Spline-Based Parameter Estimation Techniques for Two-Dimensional Convection and Diffusion Equations.

A general approximation framework based on bicubic splines is developed for estimating temporally and spatially varying parameters in two-dimensional convection and diffusion equations derived from mass transport theory. The parameter estimation problem is first cast as an abstract infinite dimensional minimization problem. Then a sequence of approximate, finite dimensional problems is defined, which yields a sequence of parameter estimates. Convergence results relating the approximate problems to the full infinite dimensional problem are presented, as well as a discussion addressing computer implementation. Finally, the technique is applied to the analysis of actual biological data from an insect dispersal experiment, in which the movement of cabbage root flies in the presence of a cabbage crop was studied. It is proposed that such a parameter estimation method can be a useful analytical tool to help develop appropriate models in population biology.