Stochastic modeling techniques

Nechval NA, Purgailis M, Berzins G, Cikste K, Krasts J, Nechval KN (2010) Invariant embedding technique and its applications for improvement or optimization of statistical decisions. In Al-Begain K, Fiems D, Knottenbelt W (eds) Analytical and stochastic modeling techniques and applications, vol 6148, LNCS. Springer, Berlin, pp 306-320... 

This chapter provides an overview of the most frequently applied numerical methods for the simulation of polymerization processes, that is, die calculation of the polymer microstructure as a function of monomer conversion and process conditions such as the temperature and initial concentrations. It is important to note that such simulations allow one to optimize the macroscopic polymer properties and to influence the polymer processability and final polymer product application range. Both deterministic and stochastic modeling techniques are discussed. In deterministic modeling techniques, time variation is seen as a continuous and predictable process, whereas in stochastic modeling techniques, a random-walk process is assumed instead. 

This section is an overview of the most important deterministic and stochastic modeling techniques to obtain the polymer microstracture as a function of monomer conversion and polymerization conditions at the microscale. It is assumed that, for this scale, the bulk concentrations and temperature are known. The simplest case is the simulation of a batch polymerization reactor on laboratory scale with perfect macromixing and isothermicity implying a reactor with spatial homogeneity of the bulk concentrations and temperature. 

A main distinction has been made between deterministic and stochastic modeling techniques. A further distinction has been proposed based on the scale for which the mathematical model must be derived (eg, micro-, meso-, and/or macroscale). Notably, the complexity of the model approach depends on the desired model output. Detailed microstractural information is only accessible using advanced modeling tools but these are associated with an increase high in computational cost. The advanced models allow one to directly relate macroscopic properties to the polymer synthesis procedure and, thus, to broaden the application market for polymer products, based on a fundamental understanding of the polymerization kinetics and their link with polymer processing. 

Stochastic modeling. Some researchers may categorize models differently as for example into numerical or analytic, but this categorization applies more to the techniques employed to solve the formulated model, rather than to the formulation per se. 

The steady-state flow numerical experiment was primarily designed to evaluate the phasic relative permeability relations. The numerical experiment is devised within the two-phase lattice Boltzmann modeling framework for the reconstructed CL microstructure, generated using the stochastic reconstruction technique described earlier. Briefly, in the steady-state flow experiment two immiscible fluids are allowed to flow simultaneously until equilibrium is attained and the corresponding saturations, fluid flow rates and pressure gradients can be directly measured and correlated using Darcy s law, defined below. 

Bayesian methods for subset selection offer several advantages over other approaches the assignment of posterior probabilities to different subsets of active effects provides a way of characterizing uncertainty about effect activity prior distributions can incorporate principles of effect dependence, such as effect heredity the identification of promising models via Bayesian stochastic search techniques is faster than all subsets searches, and more comprehensive than stepwise methods. 

Roberts, S. M. 1960b. Stochastic models for the dynamic programming formulation of the catalyst replacement problem. Conference on Optimization Techniques in Chemical Engineering, New York University, May 18, 1960, 171-188. 

If such an approach becomes widespread it will be even more necessary to calibrate and understand its merits and drawbacks by using detailed and accurate computational modelling techniques that have been thoroughly validated, such as Large Eddy Simulation methods (Rodi, 1997 [541]), stochastic simulation methods (Hort et al., 2002 [276] Turfus, 1988 [622]), and time-dependent Reynolds-averaged models. 

The state estimation technique can also be incorporated into the design of optimal batch polymerization control system. For example, a batch reaction time is divided into several control intervals, and the optimal control trajectory is updated online using the molecular weight estimates generated by a model/state state estimator. Of course, if batch reaction time is short, such feedback control of polymer properties would be practically difficult to implement. Nevertheless, the online stochastic estimation techniques and the model predictive control techniques offer promising new directions for the improved control of batch polymerization reactors. 

Several stochastic models, based on mutli-parametric regression, artificial neural networks, Kalman filter and other statistical techniques, were implemented for short-term forecast of air pollution episodes, namely high ozone concentrations (Czech Republic, Hungary, Poland, Slovenia). 

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