Structured Total Least Norm for Nonlinear Problems

• Authors: J. B. Rosen;Haesun Park;John Glick
• Venue: SIAM Journal on Matrix Analysis and Applications
• Year: 1999

An extension of the recently developed structured total least norm (STLN) problem formulation is described for solving a class of nonlinear parameter estimation problems. STLN is a problem formulation for obtaining an approximate solution to the overdetermined linear system $Ax \approx b$ preserving the given affine structure in A or [A | b], where errors can occur in both the vector b and the matrixA. The approximate solution can be obtained to minimize the error in the Lp norm, where p = 1, 2, \mbox{ or } \infty$. In the extension of STLN to nonlinear problems, the elements of A may be differentiable nonlinear functions of a parameter vector, whose value needs to be approximated. We call this extension structured nonlinear total least norm (SNTLN). The SNTLN problem is formulated and its solution by a modified STLN algorithm is described. Optimality conditions and convergence for the 2-norm case are presented. Computational tests were carried out on an overdetermined system with Vandermonde structure and on two nonlinear parameter estimation problems. In these problems, both the coefficients and the unknown parameters were to be determined. The computational results demonstrate that the SNTLN algorithm recovers good approximations to the correct values of both the coefficients and parameters, in the presence of noise in the data and poor initial estimates of the parameters. It is also shown that the SNTLN algorithm with the 1-norm minimization is robust with respect to outliers in the data.