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:

- CRM Applied Mathematics Laboratory
- GIREF (Groupe interdisciplinaire de recherche en éléments finis)

- Peter Bartello (McGill)
- Anne Bourlioux (UdeM)
- Alain Chalifour (UQTR)
- Rustum Choksi (McGill)
- Michel Delfour (UdeM)
- Jean Deteix (Laval)
- Eusebius Doedel (Concordia)
- François Dubeau (Sherbrooke)
- André Fortin (Laval)
- Robert Guénette (Laval)
- Tim Hoheisel (McGill)
- Tony Humphries (McGill)
- Guillaume Lajoie (UdeM)
- Jean-Philippe Lessard (Laval)
- Sherwin Maslowe (McGill)
- Jean-Christophe Nave (McGill)
- Adam Oberman (McGill)
- Robert Owens (UdeM)
- Georg Schmidt (McGill)
- Ronald Stern (Concordia)
- Gantumur Tsogtgerel (McGill)
- José Urquiza (Laval)
- Adrian Vetta (McGill)
- Jian-Jun Xu (McGill)

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.

- 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.
- 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).
- 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.
- 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.
- 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.
- 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.

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.

The main focus of the course is going to be on linear first and second order equations, and Sobolev spaces. Rather than trying to build everything in full generality, we will study prototypical examples in detail to establish good intuition. Roughly speaking, most of the topics from the calendar description of Math 580 and some from that of Math 581 will be covered. More precisely, the planned topics are

• First order equations, method of characteristics

• Cauchy problem for heat and wave equations

• Duhamel's, Huygens, and maximum principles

• Green's identities, harmonic functions, Harnack inequality

• Fundamental solution, Green's function, Poisson's formula

• Dirichlet problem: Perron's method, barriers, boundary regularity

• Sobolev spaces, weak and strong derivatives, Dirichlet principle

• Poisson equations: Variational formulation, boundary conditions

• Elliptic regularity, Sobolev embedding

• Laplace eigenvalues and eigenfunctions (if time permits)

Many physical processes are modelled by differential equations which involve delays. This course will provide an introduction to delay differential equations (DDEs) concentrating on the key tools needed to understand the behaviour of these equations, and also some of numerical techniques used to approximate solutions. Throughout we will emphasise the similarities and differences between DDEs and ordinary differential equations (ODEs).

Topics covered will include: DDEs as infinite dimensional dynamical systems, breaking points and smoothing of DDE solutions, continuous Runge-Kutta methods for ODEs and DDEs, linear stability of steady states, bifurcation theory. A selection of more advanced topics will also be covered. The choice of topics will depend on time and the preferences of the participants, but may include state-dependent delays, distributed delays, numerical continuation and bifurcation techniques.

This course focuses on solving Markov Decision Problems (MDP) using methods of dynamic programming and reinforcement learning. Solution approaches to finite and infinite horizon MDP through the Bellman Equation, value function iteration and policy iteration are presented. Approximate dynamic programming ideas are then introduced to increase computational speed. Methods of reinforcement learning such as temporal-difference learning, online versus offline control and eligibility traces are then illustrated. Students will have to code extensively in R throughout the course.

Algorithmic methods for big data analysis. Complexity analysis, data structure, parallel and distributed computing.

The course is organized in 4 themes that will cover various aspects of algorithmic for big data, starting from sequential programming and ending with distributed computing. In the first part of the course, the student will learn to analyze an algorithm from the computational complexity and memory requirement. The second theme in the course deals with parallel computing with shared memory. The efficiency of the parallelization and memory safety will be discussed and analysed. In the third theme, the message passing interface (MPI) will be explored, which consists in simultaneous and collaborative parallel computing without shared memory. Finally, the basics of distributed computing, its strength and requirements will be introduced. The choice of the best approach toward the resolution of a problem will depend on the problem and the nature of the data.

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.

Kinematics. Dynamics of general fluids. Inviscid fluids, Navier-Stokes equations. Exact solutions of Navier-Stokes equations. Low and high Reynolds number flow.

The main focus of the course is going to be on nonlinear problems. Sobolev spaces, the Fourier transform, and functional analytic methods will be heavily used. The planned topics are

• Tempered distributions, convolution, Fourier transform

• Fourier analytic treatment of Sobolev spaces

• Problems in half-space, shades of hyperbolicity, parabolicity, and ellipticity

• Overview of elliptic theory, regularity

• Semilinear elliptic equations, monotonicity methods

• Variational problems, compactness methods

• Semilinear evolution equations, Duhamel's principle

• The Navier-Stokes equations and related turbulence models

• Semilinear elliptic problems with critical exponents (if time permits)

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