First- and second-order constrained optimisation for inverse homogenisation of unidirectionally varying microstructure

This paper is concerned with inverse homogenisation problems that aim to identify microstructure features of a body from macroscopic measurements. The unknown microstructure is assumed to vary macroscopically in one spatial direction and affects the effective tensor in an elliptic partial differential equation that governs the behaviour of the body on the macroscale. We establish the well-posedness of the PDE-constrained minimisation problems that model the task of identifying the microstructure, derive first-order necessary optimality conditions, and prove the convergence of a gradient-projection algorithm for the studied problem class in an infinite-dimensional setting. We moreover demonstrate that recently obtained results on generalised differentiability properties of solution maps of bilateral obstacle problems make it possible to set up and analyse a novel semismooth Newton method in the considered situation. The functional analytic setting that we use for our investigation is chosen such that it covers various elliptic PDEs simultaneously (classical second-order equations, bi-Laplace problems, linear elasticity, etc.) and explicitly exploits the problem structure to obtain global convergence guarantees for the studied solution algorithms in terms of the Lipschitz constants of the involved operators. Our analysis is accompanied by numerical experiments which show that the new semismooth Newton approach requires far less expensive projection evaluations than the gradient-descent method and, thus, offers a significant speedup.