Fully discrete approximation methods for the estimation of parabolic systems and boundary parameters

A spatially and temporally discrete numerical approximation scheme is developed for the identification of a class of semilinear parabolic systems with unknown boundary parameters. The identification problem is formulated as a least squares fit to data subject to an equivalent representation for the dynamics in the form of an abstract evolution equation. Finite-dimensional difference equation state approximations are constructed using a cubic spline-based, Galerkin method and the Padé rational function approximations to the exponential. A sequence of approximating identification problems result, the solutions of which are shown to exist and, in a certain sense, approximate solutions to the original identification problem. Numerical results for two examples, one involving the modeling of biological mixing in deep sea sediment cores, and the other, the estimation of transport parameters for indoor mixing, are discussed. In both examples, the identification is based upon actual experimental data.Parts of the research were carried out while the authors were visitors at the Institute for Computer Applications in Science and Engineering (ICASE), NASA Langley Research Center, Hampton, Virginia, which is operated under NASA Contracts No. NAS1-17070 and No. NAS1-17130.Research supported in part by NSF Grant MCS-8205355, AFOSR Contract 81-0198 and ARO Contract ARO-DAAG-29-K-0029.