Iteratively Reweighted Least Squares Algorithm for Nonlinear Distributed Parameter Estimation

Abstract

In the treatment of distributed parameters estimation, the usual choice is that of minimizing the error norm using the least squares distance. This paper considers the problem of estimating distributed parameters representing properties of material that change over the spatial domain. In this case the parameters are represented by non-smooth functions. Moreover, the data may be affected by noise containing outliers that cannot be easily removed. In all these cases a great improvement in the solution can be obtained by solving a minimization problem in the $p$ norm where 0 p < < ∞ . When the norm $p=2$ is used, then the regularized nonlinear least squares problem can be efficiently solved by the Iterative Gauss Newton (IRGN) method as reported in [1,2] and references therein. The case 1 p < < ∞ is efficiently treated by the Iterative Reweighted Algorithm IRLS proposed in [3] for linear problems. We propose here a direct extension of such algorithm to the solution of non linear problems where 0 p < < ∞ . The non linear least squares and possibly non convex problem is substituted by a sequence of weighted least squares approximations which efficiently solve the non linear identification problem. The algorithm, named NL-LM-IRLS, is presented as an extension of the IRLS applied to the non linear minimization problem. Some preliminary results on one dimension identification problems are reported confirming the validity of this approach. In section 2 we explain the details of the iterative method and report the numerical experiments in section 3. ( ) ( ) 0 0 r F q y = − for k=0 , 1 , ..... r0 = F(q(0)) y for k = 0, 2,... D(k) = diag(|rk| + 100) qk+1 = argmin ||D(k)(F(q) y)||2 r(k+1) = F(q(k+1)) y