A Richardson-type iterative approach for identification of delamination boundaries

A direct problem of particular mathematical modelling is to determine the response of a system, given the governing partial differential equations, the geometry under interest, the complete boundary and initial conditions, and material properties. When one or more of the conditions for the solution of the direct problem are unknown, an inverse problem can be formulated. One of the methods frequently used for the solution of inverse problems involves finding the values of the unknowns in a mathematical formulation such that the behavior calculated with the model matches the measured response at degree evaluated in terms of the classical $L_2$ norm. Considered in this sense, the inverse problem is equivalent to an ill-posed optimization problem for the estimation of parameters whose solution in predominant part of cases is a real mathematical challenge. In this contribution, we report a novel approach which avoids the mathematical difficulties inspired by ill-posed character of the model. Our method is devoted to the computation of inverse problems furnished by second-order elliptical systems of partial differential equations and falls in the same conceptual line with the method initiated by Kozlov et al. and further extended and algorithmized by Weikl et al. We construct and employ a weak version of the algorithm found by Weikl et al. Proofs for the convergence and regularity of this version are given for the case of a single layer. The computational realization of the algorithm (called briefly AICRA) is applied and numerical results are obtained. The comparison with experiments demonstrates a good significance and representativeness.