Applied and Computational Mathematics

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.

  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.

For more information on our research programs and seminars, please consult the McGill Applied Mathematics page and the GIREF (Groupe interdisciplinaire de recherche éléments finis) page.

Courses 2011-12

Automne/Fall

Numerical Analysis of Nonlinear Equations
Concordia, COMP6361/2-DD, Sebius Doedel
An introduction to numerical algorithms for nonlinear equations, including discrete and continuous systems. The emphasis is on computer-aided numerical analysis, rather than numerical simulation. This course is suitable for scientists and engineers with a practical interest in nonlinear phenomena. Topics include computational aspects of continuation methods, fixed points and stationary solutions, asymptotic stability, bifurcations, periodic solutions, transition to chaos, travelling wave solutions, discretization techniques. A variety of applications will be considered. Numerical software packages will be available.
Fluid Dynamics
McGill, MATH 555-001, Peter Bartello
Kinematics. Dynamics of general fluids. Inviscid fluids, Navier-Stokes equations. Exact solutions of Navier-Stokes equations. Low and high Reynolds number flow.
Numerical Analysis 1
McGill, MATH 578-001, Jean-Christophe Nave
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.
Partial Differential Equations 1-First order and elliptic equations
McGill, MATH 580-001, Gantumur Tsogtgerel
Basic functional analysis and Lebesgue integration: These material will be treated piece by piece as needed, either in class or as reading projects. If you are thinking about preparing for the course these are the topics you should be reading. It would really help a lot. Generalities: Cauchy-Kovalevski theorem, characteristic surfaces, standard classifications, well posedness, approximation, estimates, coordinate transformation First order equations: Linear and quasilinear equations, characteristics, conservation laws, inviscid Burger's equation, shock waves, jump conditions Laplace operator: Fundamental solution, Green's function, mean value property and its consequences, Harnack's theorem, Dirichlet principle, well posedness Linear elliptic operators: Variable coefficients, Stokes problem, regularity theory, Sobolev spaces, Harnack estimates, spectral theory, Fredholm operators Nonlinear elliptic equations: Critical exponent, direct method of calculus of variations, regularity theory for quasilinear equations, applications of fixed point theorems As time permits we will cover topics from MATH 581, but we might as well end up doing the opposite
Topics in Applied Mathematics 1: Delay Differential Equations
McGill, MATH 761-001, Tony Humphries
Examples of Delay Differential Equations (DDEs). Fixed Delay DEs: Delayed Negative Feedback, Existence of Solutions, Linear Systems and Linearization, DDEs as semi-dynamical systems. Hopf Bifurcation for DDEs. Applications in Biology, Physics and Engineering. Introduction to Distributed Delay DEs, State-Dependent DDEs and numerical methods for DDEs.
Analyse et optimisation de forme / Shape Analysis and Optimization
UdM, MAT 6441, Michel Delfour
Plan de cours: Géométrie différentielle. Microstructures, frontières libres. Métriques de Courant et d'Hausdorff, ensembles de Federer. Calcul différentiel sur hypersurfaces. Calcul de forme. Appl.: élasticité, fluides, chaleur, images. Préalables: Quelques notions d'équations aux dérivées partielles elliptiques et d'analyse fonctionnelle. Des notes de cours complémentaires seront disponibles pour les etudiants dont la formation n'est pas tout à fait adéquate. Les projets individuels seront adaptés aux connaissances et à la formation de l'étudiant(e). Livre: M.C. Delfour et J.-P. Zolésio, Shapes and geometries: analysis, differential calculus and optimization, SIAM 2011, seconde édition (en format papier et électronique). Program of the course: Differential geometry. Microstructures, free boundaries. Courant and Hausdorff metrics, Federer's sets of positive reach. Differential calculus on hypersurfaces. Shape calculus. Appl. : elasticity, fluids mechanics, heat, images. Prerequisites: A few notions of partial differential equations and functional analysis. Complementary material will be available in the form of lecture notes. Individual projects will be adapted to the general background of the student. Book: M.C. Delfour and J.-P. Zolésio, Shapes and geometries: analysis, differential calculus and optimization, SIAM 2011., second edition (printed and electronic versions).
Équations aux dérivees partielles
UdM, MAT6110, Anne Bourlioux
Distributions et transformation de Fourier, équation de la chaleur, problème de Sturm-Liouville, espaces de Sobolev, problème de Dirichlet, valeurs et fonctions propres du laplacien, équation des ondes.
Mathématiques appliquées, sujets speciaux : Théorie de la coalescence, extensions et applications
UdM, MAT6480, Sabin Lessard

Hiver/Winter

Méthodes numériques avancées pour les EDP
Laval, MAT 7430, Hassan Manouzi
Rappel sur les E.D.P. Notions de distributions. Espaces de Sobolev. Problèmes aux limites elliptiques: formulation variationnelle, existence et unicité, exemples. Méthodes des différences finies: problèmes elliptiques, paraboliques, equation de transport. Éléments finis pour les problèmes elliptiques: dimensions 1 et 2, éléments finis de Lagrange, estimation d'erreur, intégration numérique.
Numerical Differential Equations
McGill, MATH 579 -001, Jean-Christophe Nave
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.
Partial Differential Equations 2 - Evolution equations - Systems of conservation laws and Riemann invariants
McGill, MATH 581-001, Gantumur Tsogtgerel
*Note that this is a direct continuation of MATH 580. Linear and nonlinear parabolic equations: Heat propagator, maximum principle, Harnack estimates, regularity theory, viscosity solution of first order hyperbolic systems, semilinear and quasilinear theories. Incompressible fluids: Stokes propagator, weak solution, regularity, Navier-Stokes, Euler and magnetohydrodynamics equations, regularized models, vanishing viscosity limits Hyperbolic and dispersive equations: Wave and Schroedinger propagators, weak solution, regularity, linear scattering theory, nonlinear examples.
Topics in Applied Mathematics 2: Advanced Nonlinear Dynamics and Chaos
McGill, MATH 762-001, George Haller
The internal structure of chaos: Symbolic dynamics, Bernoulli shift map, subshifts of finite type; chaos is numerical iteration. Hamiltonian dynamical systems: Conservation properties; Poincare recurrence theorem, stability of fixed points, KAM theory, integrable and near-integrable Hamiltonian systems, invariant tori, Liouville- Arnold-Jost Theorem. Normally hyperbolic invariant manifolds: Crash course on differentiable manifolds; existence, persistence, and smoothness. Geometric singular perturbation theory: Formulation in terms of invariant manifold theory; physical examples. Finite-time dynamical systems: Invariant manifolds and coherent structures in finite-time flows.
Calcul scientifique
UdM, MAT6470, R. Owens
Étude des algorithmes fondamentaux en calcul scientifique. Principes théoriques; programmation et application à des problèmes pratiques; utilisation scientifique de logiciels spécialisés.