Fast and Accurate High Dimensional Function Approximation

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This research will develop computational methods to approximate functions of many variables based on function values which may be contaminated by noise. The focus will be on kernel-based methods, such as radial basis function methods, smoothing splines, regression splines and moving least squares methods. Non-isotropic kernels, space filling designs, and new convergence proofs will be used to make these methods practical in higher dimensions. Fast transform and iterative algorithms will be developed to reduce the computational burden of kernel methods. Both computational and statistical approaches will be brought to bear on this problem. The theoretical development will provide practitioners insight into how the quality of the designs (sample points) affects the accuracy of approximation. The algorithms developed will be published as software packages for widespread use. In this information age there is an abundance of data generated from instrumental measurements and computer simulations. Strategic corporate planning and profitable product design both rely on accurate mathematical models to describe this data. As the numbers of observations and variables increase, existing methods for computing the best models fail to capture the complex relationships and fail to compute an answer in a reasonable amount of time. This research will develop a new generation of computational methods for modeling data that overcome these drawbacks. These new methods will make our manufacturing industry more competitive by facilitating more rapid prototyping of products, and they will enable our service industry to assess and respond more quickly to changing markets.