linéarisation
6:  Problèmes de localisation
7 : Problèmes de distribution (visite de clients, tournées de véhicules, ramassage des ordures, etc.)
8 : Problèmes d’horaire (employés, calendrier scolaire, etc.)
9 : Problèmes de gestion de la chaîne logistique
10 : Modèles d'optimisation à objectifs multiples
11 : Modèles de programmation par contraintes
12 : Modèles de programmation stochastique
13 : Modèles de programmation dynamique

Prof. Sylvain Perron

6-617-16

Institution: HEC Montréal

Network Optimization in Business

The aim of the course is to present the most important aspects of mathematical optimization applied to network flow problems. On the one hand, we study the specialized algorithms, and on the other hand, we take a look at the numerous applications in this field.

Prof. Jacques Desrosiers

80-682-11

Institution: HEC Montréal

Distribution Management

This course covers strategic, tactical and operational planning in distribution management systems. Long term decisions relate mainly to the location of major installations, namely transportation infrastructures. Tactical planning includes medium term operations such as route design in inter-city planning and warehouse location. Operational planning covers the design of daily pickup and delivery routes and the location of light facilities such as mail boxes. In several transportation areas operations may have to be planned in real time, like in pickups and deliveries of letters and packages in fast courier operations, in dial-a-ride services for handicapped people, and in ambulance relocation. This course introduces the main methods and applications encountered in distribution management. It is partly based on some real cases published in recent scientific articles.

Prof. Gilbert Laporte

80-655-12

Institution: HEC Montréal

Numerical Differential Equations

Numerical solution of initial and boundary value problems in science and engineering: ordinary differential equations; partial differential equations of elliptic, parabolic and hyperbolic type. Topics include Runge Kutta and linear multistep methods, adaptivity, finite elements, finite differences, finite volumes, spectral methods.

Prof. Adam Oberman

MATH 579

Institution: McGill University

Calcul scientifique

Étude des algorithmes fondamentaux en calcul scientifique. Principes théoriques; programmation et application à des problèmes pratiques; utilisation scientifique de logiciels spécialisés.

Prof. Robert G. Owens

MAT 6470

Institution: Université de Montréal