Ari Stern

Postdoctoral Researcher, UCSD Department of Mathematics

I am currently a postdoctoral scholar in the UCSD Department of Mathematics. In addition to being a member of Michael Holst's group in the Center for Computational Mathematics, I am also a fellow of the Center for Theoretical Biological Physics, where I am affiliated with the McCammon research group.

Before coming to UCSD, I received a Ph.D. from the Department of Applied and Computational Mathematics at Caltech. In what seems like the ancient past, I also got my B.A. in Mathematics and an M.A. in Mathematics of Finance from Columbia University.

Research Interests

My research is in the area of discrete geometric dynamics: developing numerical methods that preserve the essential geometric structure of the physical systems they aim to simulate. The motivating idea behind this work is that, if our discretizations respect certain structures (e.g., Lagrangian variational principles, symplectic structures, symmetries, conservation laws), then we can develop numerical integrators with better global behavior—even at lower computational cost—and with arbitrarily good accuracy.

I am particularly interested in discrete field theories, building on some of the recent advances in discrete mechanics and variational integrators for ODEs, and using “discrete differential geometry” to extend these tools into the realm of numerical PDEs. This work also draws on a wide spectrum of mathematical subjects including differential geometry, algebraic topology, Lie group theory, dynamical systems, mathematical physics, and numerical analysis.

Current applications being explored include:

computational electromagnetics,
numerical general relativity,
constrained, highly oscillatory, and multiscale dynamical systems,
simulation of mechanical systems in biophysics.

Papers Available for Download

: Stern, A. (2010), Discrete Hamilton-Pontryagin mechanics and generating functions on Lie groupoids. J. Symplectic Geom. To appear. [ bib | arXiv ]
: Stern, A., and E. Grinspun (2009), Implicit-explicit variational integration of highly oscillatory problems. Multiscale Model. Simul., 7 (4), 1779-1794. [ bib | DOI | arXiv ]
: Stern, A. (2009), Geometric discretization of Lagrangian mechanics and field theories. Ph.D. thesis, California Institute of Technology. [ bib | http ]
: Stern, A., Y. Tong, M. Desbrun, and J. E. Marsden (2009), Geometric computational electrodynamics with variational integrators and discrete differential forms. Preprint. [ bib | arXiv ]
: Stern, A., Y. Tong, M. Desbrun, and J. E. Marsden (2008), Variational integrators for Maxwell's equations with sources. PIERS Online, 4 (7), 711-715. Previously appeared in PIERS Proceedings, 443-447, July 2-6, Cambridge, USA 2008. [ bib | DOI | arXiv ]
: Stern, A., and M. Desbrun (2006), Discrete geometric mechanics for variational time integrators. In SIGGRAPH '06: ACM SIGGRAPH 2006 Courses, pages 75-80, ACM Press, New York. [ bib | DOI ]

Talks Available for Download

Variational integrators and symplectic geometry [pdf], invited lecture given at the Fourth International Young Researchers' Workshop on Geometry, Mechanics and Control in Ghent, Belgium, January 12, 2010.

Implicit-explicit variational integration of highly oscillatory problems [pdf], Department of Mathematics Colloquium, University of Notre Dame, August 28, 2009.

Implicit-explicit variational integration of highly oscillatory problems [pdf], given at the Fifth Annual Structured Integrators Workshop hosted by Caltech, May 7, 2009.

Geometric Aspects of ODEs and PDEs [pdf], given at the Computational Science, Mathematics, and Engineering Seminar at UCSD, April 9, 2009.

Discrete Differential Forms, Gauge Theory, and Regge Calculus [pdf], given at the Center for Mathematical Sciences seminar at the University of Cambridge, November 21, 2008.

Variational Integrators for Maxwell's Equations with Sources [pdf], given at the Progress in Electromagnetics Research Symposium (PIERS) in Cambridge, MA, July 3, 2008.

Geometric Numerical Integration of Lagrangian Dynamics [pdf], given at the Center for Computational Mathematics Seminar at UCSD, June 17, 2008.

Variational Implicit-Explicit Integrators: Multiple Time Scales without Resonance Instability [pdf], poster presented at the Fourth Annual Structured Integrators Workshop at Stanford, April 24–25, 2008.

Symmetry and Simulation: How Geometry Affects Scientific Computing, from the Solar System to Your Microwave Oven. Part of the Everhart Lecture Series at Caltech, April 2, 2008. Streaming webcast available at Caltech Today Theater.

Discrete Differential Forms for Numerical Relativity [pdf], given at the 24th Pacific Coast Gravity Meeting at UCSB, March 22, 2008.

Geometric Methods in Computational Electromagnetics [pdf], given at the USC Seminar Series, October 11, 2007.

Computational E&M with Variational Integrators and Discrete Differential Forms [pdf], given at the workshop Geometric Mechanics: Continuous and discrete, finite and infinite dimensional, Banff International Research Station, August 13, 2007.

DEC and AVIs for Computational Electromagnetism [pdf], given at the Third Annual Structured Integrators Workshop hosted by USC, April 30, 2007.

Variational Integrators for Problems on Multiple Time Scales [pdf], poster from DyNARUM 2007 workshop, January 23, 2007.

I participated in the course Discrete Differential Geometry: An Applied Introduction at the SIGGRAPH 2006 conference on July 30, 2006. My lecture slides [pdf] and movies [zip], are both available from the Discrete Differential Geometry Forum (which also includes resources from the past several years' courses).

Computational Resources: LaTeX on Mac OS X

MacTeX

The MacTeX distribution is probably the best and easiest way to use TeX on a Mac. In addition to all the standard TeX tools, packages, and fonts from the TeX Live distribution, MacTeX also includes the excellent front-end programs TeXShop (TeX editor and viewer), LaTeXiT (create a snippet of LaTeX text as a small PDF, for easy embedding in non-LaTeX documents), BibDesk (manage BibTeX bibliographies), and Excalibur (spell-checker that intelligently skips LaTeX commands and checks only the actual text).

Podcast: Using LaTeXiT with Keynote and Illustrator

LaTeX is great for typesetting large documents, or those with lots of mathematical content. But what if you don't want to use LaTeX for your whole document? What if you just want to insert a snippet of LaTeX text into your Apple Keynote presentation, or Adobe Illustrator figure (e.g., adding text labels to a mathematical diagram)? In this demo, I show how to do this using the LaTeXiT utility. These movies are based on a presentation I gave at the CalTeX Mac Fest and at TUG 2007. (Unfortunately, my TUG demo was plagued by technical difficulties, so if you attended, these movies will show you what it should have looked like!)

Note: when using LaTeXiT with Illustrator, be sure LaTeXiT is set up to export the format "PDF with outlined fonts" — otherwise Illustrator won't be able to import the correct LaTeX fonts. This option can be found in the "General" tab of the LaTeXiT Preferences window.

FasTeX

FasTeX is a system of abbreviations for LaTeX commands: for example, beq becomes \begin{equation}; lel becomes \left\langle; lld ("left langle double") becomes \left\langle\!\left\langle. These abbreviations are automatically expanded as you type, without the need to input any special commands or escape keys; effectively, it lets you type LaTeX documents in shorthand. Once learned, it can dramatically increase speed and reduce typos for people who do a lot of work in LaTeX. Although TeXShop users typically have to purchase abbreviation software like TypeIt4Me to use FasTeX, Emacs (my editor of choice) has built-in abbreviation facilities.

Emacs and LaTeX

While many Mac users like the excellent TeXShop software for working in LaTeX, I'm a big fan of the Emacs editor for just about any work I do. I use Carbon Emacs, which is a version of Emacs for Mac OS X that includes a number of useful pre-configured packages, such as the powerful AUCTeX package for LaTeX editing.

I have also integrated FasTeX into my Emacs environment, which I think is great for anyone who uses Emacs and LaTeX. If you'd like to do this, download the abbreviation file latex-abbrev.el, put the file in your Emacs load-path, and add the following lines to your .emacs file:

      (add-hook 'LaTeX-mode-hook
      (lambda ()
      (quietly-read-abbrev-file "latex-abbrev.el")
      (setq local-abbrev-table LaTeX-mode-abbrev-table)
      (abbrev-mode t)
      ))

Contact

Ari Stern
Department of Mathematics
University of California, San Diego
9500 Gilman Dr #0112
La Jolla CA 92093-0112

You can also email me at astern@math.ucsd.edu.