Nonlinear Marquardt algorithm

Computational the most demanding task is locating the minimum of the function (3.89) at step (ii). Since the Gauss-Newtan-Marquardt algorithm is a robust and efficient way of solving the nonlinear least squares problem discussed in Section 3.3, we would like to extend it to error—in-variables models. First we show, however, that this extension is not obvious, and the apparently simplest approach does not work. 

Least-squares methods are usually used for fitting a model to experimental data. They may be used for functions consisting of square sums of nonlinear functions. The well-known Gauss-Newton method often leads to instabilities in the minimization process since the steps are too large. The Marquardt algorithm [9 1 is better in this respect but it is computationally expensive. 

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 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]. 

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

When the equations are nonlinear in the parameters, the parameter estimates are obtained by minimizing the objective function by methods like that of Newton-Raphson or that of Newton-Gauss or an adaptation of the latter such as the Marquardt algorithm [1963], In the latter case parameters are iteratively improved by the following formula ... 

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. 

To calculate the values of AH and that best describe the folding curve, initial values of AH, ACp, Of and On are estimated, and Equation [6] is fitted to the experimentally observed values of the change in ellipticity as a function of temperature, by a nonlinear least-squares curve-fitting routine such as the Levenberg-Marquardt algorithm. Similar equations can be used to estimate the thermodynamics of folding of proteins and peptides that undergo folded multimer to unfolded monomer transitions. 

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. 

Marquardt, D.W., "An Algorithm for Least-Squares Estimation of Nonlinear Parameters", J. Soc. Indust. Appl. Math., 11(2), 431-441 (1963). 

Marquardt, D. W. (1963) An algorithm for the Least-Squares Estimation of Nonlinear parameters, J. Soc. Appl. Math 11,431. 

D. Marquardt. An algorithm for least squares estimation of nonlinear parameters. SIAM Journal on Applied Mathematics, 11 431—441, 1963. 

---