Levenberg-Marquardt nonlinear

LM denotes that the kinetic parameters were evaluated using the Levenberg-Marquardt nonlinear least-squares method. 

Once the reactor equations and assumptions have been established, and HDS, HDN, HDA, and HGO reaction rate expressions have been defined, the adsorption coefficient, equilibrium constants, reaction orders, frequency factors, and activation energies can be determined from the experimental data obtained at steady-state conditions by optimization with the Levenberg-Marquardt nonlinear regression algorithm. Using the values of parameters obtained from steady-state experiments, the dynamic TBR model was employed to redetermine the kinetic parameters that were considered as definitive values. The temperature dependencies of all the intrinsic reaction rate constants have been described by the Arrhenius law, which are shown in Table 7.4. 

The Levenberg-Marquardt method is used when the parameters of the model appear nonlinearly (Ref. 231). We stiU define... 

We perform nonlinear fitting using the Levenberg-Marquardt method implemented in the MRQMIN routine [75], From the experimental end, eight families of data are involved, namely, x (T) and x jT) at four frequencies, taken from Ref. 64. From the theory end, we employ formulas (4.121)-(4.124) with the numerical dispersion factors. The results of fitting are presented in Figures 4.6 and 4.7 and Table I. 

Nonlinear least-squares fitting in the time domain (Levenberg-Marquardt method)... 

The procedure was tested on simulated time domain MRS data where the model data consisted of metabolite peaks at 3.2, 3.0 and 2.0 ppm representing choline, creatine and IV-acetylaspartate (NAA) respectively, with corresponding values of Ak of 1.0, 1.0 and 3.0 units.89 White noise of specified standard deviation, crt, was then added. The Levenberg-Marquardt method requires suitable initial values for each of the nine parameters being fitted. The initial values of the three frequencies were taken as their known values. An exponentially decaying curve with a constant offset parameter was fitted, using a nonlinear least-squares fit, to the envelope of the free induction decay, Mv(t), in order to obtain an initial value for T and for the amplitudes, each of which was taken to be one-third of the amplitude of the envelope. The constant offset was added to account for the presence of the noise. 

A nonlinear, multiparameter regression procedure (Levenberg-Marquardt method) was applied to estimate the kinetic parameters involved in Equations (51)-(54). The experimental concentrations of the pollutant (4-CP) and of the main intermediate species (4-CC and HQ) at different reaction times were compared with model predictions. Under the operating conditions of the experimental runs, it was found that the terms a Ci- cp(f)/ aiC4-cc(f)/ and 02CHQ(t) were much lower than 1. As a result, the final expressions employed for the regression of the kinetic parameters are the following ... 

The three model parameters (rjQ, t and n) are often selected with a nonlinear least-squares algorithm which minimizes the squared difference between the measured and modeled ln for all co at temperature T. Application of a Levenberg-Marquardt algorithm [41,42] to the SAN copolymer data in Figure 13.4 yields fit parameters summarized in Table 13.4. Error bars are reported to two standard deviations. 

In the case of complex stoichiometries, and when several complexes can coexist in solution, data must be processed using several wavelengths simultaneously. This requires specific software. For instance, the commercially available SPEC FIT Global Analysis System (V3.0 for 32-bit Window Systems) deserves attention. This software uses singular value decomposition and nonlinear regression modeling by the Levenberg-Marquardt method [8]. 

Figure 3.8 Example of parameter redundancy in nonlinear models. Symbols were generated using the model given by Eq. (3.97). Solid line is the predicted fit using Eq. (3.98). The biexponential model predicted values cannot be distinguished from data generated using a triexponential equation. Starting values were 10,1,10, and 0.25. Model was fit using the Levenberg— Marquardt method within the NLIN procedure in SAS.

The results of open-system pyrolysis (Rock-Eval II) have been used to specify the kinetic parameters controlling maturation. Hydrocarbon yield rates as determined by these experiments are shown in Fig. 6.9a. Both nonlinear optimization technique (Levenberg-Marquardt method Press et al. 1986 Issler and Snowdon 1990) and linear methods are used to determine the values of the reaction parameters Aj, Ej, andX, . This technique minimizes an error function by comparing the hydrocaibon release rates, Sj, calculated by Eq. 6.9 and those rates measured in open-system pyrolysis. An example of the spectrum of activation energies obtained from this analysis is shown in Fig. 6.9b. 

Many approaches have been devised to solve equation (14) or (15). In this study, the Levenberg-Marquardt (LM) method was used. This is because the LM method works well in practice and has become the standard of nonlinear least squares routines [31]. 

In general, the error e tic-q-i+j, 0) is a non-linear function of the parameter vector 0. Therefore, the above problem is a well-known nonlinear least squares problem (NLSP) that may be solved by various optimisation algorithms such as the Levenberg-Marquardt algorithm [2], the quasi-Newton method or the Gauss-Newton (GN) algorithm [3]. 

The merit function is minimized by picking the best parameters P for 5 Some adjustable parameters hk) are related linearly as seen in (40) and (51), but others (Ajt and have a nonlinear relationship. A natural way to solve the problem is to use some nonlinear minimization program (e.g., Levenberg-Marquardt) for adjusting 4 and a/, and to solve h in the subprogram using standard linear techniques for minimizing... 

All nonlinear regression approaches use numerical methods, such as the Gauss-Newton or Levenberg-Marquardt algorithm optimisaticai algorithms, to search for the optimal point. 

The K values were analysed in KaleidaGraph using nonlinear (Levenberg-Marquardt algorithm) curve fitting. The errors reported are the standard errors obtained from the best fit. 

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