Scholarship

Industrial Experience

I worked at Blount Industries in Portland, Oregon from 1984 to 1990. I did a lot of statistics, signal processing/time series analysis, and mathematical modeling, including developing a model for a running chain saw!

Collaborators and Recent Papers

Some recent papers are Much of my work has been with Michael Vogelius at Rutgers and Lester Caudill at the University of Richmond.

I've begun some collaborations with Chris Earls at Cornell University and John Kymissis and his post-docs (and former post-docs) at Columbia University.

I've also written a nice series of papers suitable for undergraduates, with Tanya Leise at Amherst College. They have appeared in (or been submitted to) the Education Section of SIAM Review.

Other coauthors include REU students Melissa Vellela (now Melissa Nivala), Ron Ogborne, Nic Trainor, Rachel Krieger, Janine Haugh (now a full-fledged math professor at UNC Asheville), David McCune, and Professor Valdis Liepa (EE, University of Michigan),

Here is a complete list of papers, with abstracts.

Impedance Imaging

I've done quite a bit of work on the inverse problem of finding small defects (cracks, inclusions) in a conductive material from input current/boundary voltage data pairs. Some of it is theoretical [1,3], a bit experimental [6], most a blend of theory and computation [2,4,6,18,19,21,23]. The last couple papers have very nice and fast algorithms, obtained by using a ``small volume asymptotic expansion'' to represent the effect of a crack on boundary data. Papers [19,21,23] are with summer REU students.

Thermal Imaging for Nondestructive Testing

I've written a number of papers [5,7,8,9,10,14,15,18,20,24], many with Lester Caudill, on various aspects of the problem of detecting corrosion of other damage to some inaccessible portion of the boundary of an object using thermal methods (heat goes in, watch surface temperature evolve in time, infer the damage profile). Again, a blend of theory, experimental data, and computation.

Partial Differential Equations

Michael Vogelius and I have written a paper [13] on the application of homogenization theory to electrical conductors with periodic arrays of small cracks, and some papers [16,26,27] on the analysis of elliptic and parabolic PDE's with nonlinear "blow-up" boundary conditions.

Image Processing and Wavelets

Allen Broughton and I wrote a book, Discrete Fourier Analysis and Wavelets: Applications to Signal and Image Processing, published by Wiley in November 2008. You should buy a copy!

Educational

Tanya Leise and I have written papers aimed at undergraduates and/or nonspecialists, explaining Google's PageRank algorithm [22], the idea behind one approach to cloaking and invisibility [25], and a paper on the hot topic of compressed sensing [29]. I also wrote a paper [28] on tuned-mass dampers for PRIMUS, celebrating Brian Winkel's twenty years of service to that journal (in addition to starting it!)

Here's 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'm going to submit this to the Monthly relatively soon.

Other Writings

Other Stuff