Optimization problems in the presence of uncertainty

by (Sarah Lynn), 1981- Grove

GROVE, SARAH L. Optimization Problems in the Presence of Uncertainty. (Under the direction of Professor H.T. Banks). We consider optimization problems in the presence of uncertainty for three different scientific examples. The first optimization problem addressed is in the area of parameter estimation. We review the asymptotic theory for standard errors in classical ordinary least squares (OLS) inverse or parameter estimation problems involving general nonlinear dynamical systems where sensitivity matrices can be used to compute the asymptotic covariance matrices. We discuss possible pitfalls in computing standard errors in regions of low parameter sensitivity and/or near a steady state solution of the underlying dynamical system. Next we consider electromagnetic evasion-interrogation games where the evader can use ferroelectric material coatings to avoid detection while the interrogator can manipulate the interrogating frequencies and angles of incidence to enhance detection. Each player in this two-player game wishes to change the amount of reflected signal created in a way that will benefit them the most. Thus both players will attempt to optimize their chances of either remaining undetected or detecting their opponent. With the introduction of uncertainty, the resulting game is carried out over spaces of probability measures. Finally, a one player dynamical game is formulated. The premise here is for the evader to manipulate the ferroelectric material coatings to avoid detection based upon updated information about the frequencies being sent out. The uncertainty is found in the frequencies that the interrogator will employ. We incorporate both drift and diffusion into this optimization problem. Optimization Problems in the Presence of Uncertainty by Sarah L. Grove A dissertation submitted to the Graduate Faculty of North Carolina State University in partial fulfillment of the requirements for the Degree of Doctor of Philosophy Mathematics Raleigh, North Carolina 2007