报告题目：Optimal convergence of finite element approximation to an optimization problem with PDE constraint

报 告 人：龚伟 副研究员 中国科学院数学与系统科学研究院

报告时间：2021年12月8日 15:00-16:00

报告地点：腾讯会议 ID：261442150 会议密码：9999

校内联系人：宋海明 songhaiming@jlu.edu.cn

报告摘要：In this talk we consider the optimal convergence of finite element approximation

to an optimization problem with PDE constraint. Specifically, we consider an elliptic distributed optimal control problem without control constraints, which can also be viewed as a regularized inverse source problem. The main contributions are two-fold. First, we derive a priori and a posteriori error estimates for the optimization problems, under an appropriately chosen norm that allows us to establish an isomorphism between the solution space and its dual. These results yield error estimates with explicit dependence on the regularization parameter so that the constants appeared in the derivation are independent of it. Second, we prove the contraction property and rate optimality for the adaptive algorithm with respect to the error estimator and solution errors between the adaptive finite element solutions and the continuous solutions. Extensive numerical experiments are presented that confirm our theoretical results.

龚伟，中国科学院数学与系统科学研究院副研究员，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。承担及参与国家自然科学基金、国家重点研发计划项目、973计划项目等多个项目。主要研究方向包括：有限元方法，偏微分方程约束最优控制问题的分析与计算，形状及拓扑优化，数据同化等。