Shape Preserving Interpolation. Revision 1

摘要

Fast, accurate interpolation algorithms are needed in virtually all areas of high speed scientific computing. As computer power has grown, physical, mathematical and computational models have become more complicated in an attempt to achieve more realistic simulations of the underlying physical processes. These changes have influenced the trends in developing new and more sophisticated interpolation methods. In the past an important criterion used to select an interpolation algorithm was the accuracy of a method as measured by the rate of convergence of the interpolant as the mesh size is decreased to zero. However, in most practical problems one has little or no control over the number and/or location of the data points, and the mesh never tends to zero. Instead, the user demands that the interpolant provide an accurate approximation to ''physical reality'', or at least to his/her perception of that reality. In attempting to increase fidelity to the underlying physical processes, two phrases have come into vogue when describing interpolation methods: ''visually pleasing'' and ''shape preserving''. Visually pleasing means the end result must look ''right'' to the user. While this is a highly desirable goal, it is subjective and has not yet been characterized mathematically. In contrast, shape preserving refers to the preservation of one or more mathematical properties (called shape characteristics) during the interpolation process. These shape characteristics often represent physical properties of the system being modeled. This paper is about shape preserving interpolation methods as applied to solving real-world scientific problems. (ERA citation 10:050700)