Tikhonov’s regularization method

Honerkamp, (. and Weese, J. (1990) Tikhonov s regularization method for ill-posed problems a comparison of different methods for the determination of the regularization parameter. Continuum Mech. Thermodyn.. 2.17-30. 

On the basis of experimental molecular-mass distributions of the polymers, the inverse task was solved via Tikhonov s regularization method [3] as a result, the functions of distribution of active sites over the probability of chain termination were obtained (Fig. 4.3,4.4). The titanium catalytic system is characterized by a polymodal distribution function, where each maximum corresponds to a certain type of polymerization sites. Under our conditions of polybutadiene synthesis, there are four types of active sites that form macromolecules with the following most probable molecular masses InM = 9.2-10.4 (I), 11.2-11.4 (II), 12.9-13.2 (III), and 14.1-14.7 (IV) (Fig. 4.3). The position of maxima on the curves is conversion independent. This implies that each type of site forms macromolecules of a certain length throughout the process. 

In 1990, Honerkamp and Weese published a seminal paper on the use of Tikhonov s regularization for the determination of material functions. The developed method of data treatment was found particularly useful for the computation of the relaxation and retardation spectra [Elster et al, 1991 Honerkamp and Weese, 1993]. It has also been used to compute the sphere-size distribution of the dispersed phase in binary blends [Gleinser et al, 1994a], as well as the ratio of the dispersed drop diameter divided by the interfacial tension coefficient, d/Vj [Gleinser et al, 1994b]. 

Our goal is to find the parameters inverse problem. The inverse problem is ill-posed. To solve it, we use the Tikhonov regularization method and minimize the parametric functional with the appropriate stabilizer s cr) ... 

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