proposal number: 0306612
This project centers on numerical investigation of multi-dimensional functions. The first application is to integration of these functions. At present digital nets do not exploit much smoothness in the integrand. New constructions of randomized digital nets are developed to exploit increased smoothness in the integrand. While digital nets and integration lattices have been randomized, there is little or no work on randomizing Smolyak rules. This project develops such randomized Smolyak rules. The quality of the resulting techniques is studied theoretically and in computational investigations. The second application is approximation and interpretation of multi-dimensional functions by quasi-regression. The primary application of interest is to determine the relative importance of various inputs and their interactions to provide diagnostics and interpretation of prediction functions fit in machine learning applications.
A multidimensional function is one that depends on more than one input and often on many inputs. In many applications one wants to average such a function over all possible inputs. In finance, the function might be the payoff of a security, the inputs are random future prices, and the average determines a fair price. In computer graphics, images are often generated by averaging the contributions of many sampled paths of light. Recently, methods that mix deterministic strategic sampling with random sampling have been successfully applied. This project develops some new methods of this type and investigates how well they work. The second application is to determine numerically which of perhaps dozens of input variables to a function is most important and whether the variables act individually or synergistically.
