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

作     者：Kunisch, Karl Simon, John Sebastian H. 

作者机构：Johann Radon Institute for Computational and Applied Mathematics Austrian Academy of Sciences Altenberger Strasse 69 Linz4040 Austria Institute of Mathematics and Scientific Computing University of Graz Heinrichstrasse 36 GrazA-8010 Austria 

出 版 物：《arXiv》 (arXiv)

年 卷 期：2024年

核心收录：

主　　题：Shape optimization 

摘      要：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. © 2024, CC BY.