Certified Descent Algorithm (CDA) for shape optimization

Gradient-based shape optimization strategies rely on the computation of the so-called shape gradient. In many applications, the objective functional depends both on the shape of the domain and on the solution of a PDE which can only be solved approximately (e.g. via the Finite Element Method). Hence, the direction computed using the discretized shape gradient may not be a genuine descent direction for the objective functional. In this talk, we account for the numerical error introduced by the Finite Element approximation of the shape gradient by means of a goal-oriented procedure and we derive a fully-computable certified upper bound for it. The resulting Certified Descent Algorithm (CDA) for shape optimization is able to identify a genuine descent direction at each iteration and features a reliable stopping criterion based on the norm of the shape gradient. Some numerical simulations are presented to test the discussed method. This is a joint work with Olivier Pantz and Karim Trabelsi.

contact: marco.verani@polimi.it.