Mathematical modeling and optimized algorithms

Creative problem solving and breakthrough innovation

Dr. Gregory Beylkin is a co-founder and Managing Principal at TNG. He received his M.S. in mathematics from Leningrad (Saint Petersburg) University and his Ph.D. degree in mathematics from New York University (NYU), and was an Associate Research Scientist there in 1983.

Dr. Lucas Monzon is a co-founder and Managing Principal at TNG. He received his bachelor degree in Mathematics from the University of Buenos Aires, Argentina and the M. S. and Ph.D. degrees in Mathematics from Yale University.

All analog devices introduce phase distortion to the signals that pass through them. As of today, most digital signal processing systems that operate on these signals do not correct for the phase distortion introduced by analog hardware since it is remarkably difficult to use standard filter design algorithms to automatically and reliably construct compensating digital filters. Consequently, hardware designers work hard to minimize the distortion caused by their devices, effectively sacrificing performance elsewhere.

TNG has developed a new technology for robust automatic design of accurate and efficient filters which, in particular, can be applied to phase compensation filters. The entire design process is performed by robust and non-iterative algorithms so that the resulting filter achieves the user-selected accuracy and can be embedded into any device without supervision from an expert.

A ubiquitous and critically important problem in signal processing is the digital resampling of signals. For example, resampling is needed for signals distorted by Doppler effects due to emitter and receiver motion. Although the problem of digital resampling has a long history and it is straightforward to convert between two sample rates that are related by a simple rational factor, existing algorithms make it very challenging to resample an input signal at an arbitrary time-varying sampling rate. Existing solutions to this problem generally use a polyphase filter bank to produce an approximate solution, but this approach sacrifices accuracy and bandwidth in order to achieve a reasonable computational cost. TNG has developed and implemented a new fast algorithm that completely solves this problem yielding high accuracy and preserving almost the entire bandwidth, with a computational cost that is superior to traditional polyphase solutions.

Spatial data is usually collected on irregular grids which are often undersampled in some locations. For example, due to orbital constraints, satellite data are generally collected on irregular grids which contain large missing regions, or cloud cover may prevent obtaining information of different sections of the target area; dealing with such data is a major processing challenge for both defense and scientific missions. We have developed a multistage recursive algorithm that constructs a functional representation of this type of data by adaptively refining the representation. At each stage, we match the band limit of the interpolating function to the mesh granularity, yielding a stable algorithm. The resulting interpolating function is available in a highly efficient functional form, so it can be rapidly and accurately evaluated at arbitrary output locations. TNG is successfully using this new algorithm for sea surface temperature data processing and is currently developing additional novel algorithms to perform a variety of tasks for that type of data.

Dr. Gregory Beylkin is a co-founder and Managing Principal at TNG. He received the M.S. in mathematics from Leningrad (Saint Petersburg) University and the Ph.D. degree in mathematics from New York University (NYU) and was an Associate Research Scientist there in 1983. From 1983 to 1991, Greg was a Member of the Professional Staff at Schlumberger-Doll Research, Ridgefield, CT. He worked on seismic imaging and developed novel approaches to linearized inverse problems. In 1988-1989 during his sabbatical at Yale University he participated in developing key initial applications of wavelet analysis which are widely used. Since 1991 Greg has been a Professor with the Department of Applied Mathematics, University of Colorado at Boulder (UC). His research activity includes harmonic and numerical analysis, wave propagation, inverse problems, quantum chemistry, gravity modeling, and signal processing. He has published 100+ papers, holds 1 patent and many of his publications have been highly cited. Greg is a Fellow of the American Mathematical Society, the Society for Industrial and Applied Mathematics and in 2011 received the UC Faculty Assembly Award for Excellence in Research, Scholarly and Creative Work. He is a member of the Advisory Editorial Board of Applied & Computational Harmonic Analysis and of the Editorial Board of SIAM Journal on Matrix Analysis and Applications.

Dr. Lucas Monzon is a co-founder and Managing Principal at TNG. He received the bachelor degree in Mathematics from the University of Buenos Aires, Argentina and the M. S. and Ph.D. degrees in Mathematics from Yale University. Lucas has many years of experience working as a private consultant, has been Principal Investigator of several National Science Foundation grants and had participated in several projects for the Department of Energy and the Department of Defense. He has published 21 refereed papers in highly recognized professional journals; several of these papers have been highly cited. He also holds a patent in techniques for seismic survey data and a pending patent application on an inversion method for Nuclear Magnetic Resonance measuring tools. Since 2001, together with Greg Beylkin, he has been developing a new theory and key algorithms for nonlinear approximation of functions which has found numerous applications. Lucas has a part time position at the University of Colorado at Boulder. His research activity includes harmonic and numerical analysis, nonlinear algorithms, inverse problems, and signal processing.