Identification of Mathematical Models

Chapter 3 provides an introduction to the identification of mathematical models for reactive systems and an extensive review of the methods for estimating the relevant adjustable parameters. The chapter is initiated with a comparison between Bayesian approach and Poppers falsificationism. The aim is to establish a few fundamental ideas on the reliability of scientific knowledge, which is based on the comparison between alternative models and the experimental results, and is limited by the nonexhaustive nature of the available theories and by the unavoidable experimental errors. 

This paper describes application of mathematical modeling to three specific problems warpage of layered composite panels, stress relaxation during a post-forming cooling, and buckling of a plastic column. Information provided here is focused on identification of basic physical mechanisms and their incorporation into the models. Mathematical details and systematic analysis of these models can be found in references to the paper. 

Parameter estimation and identification are an essential step in the development of mathematical models that describe the behavior of physical processes (Seinfeld and Lapidus, 1974 Aris, 1994). The reader is strongly advised to consult the above references for discussions on what is a model, types of models, model formulation and evaluation. The paper by Plackett that presents the history on the discovery of the least squares method is also recommended (Plackett, 1972). 

Frequently in theoretical work on the subject, whether dealing with the steady or nonsteady state, the mathematical development of an adopted model is followed by a descriptive summary of the results which is rarely traced back clearly to the assumptions inherent in the model. This has often resulted in extravagant identification of the model with the actual phenomenon and has been an obstacle in the task of reconciling conflicting views. There is need, therefore, of a purely descriptive exposition, stripped as completely as possible of mathematical language, to clarify the physical concepts of the combustion wave phenomenon. 

No single method or algorithm of optimization exists that can be applied efficiently to all problems. The method chosen for any particular case will depend primarily on (1) the character of the objective function, (2) the nature of the constraints, and (3) the number of independent and dependent variables. Table 8-6 summarizes the six general steps for tne analysis and solution of optimization problems (Edgar, Himmelblau, and Lasdon, Optimization of Chemical Processes, 2d ed., McGraw-Hill, New York, 2001). You do not have to follow the cited order exactly, but you should cover all the steps at some level of detail. Shortcuts in the procedure are allowable, and the easy steps can be performed first. Steps 1,2, and 3 deal with the mathematical definition of the problem identification of variables, specification of the objective function, and statement of the constraints. If the process to be optimized is very complex, it may be necessary to reformulate the problem so that it can be solved with reasonable effort. Later in this section, we discuss the development of mathematical models for the process and the objective function (the economic model) in typical RTO applications. 

However kind of solutions depends on parameters of model and initial conditions. That is why while studying the processes proceeding in real systems by methods of mathematical modelling the problems of parametrical identification, i.e. problems of model s parameters determination by observations data are appeared. 

Some Aspects of Parameters Identification in Mathematical Modelling... 

The notion of model parameters defines one or more numerical values that are contained as symbolic notations in the mathematical model of a process. These numerical values cannot be obtained without any experimental research. In reality, the most important part of experimental research is dedicated to the identification of the models parameters. Generally, all the experimental works, laboratory... 

Some Aspects of Parameters Identification in Mathematical Modelling 147 with the constraint that imposes a fixed building volume ... 

On the basis of mathematical models, new experiments can be suggested and their outcome can be predicted. Especially, the control of networks can be investigated in silico. This enables identification of targets for medical intervention. 

Regression analysis models and estimation theory models are very useful for the identification of mathematical relations and parameter values in these relations from sets of data or measurements. Regression and estimation methods are used frequently in conjunction with mathematical modehng, in particular with trend extrapolation and time series forecasting, and with econometrics. These methods are often also used to validate models. Often these approaches are called system identifi-... 

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