# The Fields Institute

Industrial Mathematics Seminar Series

#### Thursday, October 15, 1998

5:00 - 7:30 p.m.

Room 230, The Fields Institute

**SCHEDULE**

5:00 - 6:00 p.m.

*"Mathematical Successes and Challenges on the Network Edge"*
André Van Schyndel, Network Edge Technology Laboratories, Northern Telecom (Nortel)

6:30 - 7:30 p.m.

*"Inverse Problems: Formulation and Solution"*

Henning Rasmussen, Department of Applied Mathematics, University of Western
Ontario

## ABSTRACTS

*"Mathematical Successes and Challenges on the Network Edge"*

André Van Schyndel

The "Network Edge" labs at Nortel translate the bits and bytes of network
communication into the real world of eyes, ears, touch, and all human cognition.
A successful construction of this interface needs expertise in psychology,
human factors, industrial design and ergonomics as well as mathematics and
physics. One crucial realization of our team has been the value of the experts
in universities and other centers of expertise (like the Fields Institute)
for fresh input into the myriad of design opportunities afforded by today's
technology. In this industrial seminar, I would like to present a few of our
many challenges where mathematics has played a key role in both understanding
the problems and producing novel solutions. I would also like to present a
few of the many potential products where we have yet to solve the mathematics
needed for successful implementation. I have chosen some current unsolved
mathematics involving inverse boundary value problems (where we know the differential
equation but need to find the boundary values which will produce solutions
with particular characteristics) and generalized Fourier series (where we
need to identify integral transforms amenable to modern digital signal processing
which produce optimal inverses in the presence of noise). We consider ourselves
very fortunate to have the opportunity to discuss these ideas with such a
distinguished audience of students, professors, experts and onlookers that
the Fields Institute has the unique capability of attracting; especially my
co-presenter Professor Henning Rasmussen, a recognized expert in mathematics
related to these challenges.

*"Inverse Problems: Formulation and Solution"*

Henning Rasmussen

Many engineering problems can be modeled by one or more differential equations
and associated boundary and initial conditions; the difficult part is to ensure
that the model contains enough detail that it represents the engineering problem
accurately enough but is simple enough that it can be solved. Usually the
solution procedure results in a computer program since very few mathematical
models of practical problems can be solved analytically.

However, there exists a large class of very important engineering problems
in which some of the very important details of the physical situation are
not known but must be deduced from measurements. As an example one can mention
the non-destructive testing of welding seams. In this case it is not known
if the weld forms a continuous seal between two metal pieces, and in fact
this is the information which is required. The mathematical models of such
problem contain parameters that during the modeling process are unknown but
which must be determined, usually in an iterative manner by comparing the
output of the model to measured data.

In order to illustrate the process by which such problems can be treated
I will discuss the modeling and development of computational programs for
two inverse problems. The first one consists of estimating the extent of a
metallic ore body without any additional drilling while in the second one
we will show how electromagnetic waves might be useful in the monitoring of
the treatment of leukemia.

## SPEAKERS

*André Van Schyndel* is Senior Staff Scientist in Nortel's
Network Edge Technology laboratories in Ottawa. His primary duties include
both creating and bringing in new ideas and technologies to interface communication
networks to their human users. He has a Ph.D. in solid state physics and acoustics
from the University of British Columbia, but has a reputation for being interested
in everything. Dr. Van Schyndel has recently been involved with Fields Institute
initiatives to encourage students with an aptitude in mathematics and science
to explore their creativity and perhaps pursue related careers.

*Henning Rasmussen* is a professor in the Department of Applied
Mathematics at the University of Western Ontario and past-President of the
Canadian Applied Mathematics Society. After receiving his Ph.D. in Applied
Mathematics from the University of Queensland, Australia, he held postdoctoral
positions at Monash University and Southampton University and taught at Technical
University of Denmark. His research interests include numerical methods, free
boundary problems, fluid mechanics and industrial mathematics. An active researcher,
Henning Rasmussen has 80 papers published in refereed journals.

**OTHER INFORMATION**

The Industrial Mathematics Seminar is offered to any interested participant
-- no reservation is necessary.