Marquardt’s algorithm

The minimization of the objective fimction, based on the sum of square errors between experimental and calculated product yields, was sq)plied to find the best set of kinetic parameters. This objective function was solved using the least squares criterion with a nonlinear regression procedure based on Marquardt s algorithm [6]. 

The least squares criterion is the basis of the objective function. The parameters are determined using Marquardt s algorithm [7]. Most of the initial values of the kinetic parameters were those reported by Krane et al [3]. The best values of all kinetic constants are... 

The elements of the covariance matrix of the parameter estimates are calculated when the minimization algorithm has converged with a zero value for the Marquardt s directional parameter. The covariance matrix of the parameters COV(k) is determined using Equation 11.1 where the degrees of freedom used... 

As mentioned in Chapter 4, although this is a dynamic experiment where data are collected over time, we consider it as a simple algebraic equation model with two unknown parameters. The data were given for two different conditions (i) with 0.75 g and (ii) with 1.30 g of methanol as solvent. An initial guess of k =1.0 and k2=0.01 was used. The method converged in six and seven iterations respectively without the need for Marquardt s modification. Actually, if Mar-quardt s modification is used, the algorithm slows down somewhat. The estimated parameters are given in Table 16.1 In addition, the model-calculated values are... 

Several options are now available to the user in the main menu of the program. Probabilities can be calculated using an iterative method. Brown s modified version of the Levenberg-Marquardt algorithm (14-16). by substi futing values for P1-P4 in Equation 1 to calculate the peak integral which are then used in Equation 2 to simulate spectra until a good match between experimental and simulated data is achieved. 

Becker, S., Heller, M., Jarke, M., Marquardt, W., Nagl, M., Spaniol, O., Thifien, D. Synergy by Integrating New Functionality. This volume (2008) Becker, S., Herold, S., Lohmann, S., Westfechtel, B. A Graph-Based Algorithm for Consistency Maintenance in Incremental and Interactive Integration Tools. Journal of Software and Systems Modeling (2007)... 

Excel s multi-parameter non-linear least-squares routine. Solver, an implementation of the Levenberg-Marquardt algorithm, is a generally useful tool. In section 10.8a we describe an addition that can make it even more useful, and in section 10.8b we briefly indicate how you can call Solver from your macro. 

Fig. S.Sl. Bi-exponential fit to fluorescence data using the Marquardt algorithm measured for...

Besides being an artifice to reduce Jacobian illLevenberg-Marquardt method is an algorithm that couples Newton s method with the gradient one. 

Figure 4 illustrates how the Carreau-Yasuda model meets the shear viscosity data of Fig. 2. A non-linear fitting algorithm (i.e. Marquardt-Levenberg) was used to obtain the parameters given in the inset. As can be seen the fit curve provides a shear viscosity function that corresponds reasonably well with experimental data so that the high shear behavior is asymptotic to a power law and the very low shear behavior corresponds to the pseudo-Newtonian viscosity po- The characteristic time X (56.55 s) can be considered as the reverse of a critical shear rate (i.e. = Yc = 0.0177 s ) that corresponds to the intersection between the high shear power...

Any of a number of non-linear optimisation algorithms may be used to minimise S(x) and convergence is usually guaranteed from good starting estimates. A modified Marquardt algorithm [ ] is to be particularly recommended. 

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