Two topics are proposed for study: (1) the variational analysis of eigenvalue functions and (2) non-parametric population analysis. In the first topic the principal investigator will apply modern techniques of nonsmooth analysis and variational geometry to study the variational properties of certain functions of the spectrum on the space of n x n complex valued matrices, e.g., the spectral abscissa and the spectral radius. In the second topic, the principal investigator intends to analyze and develop algorithmic solution techniques for the non-parametric version of the basic problem of population analysis. Specifically, the principal investigator intends to use maximum likelihood techniques to estimate the underlying probability measure associated with population variability.
Understanding the variational behavior of the spectrum of matrix valued mappings is essential to our understanding and control of discrete and continuous dynamical systems. Such systems arise in numerous practical applications ranging from the design of structures that can withstand a major earthquake to flight control for modern aircraft. The results of the principal investigator's research program are intended to make possible for the first time the derivation of optimality and/or equilibrium conditions for numerous problems associated with spectral variations that occur in numerous engineering applications. On the other hand, population analysis is the statistical methodology used to understand inter-subject variability in studies designed to analyze a phenomenon associated with a targeted population. For example, the methodology is widely used in pharmocokinetic studies since it is the key to understanding how drugs behave in humans and animals. The principal investigator intends to analyze and develop algorithmic solution techniques for the non- parametric version of this problem. The goal is to incorporate these algorithms into a software package now being developed by the Resource Facility for Population Kinetics at the University of Washington.
