Abstract

The goal of my Masters thesis was to develop a theoretical foundation for the analysis of the scalable smoothing algorithm which I have developed with the assistance of my supervisors. This theoretical work applies in any dimension but the smoother is only practical up to about 8 dimensions. In future work we intend to improve the smoother using the theory of adaptive, sparse grids. Currently the smoother minimizes a functional on a regular rectangular grid. The functional consists of the sum of a smoothing seminorm term and a constraining least squares term. The minimal smoother is known as a basis function smoother and consists of a linear combination of basis functions translated by the data points plus a polynomial. The basis function is defined directly in terms of the components of the seminorm.

In fact, the function spaces used for smoothing are the same as those used for the minimal interpolation problem, where the functional only has a seminorm term. My thesis only studies the minimal seminorm interpolant problem.

In the seventies Duchon developed interpolation theory using seminorms constructed from a positive weight function. These weight functions generated thin plate spline basis functions. For these interpolants he was also able to obtain rates of pointwise convergence as the data becomes denser.

In the 90s Will Light and Henry Wayne extended the class of weight functions used to form the seminorm.

Unfortunately, our smoothing algorithms involve using tensor product hat basis functions, and these cannot be generated using the weight functions of Light and Wayne. However, we have been able to devise an extension of their class of weight functions to overcome this limitation and so allow the theory of Light and Wayne to be applied. Pointwise convergence results for the hat basis function interpolant have thereby been obtained.