What Do You Need to Solve Problems with Many Variables?

Abstract

In many practical problems the number of independent variables is large compared to the number of data.  For example, to determine the fair price of an exotic option one must compute the average or integral of the payoff function, which may depend on hundreds of variables.  In laboratory or computer experiments one may wish to estimate or optimize the response as a function of a number of input variables or parameters.  It is normally prohibitively expensive to perform the experiment for all possible combinations of the different levels of the input variables.  The numerical solution of problems with many variables requires: i) a good design, ii) a good algorithm, and iii) an easy problem.  This talk describes recent interdisciplinary research on numerical methods for solving problems with many variables.  This research involves linear and abstract algebra, numerical analysis, probability and statistics, computational complexity, software design, parallel computing, and various application areas.