SC 99-34 Peter Deuflhard: Differential Equation Problems in Real Life. Computational
Concepts, Adaptive Algorithms, and Virtual Labs.
Abstract:
This series of lectures has been given to a class of
mathematics postdocs
at a European summer school in Martina Franca (organized
by CIME). It deals with a
variety of challenging real life problems selected from
clinical cancer
therapy,
communication technology, polymer production, and
pharmaceutical drug
design. All of
these problems from rather diverse application areas share
two common
features: (a)
they have been modelled by various differential
equations - elliptic,
parabolic, or
Schrödinger-type partial differential equations,
countable ordinary
differential equations, or Hamiltonian systems, (b) their
numerical
solution has turned out
to be a real challenge to computational mathematics.
Before diving into actual computation, the
computational concepts to be applied
are carefully considered. To start with, any numerical
analyst must
be prepared to totally
remodel problems coming from science or engineering. The
computational problems
to be
treated should be well-posed, important features of any
underlying continuous
model should be passed on, if at all possible, to the
discrete model, and the
computational resources employed (computing time, storage,
graphics) should be
adequate.
Speaking
in mathematical terms, the solutions to be approximated
live in appropriate
infinite dimensional function spaces. Examples given in
the paper are
Sobolev spaces, discrete weighted sequence
spaces, or certain statistically weighted function spaces.
The
mathematical paradigm advocated herein is that - already
due to
mathematical aesthetics - any infinite
dimensional space should not be represented by just
a single finite
dimensional space (with possibly large dimension), but by
a well-designed
sequence of finite dimensional spaces, which successively
exploit the
asymptotic
properties characterizing the original function space. The
fascinating, but
(for a
mathematician) not really surprising experience is that
mathematical
aesthetics go directly with computational efficiency. In
other words, a
careful
and sufficiently
ingenious realization of the above paradigm will lead to
efficient algorithms
that actually work in challenging real life problems.
The reason for this coincidence of aesthetics and
efficiency lies in the fact
that function spaces describe some data redundancy
that can be exploited
in the numerical solution process. In order to exploit
these redundancies,
adaptivity of algorithms is one of the key
construction principles.
Typically, wherever adaptivity has been successfully
realized, algorithms
with a computational complexity close to the (unavoidable)
complexity of the
problem emerge - a feature of extreme importance
especially in difficult
problems of science, technology, or medicine.
In traditional industrial environments, however, new
efficient mathematical
algorithms are not automatically accepted - even if they
significantly
supercede
already existing older ones (if not old-fashioned ones)
within market
dominating
software systems. Exceptions do occur
where simulation or optimization is a clear market
competition factor. Due
to this experience
the authors group has put a lot of effort in the design
of virtual labs.
These specialized integrated software systems permit a
fast and convenient
switch between numerical code and
interactive visualization tools (that we also develop, but
do not touch here).
Sometimes only such virtual labs open the door to
hospitals or industrial
labs for
new mathematical ideas.
Keywords: differential equations:,
ordinary,
partial,
countable,
hamiltonian,
finite element methods:,
adaptive,
multilevel,
grid generation,
medical therapy planning,
integrated optics,
polymer chemistry,
biochemistry,
drug design
MSC: 65-02, 65C05, 65C20, 65F15, 65J10, 65L05, 65L08, 65N25, 65N30, 65N55, 65U05, 65Y25, 78-08, 78A50