Regularized parameter estimation through iterative rescaling (PETIR): an alternative to Levenberg-Marquardt's algorithm

Maillet Denis (1) , Stéphane André (2) , Rémy Benjamin (1) , Degiovanni Alain (1)

1 Thermique
2 LEMTA - Laboratoire d'Energétique et Mécanique Théorique et Appliquée

The Gauss Newton method of least squares minimization for non-linear parameter estimation is revisited for parameters with different physical units. A normalization of each parameter with respect to its nominal value, that is at iteration number k-1, is implemented, which leads to a linear tangent model. This model uses the sensitivity matrix composed of the scaled sensitivity coefficients. It is decomposed under a singular values form and the covariance matrix of iterate number k is calculated. When the scaled standard deviation of one parameter estimates takes a too large value, inversion of the tangent model becomes ill-posed. Regularization is made by giving the smallest singular values infinite levels, which allows keeping the total number of parameters to be estimated unchanged : this regularization leads to a better conditioned problem in the following iterations until convergence of the residuals is reached. The corrresponding algorithm is tested in the case of two very ill posed-examples. This type of estimation performs very well when compared to the Lebenverg Marquardt algorithm.

Mots clés: parameter estimation; dimensional analysis; Levenberg-Marquardt; regularization; singular value decomposition

Domaines: Mécanique des matériaux [physics.class-ph]; Mécanique des matériaux [physics.class-ph]; Optimisation et contrôle [math.OC]; Thermique [physics.class-ph]; Thermique [physics.class-ph]

Maillet Denis, Stéphane André, Rémy Benjamin, Degiovanni Alain. Regularized parameter estimation through iterative rescaling (PETIR): an alternative to Levenberg-Marquardt's algorithm. 2013. ⟨hal-00867608⟩