Research
Recent work has focused on aspects of machine learning, particularly
those involving optimization and approximation theory. I have a
paper with Palle Jorgensen on lower bounds on approximation errors
for sums of ridge function (ridge functions have the form
$\backslash boldsymbol\{x\}\backslash mapsto\; f(\backslash boldsymbol\{w\}^T\backslash boldsymbol\{x\})$$\backslash bm\{x\}\backslash mapsto\; f(\backslash bm\{w\}^T\backslash bm\{x\})$ where f
is any scalar function of a single variable); accepted at SIMODS.
The focus of our paper is on nonasymptotic estimates of how the
number of ridge functions needed increases as the dimension
increases. High dimensional approximations need at least $O(d^2)$
ridge functions to enable reasonable approximations in general.
Optimization work includes work on $\backslash ell^1$
penalized optimization problems for large data sets, and Newton
methods for large data sets that improve in performance (in terms of
number of iterations) as the data set becomes larger. This includes
algorithms for lasso for large data sets, support vector machine
algorithms whose performance improves (or does not degrade) as the
number of data items increases.
Nonconvex optimization for machine learning tasks has also been a
focus: 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.

Mechanical impact & Differential Variational
Inequalities
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 1dimensional
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
viscoelastic bodies in impact over 20052009; some of this work was
done with my former students T. Wendt and Jeongho Ahn. The most
recent results published in 2014 were on nonuniqueness of solutions
of a family of viscoelastic 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. Work since then has included
numerical methods for electrical circuits with idealized diodes.
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 NPhard, 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.
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 quadtrees for these
tasks with S. Oliveira and students. In 2013 I also worked on the
stability of the numerical solution of the LaplaceBeltrami
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 timedependent 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.