My research interests are partial differential equations (PDE's), numerical
methods for PDE's, and integral equations. I am especially interested in inverse problems involving
PDE's. Most of my work has been in inverse problems for elliptic and parabolic
PDE's related to electrical and thermal imaging for nondestructive testing.
I've also dabbled a bit in homogenization techniques for PDE's, and also
in singularities in solutions to PDE's with nonlinear boundary conditions.
I also did a lot of statistics and signal processing/time series analysis when
I worked in industry.
I usually conspire with Michael Vogelius
in the mathematics department at Rutgers University
and with Lester
Caudill in the math department at the University of Richmond. I've also worked with
many undergraduate mathematicians in our
REU summer program , which I directed until 2009. I'll be mentoring research groups again in 2011 and 2012.
Here are some recent publications. Email me if you'd like a copy of anything not available as a pdf below.
Impedance Imaging, Inverse Problems, and Harry Potter's Cloak, May 2009,
to appear in the Education section of SIAM Review.
This is a paper I wrote with Tanya Leise at Amherst College.
Aimed at undergraduates, this paper gives a very accessible but rigorous presentation of the mathematical idea behind cloaking, and requires only a basic understanding of linear algebra, vector calculus, and some Fourier series. It also includes exercises and ideas for undergraduate projects.
The $25,000,000,000 Eigenvector: The Linear Algebra Behind Google,
in the Education section of the August 2006 issue of SIAM Review.
Another paper I wrote with Tanya Leise.
It gives an undergraduate-oriented explanation of the beautiful and simple linear algebra that
lies behind an important facet of Google's PageRank algorithm. The page has some demo
code.
"Reconstruction of Cracks with Unknown Transmission Condition from Boundary
Data", in Inverse Problems, February 2005, (written with two undergraduate REU students, Ron Ogborne and Melissa Vellela.)
"Reconstruction of an Unknown Boundary Portion from Cauchy Data for the Heat Equation in N Dimensions", written with Lester Caudill, in Inverse Problems, February 2005.
"Imaging of Multiple Linear Cracks Using Impedance Data", in the Journal
of Computational and Applied Math, March 2007, Volume 200 (1), p. 388-407.
(written with two undergraduate REU students, Rachel Krieger and Nic Trainor).
"Fast Imaging of Partially Conductive Linear Cracks Using Impedance Data", in
Inverse Problems 22 (2006), p 1337-1358,
written with two undergraduate REU students from summer 2004, Janine Haugh and David McCune.
The REU Inverse Problems page has
links to all of the tech reports produced by the REU inverse problems students.
A preprint of an amusing result I "rediscovered" a few summers ago: Elementary
Inversion of the Laplace Transform. It's very beautiful and
simple, yet it seems to be practically unknown---no standard book on the
Laplace Transform has it and no one I asked about it had ever heard of the
result. Ironically and tragically, I finally discovered that the
result was proved by Emil Post in 1930. I found this out quite by
accident while randomly browsing through a Dover book in a Border's bookstore!
I'll probably submit this to a journal oriented toward undergraduates,
since it deserves wider attention.
Information on our
REU program (Inverse Problems and Computational Number Theory) for summer 2009.
You should apply---in what other REU program do you
get to drive a tank?