A Parameter Identification Problem of Mixed Type Related to the Manufacture of Car Windshields

Abstract

We study the identification of a parameter in a fourth-order elliptic partial differential equation that models the optimal design of car windshields to be manufactured by the sagging process. Considered as a second-order equation for the unknown parameter, the problem is of mixed type, i.e., changing between elliptic and hyperbolic. Numerical routines for directly solving this equation are not available. In this paper we both theoretically and numerically show that the inverse problem can instead be solved in a stable way by means of a (derivative free) iterative regularization method. The course of the iteration nevertheless depends markedly on the mixed type of the second-order equation.