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!
I also spent the 2014-15 academic year at Crane Payment Innovations in Malvern PA, working on applied optics and machine learning problems.
Collaborators and Recent Papers
Some recent articles are
Much of my work has been with Michael Vogelius at
Rutgers and Lester Caudill at the University of Richmond.
- Bryan, K., and Leise, T., "Cloaking," in the Princeton Companion to Applied Mathematics, September 2015.
- Walter D., Bryan K., Stephens J., Bullmaster C., and Chakravarthy V., Localization of RF Emitters using Compressed Sensing with Multiple Cooperative Sensors, proceedings
of NAECON 2012, Dayton OH, July 25-27 2012.
- Bryan, K., Zhang, J., Pervez, N., Cox, M., Jia, X., and Kymissis, I., Inexpensive photonic crystal spectrometer for colorimetric sensing, Optics Express, Vol. 21, Issue 4, pp. 4411-4423 (2013),
- Making Do With Less: An Introduction To Compressed Sensing with
Tanya Leise at Amherst College, in SIAM Review's Education Section, Vol 55, No 3, 2013.
- Transient Behavior of Solutions to a Class of Nonlinear Boundary Value Problems, with
Michael Vogelius ,
in the Quarterly of Applied Math 69(2), June 2011, p. 261-290.
- Precise Bounds for Finite Time Blow-up of Solutions to Very General One Space-Dimensional Nonlinear
Neumann Problems, with Michael Vogelius ,
in the Quarterly of Applied Math 69(1), March 2011, p. 57-78.
- A Tale of Two Masses, PRIMUS 21(2), February 2011, p. 149-162.
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.
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 , 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  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!
Tanya Leise and I have written papers aimed at undergraduates and/or nonspecialists, explaining Google's PageRank algorithm , the idea behind one approach
to cloaking and invisibility , and a paper on the hot topic of compressed sensing . I also wrote a paper  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.