This project has two main goals. One goal is to provide rigorous mathematical foundations for a wide array of techniques developed by physicists, chemists, and other scientists over the last 10-15 years to perform computer simulations in fields ranging from theoretical chemistry and molecular physics to data mining and statistics. A common theme in these areas is the need to quickly compute approximations of functions of large and sparse matrices (such as the exponential, the square root, the logarithm, and combinations thereof). Scientists have had some success using a combination of physical intuition and heuristics, but rigorous justifications and analysis are still lacking and are sorely needed. On the other hand, computational mathematicians have until now devoted scant attention to these types of problems. The present project addresses this need. The main conceptual tool is a theory of localization, in the form of decay bounds, that the PI has been developing in recent years.
Another goal is to construct better (i.e., faster and more accurate) algorithms to compute the desired approximations. The PI will devote considerable effort to this objective, in particular using various types of polynomial approximations (interpolation, Chebyshev, Faber, etc). The resulting software will be distributed to interested parties.
The ultimate goal of research in computational mathematics is to provide scientists and engineers the algorithmic and software tools needed for the solution of challenging scientific and technical problems of increasing size and complexity. The competitiveness of American science and technology greatly benefits from (and to a large extent depends on) the creation of innovative computational methods and software and from the continuous improvement of existing techniques. Progress in the solution of the problems targeted by the PI will have a positive impact on science and engineering by enabling faster and more detailed computer simulations. In addition, several graduate students (and possibly a few undergraduates) will be impacted by this research either through direct involvement, or through the positive effects this research will have on the PI's teaching of computational and applied mathematics courses at Emory University.
