Certified Descent Algorithm for shape optimization driven by fully-computable a posteriori error estimators

TL;DR

This paper introduces a certified shape optimization algorithm that uses fully computable a posteriori error estimators to ensure reliable descent directions and stopping criteria, improving the accuracy of finite element-based shape optimization.

Contribution

The paper presents a novel certified descent algorithm with a goal-oriented error estimator that guarantees reliable descent directions in shape optimization.

Findings

The error estimator effectively bounds the shape gradient error.

The certified descent algorithm reliably identifies descent directions.

Numerical tests demonstrate improved accuracy in Electrical Impedance Tomography.

Abstract

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.