(33d) Toward Large-Scale Nonlinear Parameter Estimation with Mixed Effect Models

Researchers conducting experiments to determine reaction kinetics often replicate the experiment under similar conditions to ensure they are achieving a repeatable result. Those modeling the experimental setup can then analyze the data collected to further understand the process. Accurate modeling of the system plays an important role in estimating parameters related to the process.

During each experiment, there are sources of error that occur which fall into one of two categories: fixed and random effects. Fixed effects describe the relationship between the dependent variable and predictor variables whereas random effects usually represent random deviations from the relationships described by fixed effects (B.T. West et al. 2007). One can expect to minimize random experimental error by accounting for it in the modeling of the parameter estimation process.

Mixed effects models are used for chemical kinetics because they account for both the random error within an individual experiment as well as across a set of experiments, preventing bias in the estimation of the fixed parameters (Hickman et al. 2019).

For a given repeated set of experiments, there are parameters that are unique to an individual data set and those which are shared among all data sets. These represent local and global parameters, respectively. Kinetic parameters are global parameters and modeling them using a mixed effects approach minimizes the error across multiple data sets to achieve an accurate result.

This work focuses on using KIPET (Kinetic Parameter Estimation Toolbox) to estimate the parameters of the mixed effects problem (Short et al., 2019). KIPET uses Pyomo (Hart et al. 2017) to formulate the parameter estimation problem based on maximum likelihood principles and large-scale nonlinear programming strategies. KIPET applies an orthogonal collocation strategy on finite elements to discretize the nonlinear system (Short et al., 2019). While the resulting nonlinear system can be solved with a direct nonlinear algebra solver, as the problem size grows, the time and memory requirements can make this approach intractable (Zavala et al. 2006). This work seeks to exploit the mixed effects problem structure and use a Schur compliment decomposition and parallel solver solution technique across data sets. This work also focuses on applying these techniques to a case study (Hickman et al. 2019).

Hart, W.E., Laird, C.D., Watson, J.-P., Woodruff, D.L., Hackebeil, G.A., Nicholson, B.L., Siirola, J.D., Pyomo — Optimization Modeling in Python, Springer (2017)

Daniel A. Hickman, Michael J. Ignatowich, Michael Caracotsios, James D. Sheehan, Fabio D'Ottaviano “Nonlinear mixed-effects models for kinetic parameter estimation with batch reactor data,” Chemical Engineering Journal, DOI: 10.1016/j.cej.2018.08.203

C. Schenk, M. Short, J. S. Rodriguez, D. Thierry, L. T. Biegler, S. Garcia-Munoz, W. Chen, “Introducing KIPET: A novel open-source software package for kinetic parameter estimation from experimental datasets including spectra," Computers and Chemical Engineering, 134(4), 106716 (2020

V. M. Zavala, C. D. Laird and L. T. Biegler, “Interior-Point Decomposition Approaches for Parallel Solution of Large-Scale Nonlinear Parameter Estimation Problems" Chemical Engineering Science , 63, pp. 4834- 4845 (2008)