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:
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:
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.
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.
Honours level introduction to linear optimization and its applications: duality theory, fundamental theorem, sensitivity analysis, convexity, simplex algorithm, interior point methods, quadratic optimization, applications in game theory.
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.
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.
Processus de modélisation mathématiques avancés : simulations, estimation de paramètres, interprétation. Utilisation des mathématiques dans un milieu multidisciplinaire (p. ex. oncologie, neurosciences, génétique). Étude de cas et projets appliqués.
Formulation et modélisation analytique des géométries intrinsèques de données. Algorithmes pour les construire et les utiliser en apprentissage automatique. Applications : classification, regroupement et réduction de la dimensionnalité.
Notions fondamentales de probabilités appliquées à divers domaines de l’intelligence artificielle. Réseaux bayésiens, champs markoviens, diverses méthodes d’inférence (variationnelle, par maximum a posteriori, recuit simulé, etc.), échantillonnage et méthodes de Monte Carlo par chaînes de Markov, séries chronologiques, partitionnement spectral et modèles à variables latentes. Applications en imagerie, en analyse de textes et sur les réseaux de neurones.
This course provides an introduction to Nonsmooth Analysis, beginning with proximal calculus, featuring proximal normals, subgradients, and generalizations of ordinary rules of calculus. Some specializations to classical Convex Analysis are given, and various types of tangency are studied. A main application is to nonsmooth constrained optimization.
The course reference is "Nonsmooth Analysis and Control Theory" by F.H. Clarke, Yu. S. Ledyaev, R.J. Stern and P.R. Wolenski--Graduate Texts in Mathematics (173), Springer, 1998. (A pdf of this book will be made available.)
Foundations of game theory. Computation aspects of equilibria. Theory of auctions and modern auction design. General equilibrium theory and welfare economics. Algorithmic mechanism design. Dynamic games.
Convex sets and functions, subdifferential calculus, conjugate functions, Fenchel duality, proximal calculus. Subgradient methods, proximal-based methods. Conditional gradient method, ADMM. Applications including data classification, network-flow problems, image processing, convex feasibility problems, DC optimization, sparse optimization, and compressed sensing.
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.
Flots discrets et continus. Équations différentielles non linéaires, techniques classiques d’analyse de dynamique, existence et stabilité de solutions, variétés invariantes, bifurcations, formes normales, systèmes chaotiques. Applications moderne.
Examen de modèles fondamentaux utilisés en biologie mathématique et de leur analyse utilisant des outils modernes de calcul scientifique. Systèmes dynamiques discrets et continus, procédés stochastiques, modèles statistiques et simulation numérique. Enquête des publications récentes en biologie mathématique par journal club.
Virgule flottante. ÉDOs. Méthodes directes et itératives pour la résolution de systèmes linéaires et non-linéaires. Valeurs propres. ÉDPs elliptiques et paraboliques. Équation de Black-Scholes. Optimisation sans contraintes (MAT 6473 uniquement), Décomposition en valeurs singulières (SVD, MAT 6473 uniquement).