Statistical Integration and Approximation

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Owen 9704495 This research investigates the use of statistical sampling ideas in high dimensional numerical problems, particularly numerical integration and approximation. Statistical thinking enters the picture because in high dimensional problems one can get at most a sparse sample of the input space. Surprising benefits can accrue: for example it is known that randomizing a low discrepancy sequence can increase it's accuracy. For the case of numerical integration, this project looks at adaptive importance sampling techniques in which the function values themselves are used to adapt the sampling towards the most relevant region of the input space, while maintaining a data based estimate of integral accuracy. In the case of numerical approximation, this project will apply recent progress in hybrid numerical integration methods to finding ways of building sample based approximations of functions with data determined error estimates. Numerical integration is a fundamental computational task with applications in physics, chemistry, finance, statistics, and numerical approximation. Numerical approximation is also fundamental, and the advent of powerful and flexible computer simulators of scientific and engineering phenomena can only increase the demand for computer experiment techniques based on approximation. At present these computer models are widely used in the design of computer chips, automobile parts, airplanes, and in global climate modeling. These functions may take hours or even days of computer time and so finding fast approximations (and estimating their accuracy) is important. Better exploration of these functions, call it function mining, can lead to faster introduction and greater reliability for new products. It can also be used to thoroughly explore the predictions of models for environmental change and storage of nuclear wastes.