LLNL Basic Energy Sciences Program in numerical methods for interpolation and approximation FY - 1984

The use of approximations in solving real-world problems is probably inevitable. Mathematical models approximate physical reality; computational models approximate the underlying mathematics; and the computer itself only approximates the necessary arithmetic calculations. The numerical methods described in this report involve approximating smooth functions. These approximations are usually found in computational models and take into account the underlying mathematics and the computational realities of working with digital computers. As computer power increases, computational models tend to grow in complexity and size. Consequently, new demands are placed on the approximations to ensure that they faithfully reproduce the properties of the underlying functions. Typical examples of such demands include preserving monotonicity, convexity, and smoothness. Also typical is the need to process larger and larger data sets. Thus, speed and storage constraints never seem to disappear. The goal of this research is to develop sophisticated interpolation/approximation algorithms to meet these new challenges. Some recent results are described.