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Reduced Basis Method for Parametrized Elliptic Optimal Control Problems

机译:参数化椭圆最优控制问题的简化基方法

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摘要

We propose a suitable model reduction paradigm-the certified reduced basis method (RB)-for the rapid and reliable solution of parametrized optimal control problems governed by partial differential equations. In particular, we develop the methodology for parametrized quadratic optimization problems with elliptic equations as a constraint and infinite-dimensional control variable. First, we recast the optimal control problem in the framework of saddle-point problems in order to take advantage of the already developed RB theory for Stokes-type problems. Then, the usual ingredients of the RB methodology are called into play: a Galerkin projection onto a low-dimensional space of basis functions properly selected by an adaptive procedure; an affine parametric dependence enabling one to perform competitive offline-online splitting in the computational procedure; and an efficient and rigorous a posteriori error estimate on the state, control, and adjoint variables as well as on the cost functional. Finally, we address some numerical tests that confirm our theoretical results and show the efficiency of the proposed technique. Copyright © by SIAM. Unauthorized reproduction of this article is prohibited.
机译:我们提出了一种合适的模型简化范例-认证的简化基础方法(RB)-用于快速可靠地解决由偏微分方程控制的参数化最优控制问题。特别是,我们开发了以椭圆方程为约束和无穷维控制变量的参数化二次优化问题的方法。首先,我们在鞍点问题的框架内重铸最优控制问题,以利用已经开发的针对斯托克斯型问题的RB理论。然后,RB方法的通常要素被发挥作用:一个Galerkin投影到基函数的低维空间上,该基函数通过自适应过程适当地选择;一种仿射参数依赖性,使人们能够在计算过程中进行有竞争力的离线-在线分割;以及对状态,控制和伴随变量以及成本函数的高效且严格的后验误差估计。最后,我们进行一些数值测试,这些实验证实了我们的理论结果并显示了所提出技术的效率。 SIAM版权所有©。未经授权不得转载本文章。

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