A parallel implementation for parameter estimation with implicit models

Parameter estimation and data regression represent special classes of optimization problems. Often, nonlinear programming methods can be tailored to take advantage of the least squares, or more generally, the maximum likelihood, objective function. In previous studies the authors have developed large-scale nonlinear programming methods that are based on tailored quasi-Newton updating strategies and matrix decomposition of the process model. The resulting algorithms converge in a reduced space of the parameters while simultaneously converging the process model. Moreover, tailoring the method to least squares functions leads to significant improvements in the performance of the algorithm. These approaches can be very efficient for both explicit and implicit models (i.e. problems with small and large degrees of freedom, respectively). In the latter case, degrees of freedom are proportional to a potential large number of data sets. Applications of this case include errors-in-all-variables estimation, data reconciliation and identification of process parameters. To deal with this structure, the authors apply a decomposition approach that performs a quadratic programming factorization for each data set. Because these are small components of large problems, an efficient and reliable algorithm results. These methods have generally been implemented on standard von Neumann architectures and few studies exist that exploit the parallelism of nonlinear programming algorithms. It is therefore interesting to note that for implicit model parameter estimation and related process applications, this approach can be quite amenable to parallel computation, because the major cost occurs in matrix decompositions for each data set. Here the authors describe an implementation of this approach on the Alliant FX/8 parallel computer at the Advanced Computing Research Facility at Argone National Laboratory. Special attention is paid to the architecture of this machine and its effect on the performance of the algorithm. This approach is demonstrated on five small, undetermined regression problems as well as a larger process example for simultaneous data reconciliation and parameter estimation.