Verified Global Optimization for Estimating the Parameters of Nonlinear Models

Abstract

Nonlinear parameter estimation is usually achieved via the minimization of some possibly non-convex cost function. Interval analysis allows to derive algorithms for the guaranteed characterization of the set of all global minimizers of such a cost function when an explicit expression for the output of the model is available or when this output is obtained via the numerical solution of a set of ordinary differential equations. However, cost functions involved in parameter estimation are usually challenging for interval techniques, if only because of multi-occurrences of the parameters in the formal expression of the cost. This paper addresses parameter estimation via the verified global optimization of quadratic cost functions. It introduces tools instrumental for the minimization of generic cost functions. When an explicit expression of the output of the parametric model is available, significant improvements may be obtained by a new box exclusion test and by careful manipulations of the quadratic cost function. When the model is described by ODEs, some of the techniques available in the previous case may still be employed, provided that sensitivity functions of the model output with respect to the parameters are available.

Reliable uncertainties in indirect measurements

“…On the other hand, when the said outlier is 'small' enough, then the fitted curve will be unrealistically narrow, thus suggesting incredibly good precision of its parameters. No wonder that this approach didn't gain much popularity in scientific laboratory practice (notable exceptions are proposals given by Zhilin 3,4 or Kieffer 5 ), nevertheless it is developed in the field of various engineering applications under the keywords like tolerance problem 6,7 , confidence regions 8,9 , or guaranteed state estimation 10,11 .…”

“…This fact was most likely first recognized by Shary, who also proposed a classification scheme for solutions of interval systems of linear equations (applicable to non-linear systems as well). The literature concerning linear interval equations is now quite impressive 3,6,7,10,12,13,14,15,16,17,18,19,20,21 while non-linear systems are investigated less frequently 5,8,11,22,23 and often presented at conferences only 24 . Needless to say that majority of mentioned papers is of purely mathematical nature, without explicit relations to physical problems.…”