Low-regret shape optimization in the presence of missing Dirichlet data

TL;DR

This paper introduces a low-regret shape optimization method for elliptic PDEs with missing Dirichlet boundary data, providing robust deformation fields through a novel formulation and numerical analysis.

Contribution

It formulates a no-regret shape optimization problem with missing boundary data using Fenchel transform and analyzes the convergence and effectiveness of low-regret solutions.

Findings

Low-regret solutions are robust against missing boundary data.

The proposed method converges to the no-regret solution.

Numerical examples demonstrate the effectiveness of the approach.

Abstract

A shape optimization problem subject to an elliptic equation in the presence of missing data on the Dirichlet boundary condition is considered. It is formulated by optimizing the deformation field that varies the spatial domain where the Poisson equation is posed. To take into consideration the missing boundary data the problem is formulated as a no-regret problem and approximated by low-regret problems. This approach allows to obtain deformation fields which are robust against the missing information. The formulation of the regret problems was achieved by employing the Fenchel transform. Convergence of the solutions of the low-regret to the no-regret problems is analysed, the gradient of the cost is characterized and a first order numerical method is proposed. Numerical examples illustrate the robustness of the low-regret deformation fields with respect to missing data. This is likely the first time that a numerical investigation is reported on for the level of effectiveness of the low-regret approach in the presence of missing data in an optimal control problem.