Mesh refinement for inverse problems with finite element models

Abstract

Many inverse problems arising in experimental mechanics involve solutions to partial differential equations in the forward problem, typically using finite element methods for those solutions. Given that iterative solutions to the inverse problem then involve repeated evaluations of the finite element model, it is useful to carefully consider the mesh to be used and its effect on the trade-off between accuracy and computational cost of the solution. We show that approximation theory can be applied directly to the inverse problem, not merely to the finite element model contained in the forward problem, to give bounds for the error made by using a given mesh to approximate the solution to the partial differential equation. Adaptive mesh refinement allows to focus computational effort on goals set by the quantities of interest in the inverse problem, rather than on the overall accuracy of the solution to the forward problem.