Fast Multidimensional Interpolations

We have developed a high-performance, flexible scheme for interpolating multi-dimensional data. The technique can reproduce exactly the results obtained, for example, from Ordinary Kriging and related techniques in 3D, or from Thin-Plate Splines (Briggs' minimum-cuwature algorithm) in 2D. Moreover, com- pared to traditional implementations of these algorithms, our method enjoys large computational-cost savings. The new approach produces an interpolation that obeys a Partial Differential Equation (PDE). The PDE may arise from physically based arguments, but its form can vary widely. It might be specified only implicitly (as through a Model Variogram), or be nonlinear (although a performance penalty could then apply). While a formal equivalence between Kriging and Splines has been known for some time (Matheron, 1980), the present derivation, from radial basis functions, further illuminates this connection. Thus, for example, we can make explicit the PDEs that underlie some of the Model Variograms most often used in Geostatistics. Besides its practical utility, the work thereby ac- quires a theoretical interest.