Effective parameter estimation within a multi-dimensional population balance model framework

This study considers optimization problems with multi-dimensional population balance models embedded. The objective function is formulated as a least-squares problem, minimizing the difference between target data and simulated model output and the goal is to find model parameter values that best fit the data. Results show that derivative-free methods, such as the Nelder–Mead simplex method, fail to converge to an optimal solution. A similar result was obtained with gradient-based methods such as BFGS, quasi-Newton, Newton, Gauss–Newton, Levenberg–Marquardt and SQP, and with a stochastic genetic algorithm. It was hypothesized that three main issues could contribute to these convergence failures: (1) gradients were calculated based on finite differences, and as a result of improper step size determination, the numerical error could be prohibitive resulting in inaccurate derivative information, (2) the parameters may not be identifiable and (3) numerical instability could occur during the course of optimization. To circumvent these issues, this work addresses the calculation of derivative information based on automatic differentiation and sensitivity analysis to ensure increased accuracy. Issues such as parameter identifiability are also dealt with by analyzing an accurate Fisher information matrix. Given the computational burden in calculating accurate Jacobians and Hessians, compounded by the potential nonsmoothness introduced into the objective function as a result of granule nucleation, other optimization strategies may be warranted and this work addresses those accordingly. Overall, by systematically assessing the problem formulation and mechanisms, the results show that substantial improvements in convergence can be achieved by utilizing appropriate optimization techniques, thus leading to more successful and optimal parameter estimation.