报告题目：On discrete shape gradients of boundary type for PDE-constrained shape optimization

报告时间：2020年12月21日（星期一） 下午15:00 - 18:00

报告平台：腾讯会议，ID: 291 362 559

报告人： 龚伟 研究员、博士生导师（中国科学院数学与系统科学研究院）

摘要：Shape gradients have been widely used in numerical shape gradient descent algorithms for shape optimization. The two types of shape gradients, i.e., the distributed one and the boundary type, are equivalent at the continuous level but exhibit different numerical behaviors after finite element discretization. To be more specific, the boundary type shape gradient is more popular in practice due to its concise formulation and convenience to combine with shape optimization algorithms but has lower numerical accuracy. In this talk we introduce a simple yet useful boundary correction for the normal derivatives of the state and adjoint equations, motivated by their continuous variational forms, to increase the accuracy and possible effectiveness of the boundary shape gradient in PDE-constrained shape optimization. We consider particularly the state equation with Dirichlet boundary conditions and provide a preliminary error estimate for the correction. Numerical results show that the corrected boundary type shape gradient has comparable accuracy to that of the distributed one. Moreover, we give a theoretical explanation for the comparable numerical accuracy of the boundary type shape gradient with that of the distributed shape gradient for Neumann boundary value problems.

报告人简介：龚伟，中国科学院数学与系统科学研究院研究员，博士生导师，2009年获中国科学院数学与系统科学研究院理学博士学位，2010年获德国洪堡基金会资助赴德国汉堡大学做博士后研究，2017年受“陈景润未来之星”特优人才计划资助。在偏微分方程约束优化及最优控制问题的数学理论及数值算法等方面取得一系列重要成果。承担及参与国家重点研发计划项目、973计划项目及国家自然科学基金等多个项目。

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