9705380 Jonathan Goodman ABSTRACT OF PROPOSAL BY JONATHAN GOODMAN APPLIED ANALYSIS AND COMPUTATION This proposal covers several areas of applied mathematics, applied analysis, and computational science. One area is the mathematical analysis of nonlinear waves and fronts, particularly multidimensional shocks. Another area is control of systems governed by hyperbolic partial differential equations (wave propagation equations) and discrete ``lumped parameter'' approximations to them that would be used in numerical computation of optimal controls. Two main computational areas are: 1: anisotropic adaptive refinement methods for multidimensional approximation and finite element computation, and 2: Monte Carlo methods for computing quantum mechanical properties of systems of interacting electrons. Several other areas, including the research of graduate students under my supervision are discussed. There are several projects described here, most involving collaborations with colleagues, postdoctoral trainees, or graduate students. The project on control of systems governed by hyperbolic differential equations is about methods for removing acoustic noise from structures. This has applications in aircraft and submarine technology and in other places. It fits in with a larger effort to design "smart materials" and "smart structures". Modern sensors, actuators, and computers are fast enough to react to individual sound waves. The mathematical problem is to design good computational "control strategies" that use this ability effectively. The existing mathematical theory of control was developed with smaller, simpler systems in mind and does not apply directly to control of systems where acoustic waves (sound waves) are propagating. We hope that our theory of control will apply to such problems. A more computational project is the attempt to compute the "electronic structure" of molecules from quantum mechanics. The equation to be solved (the Schrodin ger equation) has been known since 1926. Still in 1997, there is no reliable way to solve the Schrodinger equation for systems involving more than one electron (an oxygen atom has 8). The ability to do this would have enormous scientific and technological impact, with applications ranging from superconductivity to drug design.
