David Stewart, Professor
Department of Mathematics
University of Iowa

Picture of David Stewart
Professor David Stewart
McLean Hall 325B
Email: david-e-stewart (at) uiowa.edu
Paper Mail
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

Email: david-e-stewart@uiowa.edu
Office hours: MTuW 10:30-11:30am

Research Areas: Numerical Analysis, Mathematical Modeling, Scientific Computing, Optimization,  Optimal Control
Other Publications here.

Other activities:

More Details on Research:
I have worked on mathematical and computational models of rigid body mechanics (including friction and collisions). This involves pure mathematics (measure differential inclusions, J.J. Moreau, 1983) as well as numerical analysis and more classical applied mathematics. This relates to my PhD work (1990) on discontinuous ODE's and differential inclusions. One of the major outstanding issues in the area is the resolution of Painlevé's paradoxes. In 1998 I proved 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 was the first general result on rigid body dynamics to do so. This work led me to investigations and results about measure differential inclusions and equations. More details here. Going beyond rigid body models, I worked on elastic body mechanics with impact. I studied a number of different models of elastic and visco-elastic bodies in impact over 2005-2009; some of this work was done with my former students T. Wendt and Jeongho Ahn. The most recent results published in 2014 were on non-uniqueness of solutions of a family of visco-elastic bodies in impact.
Differential Variational Inequalities (DVIs) are a means of modeling dynamical systems which have hard constraints or limits.  These extend the idea of differential equation, are closely related to differential inclusions, and are useful for modeling a wide variety of systems arising in mechanics, biology, economics, and engineering. I have explored questions of when solutions exist (with J.-S. Pang), and when solutions are also unique.

Machine learning and data mining are important topics with the vast amount of data that is being collected through many many means. I am interested in efficient and useful algorithms for data mining and machine learning. In 2018, with  S. Oliveira and  J. Hajewski  I have developed an algorithm for efficiently solving Support Vector Machine optimization problems with an l1 penalty term. We are designing parallel algorithms for this type of classification.  I have also worked with P. Jorgensen on the approximation theory of sums of ridge functions.

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.  In 2006 I have worked on optimal control problems where the dynamics are discontinuous with M. Anitescu.  In these systems the adjoint variables (essentially Lagrange multipliers) satisfy the usual differential equations, except that at times they have jumps.

In 2017, I developed an algorithm with C. Stiegler for minimizing quadratic functions over permutations. This was used to improve an algorithm ELMVIS+ which used Extreme Learning Machines for visualizing complex data sets.
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. I have worked on a variety of problems in scientific computing including: 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. However, this means that there are more geometric tasks that have to be performed in the basic meshfree methods. I developed some computational geometry algorithms using quad-trees for these tasks with S. Oliveira and students. In 2013 I also worked on the stability of the numerical solution of the Laplace-Beltrami operation using meshless methods on curved manifolds with A. Paiva Neto, L.G. Nonato and others.

In 2014, I analyzed the stability of a numerical method for finite difference problems arising from time-dependent diffusion problems with C. Oishi and J. Cuminato. During 2017 I consulted with a biomedical lab to help understand the mechanical behavior of mucus. I have a passion for mathematical modeling of biological and medical phenomena, and plan to continue this work in the future.

Current PhD students

Former PhD students (reverse chronological order):

Places I have worked
1998-present      Mathematics Dept., University of Iowa
1996-1997          Mathematics Dept., Virginia Polytechnic Institute and State University
1995-1996          Mathematics Dept., Texas A&M University 

1991-1994          Australian National University, School of Mathematical Sciences and  advanced computation group
1990-1991          Mathematics Department, University of Queensland,Australia

Some Pictures and Trips:

Here is a picture from the DaVinci/SICONOS meeting at Grenoble in 2005 showing the DaVinci group.

Here is a picture of some of us relaxing after the International Conference on Complementarity Problems (ICCP) in Berlin in 2014.
(Thanks to Uday Shanbhag -- not shown -- for the picture.)

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
Here is the list of the attendees.

My Erdös number is less or equal to 3.  Here is the proof.

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