Certified Descent Algorithm for shape optimization driven by fully-computable a posteriori error estimators
For practitioners of shape optimization, this work addresses the critical issue of numerical errors in gradient computation, offering a certified approach that ensures descent directions and stopping criteria are reliable.
The paper introduces a Certified Descent Algorithm (CDA) for shape optimization that accounts for finite element approximation errors in the shape gradient, providing a certified upper bound and fully-computable error estimator. Numerical simulations on Electrical Impedance Tomography demonstrate the method's ability to identify genuine descent directions and reliable stopping criteria.
In this paper we introduce a novel certified shape optimization strategy - named Certified Descent Algorithm (CDA) - to account for the numerical error introduced by the Finite Element approximation of the shape gradient. We present a goal-oriented procedure to derive a certified upper bound of the error in the shape gradient and we construct a fully-computable, constant-free a posteriori error estimator inspired by the complementary energy principle. The resulting CDA is able to identify a genuine descent direction at each iteration and features a reliable stopping criterion. After validating the error estimator, some numerical simulations of the resulting certified shape optimization strategy are presented for the well-known inverse identification problem of Electrical Impedance Tomography.