
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 AdvectionDiffusionReaction
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 semiiteration
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:
http://www.cs.toronto.edu/~ccc/Courses/cs2321.html
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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 lognorm
2. General Properties of Numerical Methods [3.13.3]
3. Standard Classes of Methods [Chapters 4 and 5]
 One step methods, Taylor series and RungeKutta
 Derivation of RungeKutta formulas
 Local error estimates for RungeKutta formulas
 Multistep methods, Adams formulas
 Derivation of variable step formulas
 Implementation issues for multistep formulas
 Survey of existing software
4. Difficulty of Stiffness [3.43.6]
 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 RungeKutta 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 DifferentialAlgebraic Equations,
U. M. Ascher and L. R. Petzold; SIAM, 1998.
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CANCELLED
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 RungeKutta methods
 stochastic linear multistep methods
 difficulties with lack of commutativity in the problem the magnus
formula
 numerical results
 Bseries and convergence of methods
3. Stability properties and implicit methods
 Astability
 MSstability
 Tstability
 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.
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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, 1:303:00 pm and 3:305:00 pm
Start date: January 7 April 8, 2002
Location: The Fields Institute, room 230
This course provides a rigorous uptodate treatment of topics in Continuous
Optimization (Nonlinear Programming). This includes a handson approach
with exposure to existing software packages.
See Course
overview
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Numerical Solution of PDEs
Instructor: Robert
Almgren, University of Toronto
Day/time: Wednesday, 1:304:30 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.
Topics
 Parabolic equations
 Explict and implicit discretizations
 Consistency, stability, and convergence
 Von Neumann stability (Fourier analysis)
 Variational inequalities and free boundaries
 Multidimensional 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 NavierStokes equations of fluid dynamics
 Adaptive mesh refinement techniques
 Spectral and pseudospectral methods
 Special techniques: random walkers, lattice gas
Prerequisites
Applied mathematical knowledge at the level of a firstyear 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
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