Foundations of Multivariate Functional Approximation for Scientific Data

The size and complexity of scientific data are rapidly outpacing our ability to manage, analyze, visualize, and draw conclusions from the results. One way to streamline scientific data processing is to transform the data model to one that is not only more efficient but that facilitates analytical reasoning. To this end, we are investigating modeling discrete scientific data by a functional basis representation based on a tensor product of nonuniform rational B-spline functions (NURBS). The functional model is an approximation that is more efficient to communicate, store, and analyze than the original form, while having geometric and analytic properties making it useful for further analysis without translating back to the discrete form. This paper presents four main contributions. The first is modeling in high dimensions, not just curves or surfaces but volumes and hyper-volumes. The second is an adaptive algorithm that adds knots and control points until the maximum error is below a threshold set by the user. The adaptive algorithm also works in high dimensions. The third is precise evaluation of values and high-order derivatives anywhere in the domain, again in high-dimensional space. The fourth contribution is parallelization on high-performance supercomputers. We evaluate our model using both synthetic and actual scientific datasets.