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Abstract
The identification of reaction kinetics represents the main challenge in building models for reaction systems. The identification task can be performed via either simultaneous model identification (SMI) or incremental model identification (IMI), the latter using either the differential (rate-based) or the integral (extent-based) method of parameter estimation. This contribution presents an extension of extent-based IMI that guarantees convergence to globally optimal parameters.
In SMI, a rate law must be postulated for each reaction, and the model concentrations are obtained by integration of the balance equations. The procedure must be repeated for all combinations of rate candidates. This approach is computationally costly when there are several candidates for each reaction, and convergence problems may arise due to the large number of parameters.
In IMI, the identification task is decomposed into several sub-problems, one for each reaction [1]. Since IMI deals with one reaction at a time, only the rate candidates for that reaction need to be compared. In addition, convergence is facilitated by the fact that only the parameters of a single reaction rate are estimated. In rate-based IMI, the parameters are estimated by fitting the simulated rates to the experimental rates obtained by differentiation of measured concentrations. In extent-based IMI, the simulated rates are integrated to yield extents, and the parameters are estimated by fitting the simulated extents to experimental extents obtained by transformation of measured concentrations [2]. The simulated rates are functions of concentrations. Hence, since each reaction is simulated individually, the simulated rates must be computed from measured concentrations.
Most parameter estimation methods converge to local optimality, which may result in an incorrect model. It turns out that extent-based IMI is particularly suited to global optimization since each estimation sub-problem (i) involves only a small set of parameters, and (ii) can be rearranged as an algebraic problem, where the objective function is polynomial in the parameters with coefficients computed only once prior to optimization using a Taylor expansion. These features facilitate the task of finding a global optimum for each reaction. Instead of the classical branch-and-bound approach, this technique relies on reformulating the estimation problem as a convex optimization problem, taking advantage of the equivalence of nonnegative polynomials and conical combination of sum-of-squares polynomials on a compact set to solve the problem as a semidefinite program [3].
A simulated example of an identification problem with several local optima shows that extent-based IMI can be used to converge quickly to globally optimal parameters.
References:
[1] Bhatt et al., Chem. Eng. Sci., 2012, 83, p. 24
[2] Rodrigues et al., Comput. Chem. Eng., 2015, 73, p. 23
[3] Lasserre, SIAM J. Optim., 2001, 11(3), p. 796
In SMI, a rate law must be postulated for each reaction, and the model concentrations are obtained by integration of the balance equations. The procedure must be repeated for all combinations of rate candidates. This approach is computationally costly when there are several candidates for each reaction, and convergence problems may arise due to the large number of parameters.
In IMI, the identification task is decomposed into several sub-problems, one for each reaction [1]. Since IMI deals with one reaction at a time, only the rate candidates for that reaction need to be compared. In addition, convergence is facilitated by the fact that only the parameters of a single reaction rate are estimated. In rate-based IMI, the parameters are estimated by fitting the simulated rates to the experimental rates obtained by differentiation of measured concentrations. In extent-based IMI, the simulated rates are integrated to yield extents, and the parameters are estimated by fitting the simulated extents to experimental extents obtained by transformation of measured concentrations [2]. The simulated rates are functions of concentrations. Hence, since each reaction is simulated individually, the simulated rates must be computed from measured concentrations.
Most parameter estimation methods converge to local optimality, which may result in an incorrect model. It turns out that extent-based IMI is particularly suited to global optimization since each estimation sub-problem (i) involves only a small set of parameters, and (ii) can be rearranged as an algebraic problem, where the objective function is polynomial in the parameters with coefficients computed only once prior to optimization using a Taylor expansion. These features facilitate the task of finding a global optimum for each reaction. Instead of the classical branch-and-bound approach, this technique relies on reformulating the estimation problem as a convex optimization problem, taking advantage of the equivalence of nonnegative polynomials and conical combination of sum-of-squares polynomials on a compact set to solve the problem as a semidefinite program [3].
A simulated example of an identification problem with several local optima shows that extent-based IMI can be used to converge quickly to globally optimal parameters.
References:
[1] Bhatt et al., Chem. Eng. Sci., 2012, 83, p. 24
[2] Rodrigues et al., Comput. Chem. Eng., 2015, 73, p. 23
[3] Lasserre, SIAM J. Optim., 2001, 11(3), p. 796