Marquardt parameter

The pseudo-inverse for the calculation of the shift vector in equation (4.67) has been computed traditionally as J+= (J Jp1 J. Adding a certain number, the Marquardt parameter mp, to the diagonal elements of the square matrix J J prior to its inversion, has two consequences (a) it shortens the shift vector 8p and (b) it turns its direction towards steepest descent. The larger the Marquardt parameter, the larger is the effect. In matrix formulation, we can write ... 

Figure 4-37. Appending the Marquardt parameter mp to the Jacobian J and residual vector r...

Depending on the change of the sum of squares the Marquardt parameter is adjusted the parameter is reduced upon convergence and increased otherwise. There are no general rules on how exactly this should be done in detail it depends on the specific case. Usually convergence occurs with no Marquardt parameter at all in the Matlab program Ma in Chrom. m, it is thus initialised as zero. 

If required, the initial value for the Marquardt parameter, mp, in case of divergence, has to be chosen sensibly as well the original suggestion was to use the value of the largest diagonal element of J J. This is, however, not crucial and in the Matlab function nglm. m further below, we set this initial value, if required, to one. 

The complexity of the flow diagram shown below might be surprising. A few remarks are appropriate it is possible that the Marquardt parameter reaches a high value and this results in a very small shift vector. Consequently, the change in ssq gets very small too and the algorithm decides prematurely that the minimum has been reached. 

To prevent this sequence, one last iteration is done without a Marquardt parameter (mp= 0) if the termination criterion is satisfied but mp is not yet zero. 

Marquardt parameter % convergence limit % step size for numerical d iff... 

The right panel of Figure 4-39 shows measurements and fitted curve, the left displays graphically the development of ssq and mp. The markers represent ssq for each iteration and the markers the Marquardt parameter, both on a logarithmic scale. Increase in ssq results in an increase of mp. The final mp= 0 cannot be displayed on the logarithmic plot. 

According to the following output, the module needed six iterations to find the minimum of the objective function. The value of the Marquardt parameter PM, i.e., 7 ) gradually decrased. In iterations 5 and 6 several attempts with different Marquardt parameters are necessary to improve the objective function. In less cooperative estimation problems the module frequently needs to increase the Marquardt parameter. The current value of the sum of squares, i.e., the objective function and the relative step length SL are also printed in every iteration. 

Show that increasing the Marquardt parameter moves the correction vector Ap toward the direction of the negative gradient of the objective function while the length of the correction vector decreases. 

However, the Hessian or its approximation has to be inverted to determine the parameter step-change for the next iteration and, especially when far from the real minimum of the SSR, the matrix is not positive definite, a requirement for inversion. Levenbcrg [51] and Marquardt [52] therefore added a diagonal matrix to it and allowed this contribution to vary according to a parameter A, the Marquardt parameter. For the Ath iteration this yields... 

Levenberg-Marquardt parameter after the Arth iteration -... 

A = Levenberg-Marquardt parameter p = viscosity 0 = interfacial tension SUPERSCRIPTS... 

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