Adjoint state method in the block-parametric approach to the inverse problem analysis for elastic layered packages

Issues of efficiently calculating the gradient of the residual function play an important role in the analysis of inverse problems of any nature. One of the methods for calculating the gradient, which allows considering the “state equation” as part of the forward mapping, is the Adjoint State Method. This method is applied to inverse problems for a layered elastic package located on an elastic foundation. The materials of the package layers are generally assumed to be anisotropic and non-uniform, as are the parameters of the elastic foundation. The model/observed data of the inverse problem are the ray displacements of some points on the upper surface of the package. Formulas for the gradient components are obtained in an infinite-dimensional version using a block-parametric approach to the analysis of inverse problems. The essence of the approach lies in the special block-parametric approximation of the prior probability density and likelihood function in a set of parameters and model data of the problem. The method allows one to estimate the parameters of the prior distribution of the diagnosed values, identify and exclude outliers of measured data from the created model, and construct the estimate of the posterior probability density of unknown parameters with an acceptable resolution. The issues related to the discretization of the initial infinite-dimensional problems, comparison of the results obtained based on the adjoint state method and difference schemes of different orders of approximation for calculating the gradient components, as well as the application of the Multigrid Iterative Method for solving the forward and adjoint problems are considered.