Small-Sample Statistical Condition Estimation

The research conducted over the past four years of this AFOSR grant has focused on one of the fundamental factors influencing the accuracy of floating-point computations namely, the condition or sensitivity of the particular problem being solved. Many fundamental calculations in scientific and engineering computation in general, and control engineering in particular, can be viewed as functions that map a set of input values to a set of output values. Assuming the algorithm used in the computation is numerically stable, the sensitivity of this input-output map determines bow accurately the output values can be estimated. The methods used in our research rely on the fundamental results in the (1) and (19) wherein it was shown that significant information about large linear operators (which arise naturally as the derivative of some nonlinear function of interest) can be obtained from their action on smaller-dimensional subspaces. Such information can be determined reliably and efficiently using the small-sample statistical method. In addition to further advancing our basic theory, we have explored applications of statistical techniques to a wide variety of problems in control theory and image processing. Our research has been reported in over 40 scholarly articles.