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Guaranteed lower bounds for cost functionals of time-periodic parabolic optimization problems
Computers & Mathematics with Applications ( IF 2.9 ) Pub Date : 2020-06-16 , DOI: 10.1016/j.camwa.2020.04.021
Monika Wolfmayr

In this paper, a new technique is shown for deriving computable, guaranteed lower bounds of functional type (minorants) for two different cost functionals subject to a parabolic time-periodic boundary value problem. Together with previous results on upper bounds (majorants) for one of the cost functionals, both minorants and majorants lead to two-sided estimates of functional type for the optimal control problem. Both upper and lower bounds are derived for the second new cost functional subject to the same parabolic PDE-constraints, but where the target is a desired gradient. The time-periodic optimal control problems are discretized by the multiharmonic finite element method leading to large systems of linear equations having a saddle point structure. The derivation of preconditioners for the minimal residual method for the new optimization problem is discussed in more detail. Finally, several numerical experiments for both optimal control problems are presented confirming the theoretical results obtained. This work provides the basis for an adaptive scheme for time-periodic optimization problems.



中文翻译:

时间周期抛物线优化问题的成本函数的有保证的下界

在本文中,展示了一种新技术,用于推导两个不同成本函数在抛物线时间周期边值问题下的可计算的,有保证的函数类型下限(线性)。再加上先前关于成本函数之一的上限(主要)的结果,次要和主要都导致针对最优控制问题的函数类型的双面估计。对于具有相同抛物线式PDE约束的第二个新成本函数主题,可以导出上限和下限,但目标是所需的梯度。时间周期最优控制问题通过多谐波有限元方法离散化,从而导致线性方程组具有鞍点结构。对于新优化问题的最小残差方法,预处理器的推导将得到更详细的讨论。最后,针对这两个最优控制问题进行了一些数值实验,证实了所获得的理论结果。这项工作为时间周期优化问题的自适应方案提供了基础。

更新日期:2020-06-16
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