Parameter Estimation in Dynamical Models

This paper gives a general introduction to the parameter estimation problem for dynamical models. The basic formulation and methodology in a parameter estimation problem will be discussed and some rather simple examples will be presented. It will be shown that even for linear dynamics the parameter estimation problem becomes nonlinear and may become extremely difficult to solve. Also, to have a well-posed problem with a unique solution, care must be taken when a parameter estimation problem is formulated. The discussion leads to the conclusion that it is possible to estimate poorly known parameters in a model, at least for simple dynamical models, but care must be taken to have a consistent solution. The rule is that all parameters which will be estimated should be added in a penalty function as weak constraints measuring their distance from a first guess in some norm. Some previous works in which data assimilation methods have been used to improve estimates of poorly known model parameters or even the model bias will be briefly reviewed. that [24] replaced the first-guess values of the parameters with the current estimate in each iteration of the parameters. These should of course be kept constant. Clearly, [24] solved a different inverse problem in each iteration and did not have any real penalty of the first guesses at all. Actually, it is not clear from their figures that the iterations did converge.

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