Application of numerical interval analysis for statistical computing in a massively parallel computing environment
Ouhong Wang · 1994
Interval analysis is a relatively new method in performing scientific computation. It uses intervals as basic computing elements and has a set of arithmetic that is different from the traditional one. The shortcomings of conventional computation that motivate the development of interval analysis are the inaccuracy caused by the finite number systems of computers and the uncertainty of various approximations. Using interval analysis we can manage to have the theoretically true result after each computing step contained in the computed interval. Although the exact location of the true value is not determined (sometimes it is even not possible), we know a pair of guaranteed lower and upper bounds of it. Using various techniques we can shrink the length of the interval, hence obtain highly accurate results;This dissertation investigates the use of interval analysis in statistical computing. We focus on two major applications: bounding computational errors and global optimization. Bounding computational errors is demonstrated by high dimensional Normal and t probability computations. Both the rounding errors from the finite nature of machine computing and the approximation errors from mathematical approximation are taken into account. We are able to compute high dimensional probabilities to high accuracy, which can not be achieved using traditional methods. For global optimization, we use the capability of interval analysis to compute the range of a function over a finite region. This can give us idea of the variation of the objective function value over that region. Within a finite region, the algorithm can throw away local optima, and if there exists more than one global optimum we are able to locate them all. This application is demonstrated by several examples in Statistics and Probability;;The use of interval analysis is at the expense of running time being large. In order to make it practical to use, we develop all the algorithms on a massively parallel machine. This way we obtain guaranteed highly accurate results within reasonable running times.