Optimal determination of rate coefficients in multiple-reaction systems

An efficient methodology is developed for parameter estimation and is applied by fitting 6 unknown rate coefficients. The estimation procedure is generally applicable to any system, although development has currently been limited to first-order systems of ordinary differential equations (ODE), such as those describing multiple chemical reactions. The objective is to find parameter values so as to minimize the sum of squared error (SSE), where each error term is the difference between the calculated system solution at a point and a selected data value. Since the calculated solution is generally quite nonlinear, an iterative solution is required. At each iteration, parameter values are supplied, the system is solved, and the SSE is determined. In addition, efficient algorithms require the SSE gradient (with respect to the vector of unknown parameters) in order to provide updated parameter estimates. Using conventional techniques, determination of this gradient involves solution of an ODE system for each parameter to be estimated. ff more than a few parameters are involved, the cost could be prohibitive. However, a procedure using adjoint operators is developed in which the SSE gradient can be calculated by solving only one additional ODE system, regardless of the number of parameters being optimized. Combined with a quasi-Newton updating system, an efficient methodology results. This methodology has been applied to a set of six chemical reactions describing the aqueous speciation (hydrolysis) of iodine.