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.

2020-21 Course Listings

Fall

Topics in Statistics & Probability: Reinforcement Learning

This course is an introduction to reinforcement learning techniques. It requires extensive programming with the R language. Topics covered include: Multi-armed bandit problem, Markov Decision Problems, Dynamic Programming, Monte-Carlo solution methods, Temporal difference methods, Multi-period Approximation methods, Policy gradient.

Prof. Frédéric Godin

MAST 679/2, MAST 881

Institution: Concordia University

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. Gantumur Tsogtgerel

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

Topics in Applied Mathematics. Statistical Learning Theory: Mathematical Foundations of Machine Learning

SVM and kernel methods. Proof of generalization using concentration of measure.

Prof. Adam Oberman & Prakash Panangaden

MATH 597

Institution: McGill University

Geometric data analysis

Formal and analytic approaches for modeling intrinsic geometries in data. Algorithms for constructing and utilizing such geometries in machine learning. Applications in classification, clustering, and dimensionality reduction.

The course will accommodate anglophone students who are interested in taking it, as well as francophone students.

Prof. Guy Wolf

MAT 6493

Institution: Université de Montréal

Winter

Topics in Applied Mathematics: Sparsity and Compressed Sensing

Sparsity is a key principle in real-world applications such as image or audio compression, statistical data analysis, and scientific computing. Compressed sensing is the art of measuring sparse objects (like signals or functions) using the minimal amount of linear measurements. This course is an introduction to the mathematics behind these techniques: a wonderful combination of linear algebra, optimization, numerical analysis, probability, and harmonic analysis. 

Topics covered include: l1 minimization, iterative and greedy algorithms, incoherence, restricted isometry analysis, uncertainty principles, random Gaussian sampling and random sampling from bounded orthonormal systems (e.g., partial Fourier measurements), applications to signal processing and computational mathematics.

Prof. Simone Bruguapaglia

MAST 680-C (MAST 837-C)

Institution: Concordia University

Méthodes numériques avancées pour les EDP

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, équation de transport. Éléments finis pour les problèmes elliptiques : dimensions 1 et 2, éléments finis de Lagrange, estimation d'erreur, intégration numérique.

Prof.

MAT 7435

Institution: Université Laval

Honours Mathematical Models in Biology

The formulation and treatment of realistic mathematical models describing biological phenomena through such qualitative and quantitative mathematical techniques as local and global stability theory, bifurcation analysis, phase plane analysis, and numerical simulation. Concrete and detailed examples will be drawn from molecular, cellular and population biology and mammalian physiology.

Prof.

MATH 537

Institution: McGill University

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

Partial Differential Equations 2

Systems of conservation laws and Riemann invariants. Cauchy-Kowalevskaya theorem, powers series solutions. Distributions and transforms. Weak solutions; introduction to Sobolev spaces with applications. Elliptic equations, Fredholm theory and spectra of elliptic operators. Second order parabolic and hyperbolic equations. Further advanced topics may be included.

Prof. Gantumur Tsogtgerel

MATH 581

Institution: McGill University

Mathematical genetics and systems biology

Branching processes, Wright-Fisher and Moran models, infinite alleles model. Evolutionary factors: selection, mutation, migration, recombination. Reconstruction and inferences of genetic networks. 

The course will accommodate anglophone students who are interested in taking it, as well as francophone students.

Prof. Morgan Craig

MAT 6461

Institution: Université de Montréal

Calcul scientifique

Virgule flottante. ÉDOs. Modélisation et simulations. Méthodes directes et itératives pour la résolution de systèmes linéaires et non-linéaires. Gestion de données. Valeurs propres. ÉDPs elliptiques et paraboliques. Équation de Black-Scholes.

Prof. Robert G. Owens

MAT 6473

Institution: Université de Montréal