Stable, Efficient, Adaptive Algorithms for Approximation and Integration

Illinois Institute Of Technology, Chicago IL

Investigators

Abstract

Computational methods allow the simulation of experiments that are too costly, too dangerous, or otherwise infeasible to perform in physical situations. Examples include evaluating the safety and efficiency of designs for nuclear reactors, predicting the frequency of breakdowns in power grids, and assessing the risks and rewards of financial investments. This research project will develop new ways of computing surrogate models for time-consuming simulations. These new surrogate models will be quicker to compute, avoid catastrophic computer error, and more faithfully represent the processes that they are designed to model. More clever ways of exploring future scenarios will be developed to decrease the computational time required to find the average, or the worst possible, behavior of complex systems such as those mentioned. The new algorithms arising from this research will be made publicly available for other researchers and practitioners. Undergraduate and graduate students involved in developing these algorithms will be better prepared for scientific careers. Function approximation and integration are two fundamental problems in computational mathematics. This research project will construct algorithms for function approximation and integration that are computationally stable, avoiding catastrophic round-off error. These algorithms will be adaptive, determining the algorithm parameters based on function data to meet the user-specified error tolerance with rigorous justification. The algorithms will also be asymptotically efficient, requiring essentially the same computational effort as the best possible algorithms. Function approximation algorithms will be based on the Hilbert-Schmidt SVD decomposition that the investigators have developed for kernel methods. Integration algorithms will be developed that adaptively determine the sample size required. The investigators will continue to mentor students as well as post-doctoral scholars and will establish the usefulness of the algorithms under development through collaborations with domain specialists.