Discrete Concepts in PDE Constrained Optimization
Discrete Concepts in PDE Constrained Optimization
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DOI：
10.1007/978-1-4020-8839-1_3
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发表时间：
2009-01-01
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影响因子：
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通讯作者：
Hinze, Michael
中科院分区：
文献类型：
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作者：
Hinze, Michael
在本章中，我们将介绍带PDE约束的优化问题的离散概念。作为模型的状态，我们考虑椭圆和抛物偏微分方程，这是很好地理解从分析的角度来看。这允许在离散化中关注结构方面。我们讨论和比较的方法首先离散化，然后优化和首先优化，然后离散化，并引入了变分离散的概念，避免显式离散的控制。我们调查问题的控制一般的约束，也考虑逐点的状态，并对状态的梯度上的界限。我们提出了误差分析的变分离散的概念，并完成我们的分析结果与数值例子，证实了我们的分析结果。
In the present chapter we give an introduction to discrete concepts for optimization problems with PDE constraints. As models for the state we consider elliptic and parabolic PDEs which are well understood from the analytical point of view. This allows to focus on structural aspects in discretization. We discuss and compare the approaches First discretize, then optimize and First optimize, then discretize, and introduce a variational discrete concept which avoids explicit discretization of the controls. We investigate problems with general constraints on the control, and also consider pointwise bounds on the state, and on the gradient of the state. We present error analysis for the variational discrete concept and accomplish our analytical findings with numerical examples which confirm our analytical results.