An equilibrated fluxes approach to the certified descent algorithm for shape optimization using conforming finite element and discontinuous Galerkin discretizations
For researchers in shape optimization and inverse problems, this work provides a fully-computable, local error estimator that extends the certified descent algorithm to a broader class of discretizations, though the improvement is incremental.
The paper presents a goal-oriented error estimator for the certified descent algorithm in shape optimization, using an equilibrated fluxes approach that works for both conforming finite element and discontinuous Galerkin discretizations. The method is validated on an electrical impedance tomography inverse problem, confirming its ability to identify genuine descent directions and provide a reliable stopping criterion.
The certified descent algorithm (CDA) is a gradient-based method for shape optimization which certifies that the direction computed using the shape gradient is a genuine descent direction for the objective functional under analysis. It relies on the computation of an upper bound of the error introduced by the finite element approximation of the shape gradient. In this paper, we present a goal-oriented error estimator which depends solely on local quantities and is fully-computable. By means of the equilibrated fluxes approach, we construct a unified strategy valid for both conforming finite element approximations and discontinuous Galerkin discretizations. The new variant of the CDA is tested on the inverse identification problem of electrical impedance tomography: both its ability to identify a genuine descent direction at each iteration and its reliable stopping criterion are confirmed.