July 17, 2024

Numerical and Computational Challenges in Science and Engineering Program

Graduate Course Information

Summer 2002

July 29 - August 2 (2002)
Short Course on Numerical Solution of Advection-Diffusion-Reaction Equations,
Prof. Jan Verwer and Martin Berzins
(this course not available for credit)

Starting Winter 2002

Numerical Solution of PDEs, Prof. Robert Almgren
Numerical Solution of Optimization Problems, Prof. Henry Wolkowicz
Short Course on Numerical and Computational Challenges in Environmental Modelling, Prof. Zahari Zlatev

Starting Fall 2001

Numerical Linear Algebra, Prof. Christina Christara
Numerical Solution of ODEs, Prof.'s Wayne Enright and Ken Jackson
Numerical Solution of SDEs, Prof. Kevin Burrage * POSTPONED *
Short Course on Matrix Valued Function Theory, Prof. Olavi Nevanlinna
Short Course on Numerical Bifurcation and Centre Manifold Analysis in Partial Differential Equations, Prof. Klaus Böhmer

Numerical Linear Algebra
Instructor: Christina Christara, University of Toronto

Day/time: Tuesdays, 2:00 - 5:00 pm
Start date: Tuesday, September 11 - December 4, 2001
Location: The Fields Institute, room 230

This course focuses on the efficient solution of large sparse linear systems. Such systems may arise from the discretisation of PDE problems, approximation problems or other science and engineering problems. We briefly introduce some standard linear solvers, then proceed to study selected developments in the area of Numerical Linear Algebra, including:

  • iterative solvers
  • acceleration techniques, such as semi-iteration and conjugate gradient
  • preconditioning techniques, such as domain decomposition methods,
    (Schur complement and Schwarz splitting methods)
  • multigrid schemes and fast direct solvers, such as Fast Fourier Transform methods
  • applications to PDEs

    Prerequisites: Calculus, basic Numerical Linear Algebra, Interpolation, some knowledge of PDEs,
    programming (preferably in MATLAB or FORTRAN).

    Web page:

Numerical Solution of ODEs
Instructors: Wayne Enright, and Ken Jackson, University of Toronto

Day/time: Mondays, 10:00 am -1:00 pm
Start date: September 10 - December 3, 2001
Location: The Fields Institute, room 210

1. Mathematical Setting [1.1, Chapter 2]

  • Solution of perturbed systems
  • The defect of a numerical solution
  • General error bounds for perturbed systems
  • Tight bounds using log-norm

2. General Properties of Numerical Methods [3.1-3.3]

  • Classical properties of order/stability/convergence
  • Local error, global error, various error bounds

3. Standard Classes of Methods [Chapters 4 and 5]

  • One step methods, Taylor series and Runge-Kutta
  • Derivation of Runge-Kutta formulas
  • Local error estimates for Runge-Kutta formulas
  • Multistep methods, Adams formulas
  • Derivation of variable step formulas
  • Implementation issues for multistep formulas
  • Survey of existing software

4. Difficulty of Stiffness [3.4-3.6]

  • What is a `stiff problem' and where do they arise
  • What are the difficulties/complications that affect computation

5. Special methods for Stiff problems [4.7, 5.1.2, 5.4.3]

  • Implicit Runge-Kutta methods
  • BDF methods
  • Exploiting special problem structure
  • Survey of existing software

6. Differential/Algebraic Equations [Chapters 9 and 10]

  • Problem structure and classification
  • Two basic approaches
  • Survey of existing software

7. Delay Differential Equations

  • Classification of problems and the associated mathematical properties
  • Numerical issues
  • Survey of existing software

8. Validated Numerical Methods for ODEs

  • Guaranteed error bounds/Interval arithmetic
  • Limitations and inherent difficulties

9. Parallel Methods for ODEs

  • Special Formulas
  • Waveform relaxation
  • Other approaches

Prerequisites: We assume a solid undergraduate background in mathematics and computer science.
Such a background would normally involve two years of calculus, a year of linear algebra, a year of numerical analysis and exposure to one or more high level programming languages, preferably FORTRAN or C. A mathematical course on the theoretical or analytic properties of differential equations would be helpful, although not essential.

The textbook for the course is: Computer methods for Ordinary Differential Equations and Differential-Algebraic Equations,
U. M. Ascher and L. R. Petzold; SIAM, 1998.



Numerical Solution of SDEs
Instructor: Kevin Burrage, The University of Queensland

Day/time: Wednesdays, 10:00 am -1:00 pm
Start date: October 3 - December 5, 2001
Location: The Fields Institute, room 230

Numerical methods for stochastic differential equations by Kevin Burrage and Pamela Burrage

1. Introduction to sdes

  • models
  • different noise processes
  • stochastic integrals
  • taylor series
  • expectations
2. Numerical methods and their order properties
  • weak and strong order
  • stochastic Runge-Kutta methods
  • stochastic linear multistep methods
  • difficulties with lack of commutativity in the problem the magnus formula
  • numerical results
  • B-series and convergence of methods
3. Stability properties and implicit methods
  • A-stability
  • MS-stability
  • T-stability
  • stiffness
  • composite methods
  • implicit methods
4. An application in hydrology - the numerical solution of a stochastic partial differential equation
  • wiener processes in time and space
  • computation techniques
5. Implementation issues
  • computation of stochastic integrals
  • the brownian path
  • variable step size implementations
  • embedding, extrapolation
  • PI control

Background reading: The book by P. Kloeden and E. Platen on numerical methods for SDES.
There will also be handouts of notes.


Short Course on Matrix Valued Function Theory
Instructor: Olavi Nevanlinna, Helsinki University of Technology

Dates: October 11, 18, 19, 25 and 26th, 2001
Location: The Fields Institute, Room 210
Course overview

Numerical Solution of Optimization Problems (C&O 769 Winter Semester 2002)
Instructor: Henry Wolkowicz, University of Waterloo

Day/time: Monday, 1:30-3:00 pm and 3:30-5:00 pm
Start date: January 7- April 8, 2002
Location: The Fields Institute, room 230
This course provides a rigorous up-to-date treatment of topics in Continuous Optimization (Nonlinear Programming). This includes a hands-on approach with exposure to existing software packages.
See Course overview

Numerical Solution of PDEs
Instructor: Robert Almgren, University of Toronto

Day/time: Wednesday, 1:30-4:30 pm
Start date: January 9- April 10, 2002
Location: The Fields Institute

This course will cover basic techniques for solving partial differential equations on the computer, with emphasis on finite difference methods. Special attention will be paid to how the features of a good discretization reflect the mathematical properties of the PDE being solved.

  • Parabolic equations
    • Explict and implicit discretizations
    • Consistency, stability, and convergence
    • Von Neumann stability (Fourier analysis)
    • Variational inequalities and free boundaries
    • Multi-dimensional problems
  • Elliptic equations
    • The maximum principle
    • Solution of sparse linear systems
  • Hyperbolic equations and conservation laws
    • CFL stability
    • Flux conservation and shock capturing
  • Variational formulations and finite element methods
  • Possible special topics (if time permits)
    • The Euler and Navier-Stokes equations of fluid dynamics
    • Adaptive mesh refinement techniques
    • Spectral and pseudo-spectral methods
    • Special techniques: random walkers, lattice gas


Applied mathematical knowledge at the level of a first-year graduate student in mathematics, especially linear algebra and ordinary differential equations. Previous study of partial differential equations is very useful. Assignments will be given that require use of the Matlab programming environment.

University of Toronto students may register for this course as CSC446/2310H.

Taking the Institute's Courses for Credit

As graduate students at any of the Institute's University Partners, you may discuss the possibility of obtaining a credit for one or more courses in this lecture series with your home university graduate officer and the course instructor. Assigned reading and related projects may be arranged for the benefit of students requiring these courses for credit.

Financial Assistance

As part of the Affiliation agreement with some Canadian Universities, graduate students are eligible to apply for financial assistance to attend graduate courses. Interested graduate students must forward a letter of application with a letter of recommendation from their supervisor.
Two types of support are available:

  • Students outside the greater Toronto area may apply for travel support. Please submit a proposed budget outlining expected costs if public transit is involved, otherwise a mileage rate is used to reimburse travel costs. We recommend that groups coming from one university travel together, or arrange for car pooling (or car rental if applicable).

  • Students outside the commuting distance of Toronto may submit an application for a term fellowship. Support is offered up to $1000 per month. Send an application letter, curriculum vitae and letter of reference from a thesis advisor to the Director, Attn.: Course Registration, The Fields Institute, 222 College Street, Toronto, Ontario, M5T 3J1.

Applications for financial support should be received by the following deadlines: June 1, 2001 for the Fall term, and October 1, 2001 for the Winter term.

For more details on the thematic year, see Program Page