# Research

• #### Machine learning

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 $\boldsymbol\left\{x\right\}\mapsto f\left(\boldsymbol\left\{w\right\}^T\boldsymbol\left\{x\right\}\right)$$\bm\left\{x\right\}\mapsto f\left(\bm\left\{w\right\}^T\bm\left\{x\right\}\right)$ where f is any scalar function of a single variable); accepted at SIMODS. The focus of our paper is on non-asymptotic estimates of how the number of ridge functions needed increases as the dimension increases. High dimensional approximations need at least $O\left(d^2\right)$ ridge functions to enable reasonable approximations in general. Optimization work includes work on $\ell^1$ 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 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. 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 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.
• #### Other Research

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.