Title: A parameter estimation method for stiff ordinary differential equations using particle swarm optimisation

Authors: William Arloff; Karl R.B. Schmitt; Luke J. Venstrom

Addresses: Department of Mathematics and Statistics, Valparaiso University, Valparaiso, IN, USA ' Department of Mathematics and Statistics, Department of Computing and Information Sciences, Valparaiso University, Valparaiso, IN, USA ' Department of Mechanical Engineering, Valparaiso University, Valparaiso, IN, USA

Abstract: We propose a two-step method for fitting stiff ordinary differential equation (ODE) models to experimental data. The first step avoids integrating stiff ODEs during the unbounded search for initial estimates of model parameters. To avoid integration, a polynomial approximation of experimental data is generated, differentiated and compared directly to the ODE model, obtaining crude but physically plausible estimates for model parameters. Particle swarm optimisation (PSO) is used for the parameter search to overlook combinations of model parameters leading to undefined solutions of the stiff ODE. After initial estimates are determined, the second step numerically solves the ODE. This refines model parameter values through a bounded search. We demonstrate this method by fitting the model parameters (activation energies and pre-exponential factors) of the Arrhenius-based temperature-dependent kinetic coefficients in the shrinking core solid-state chemical kinetics model for the reduction of Cobalt (II, III) Oxide (Co\(_3\)O\(_4\)) particles to Cobalt (II) Oxide (CoO).

Keywords: optimisation; particle swarm optimisation; PSO; ODE; ordinary differential equations; stiff ODEs; solid-state kinetics; shrinking core model.

DOI: 10.1504/IJCSM.2018.10016539

International Journal of Computing Science and Mathematics, 2018 Vol.9 No.5, pp.419 - 432

Received: 15 Aug 2017
Accepted: 28 Nov 2017

Published online: 08 Oct 2018 *

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