Office
McLean Hall 325B
Addresses
- Email
- dstewart (at) math.uiowa.edu
- Paper Mail
- David Stewart
Department of Mathematics
Room 14, McLean Hall
The University of Iowa
Iowa City, IA 52242-1419
USA - Phone:
- voice: 319-335-3832
fax: 319-335-0627
I teach at the University of Iowa Mathematics Department.
Here are my classes. And here is my schedule for Spring 2008.
Places I have worked...
2006-present
- 2000-2006
- Associate Professor, Mathematics
Dept., University of Iowa.
- 1997-2000
- Assistant Professor, Mathematics
Dept., University of Iowa.
- 1996-1997
- Assistant Professor, Mathematics
Dept., Virginia Polytechnic Institute
and State University
1995-1996
- 1991-1994
- Australian National University, School of Mathematical Sciences
as Postdoctoral Fellow and then Research Fellow in the advanced
computation group
- 1990-1991
- Mathematics
Department, University of Queensland,Australia as
Lecturer in Computational Mathematics. (This is equivalent to a
visiting
assistant professor position.)
Professional Interests
Erdös number
My Erdös
number is less or equal to 3. Here is
the proof.
A long, long time ago, I can still remember (some)...
Here is a picture from the January 1993 SCADE meeting in Auckland, New
Zealand.
If you can identify anybody not already identified, let me know...
Or better yet, if you know xfig, then update the xfig file in the zip
here: auckland93.zip.
Conference picture.
Here is the list of the attendees,
thanks to John Butcher (who thanks his wife for keeping it).
Current interests/research
I am a member of the Numerical
Analysis group at the University of Iowa. Amongst other
things,
we have a seminar
series and a preprint
collection.
Research projects
I am part of the DaVinci
(Differential Algebraic and Variational Inequalities in Control and
sImulation) project. It is about how to simulate and control
non-smooth dynamical systems systems of different kinds. These
arise in the context of rigid-body dynamics (see below), electrical
circuits with switching elements such as diodes and transistors,
and hybrid
control systems, for example. Many of these systems can be
modeled using complementarity theory ; complementarity
conditions
have the form
f(x,y) = 0,
0 <= x orthogonal to y >= 0
where x and y can be vectors (in which case "x
>= 0" means "xi >= 0 for all i").
If f represents something like a differential equation, then
this
is a Dynamic Complementarity Problem (DCP). A special case is the
class of Linear
Complementarity Systems (LCS), which have the form
dx/dt = Ax + Bu,
y = Cx + Du,
0 <= u(t) orthogonal to y(t)
>= 0.
Recently, I have been working on convolution complementarity
problems which have the form: Find u(t) satisfying
0 <= u(t) orthogonal to (k*u)(t)
+ q(t) >= 0
for all t >= 0 given the functions k(t)
and q(t).
This has applications in elastic body impact problems.
Rigid body dynamics & measure differential inclusions
My current work is on mathematical and computational models of rigid
body mechanics (including friction and collisions). This involves pure
mathematics (measure differential inclusions, which were
invented
bu J.J. Moreau in the 1980's) as well as numerical analysis and more
classical applied mathematics. This relates to previous work on
discontinuous ODE's and differential inclusions.
Here are some PostScript ``movies'' showing some simulations of
bouncing balls (step-size is h = 0.01, coefficient of
restitution
is 0.9, and coefficient of friction is 0.5, with g = 9.81 m.s-2;
each ball has a radius of 10 cm). Hold your finger down on the Return
key if you are using Ghostview.
Plan view (PostScript) or (PDF)
Elevation view (PostScript) or (PDF)
Rod problem á la Painlevé (PostScript)
or (PDF)
One of the major outstanding issues in the area is the resolution of
Painlevé's paradoxes. I have recently been able to prove
rigorously in terms of measure differential inclusions & equations,
that these problems do indeed have solutions, provided the maximal
dissipation form of Coulomb's law is used with the post impact
velocity - at least for 1-dimensional frictional forces and one
contact.
This includes Painlevé's examples, and is the first general
result on rigid body dynamics to do so.
This work has led to some new investigations and results about
measure differential inclusions and equations.
Linear (and Nonlinear) Complementarity Problems
This work also relates to Linear Complementarity Problems (LCP's).
LCP's are problems where given a square matrix M and a
compatible
matrix q, the task is to find vectors z and w
such that
Mz+q = w >= 0, z >= 0, z Tw = 0
where the inequalities are understood componentwise. LCP's
(in spite of their name) are truly nonlinear, and tools of nonlinear
analysis such as degree and index theory can be successfully applied to
LCP's (and NCP's). They can be represented in various ways in terms of
nonlinear (but nonsmooth!) systems of equations.
Some recent work (with Jong-Shi Pang) has been on developing a
unified complementarity formulation of contact problems with friction.
(See the previous paragraph on rigid body dynamics as well.)
Optimization and Optimal Control
Other interests include optimization and optimal control. This includes
some work on solving optimal control problems with discrete control
values (e.g., {on, off}, or {-1, 0, +1}) with switching costs. (Without
switching costs, the optimal solutions typically ``chatter'' rapidly
between the allowed control values, which ``convexifies'' the problems
and makes into standard optimal control problems.) These problems can
be
NP-hard, but there are ways of developing good, efficient, suboptimal
algorithms.
Dynamical Systems
I am interested in good computational/numerical methods for dealing
with dynamical systems, fractals and related objects. One area of
interest is the (numerically stable) calculation of Lyapunov exponents,
which has led me to investigate singular value decompositions of
products of matrices and the notion of stable products: small
perturbations to the factor matrices should not lead to large
relative changes in the singular values.
I have also worked on algorithms that can distinguish between the fractal
and Hausdorff-Besicovitch dimensions.
Computational geometry
I have also done some work on using quadtrees and octrees to improve
the asymptotic behavior of some algorithms for meshfree methods.
Meshfree methods are Galerkin methods for solving PDE's which don't
rely
on a mesh like standard Finite Element Methods do. (This is
particularly crucial since current meshing software has a great deal of
difficulty avoiding triangles with small angles and similar pathologies
with other elements in two and three dimensions.) However, this
means that there are more geometric tasks that have to be performed in
the basic meshfree methods.
David Stewart
Last modified Jan 21, 2008.
To the Department of
Mathematics