Certified Reduced Basis Methods for Parametrized Distributed Elliptic Optimal Control Problems with Control Constraints
Certified Reduced Basis Methods for Parametrized Distributed Elliptic Optimal Control Problems with Control Constraints
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DOI：
10.1137/16m1059898
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发表时间：
2016-12
期刊：
影响因子：
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通讯作者：
E. Bader;Mark Kärcher;M. Grepl;K. Veroy
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文献类型：
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作者：
E. Bader;Mark Kärcher;M. Grepl;K. Veroy
本文用约化基方法有效可靠地解决了标量强制椭圆型偏微分方程参数化最优控制问题。我们考虑了具有分布控制和单向控制约束的标准线性二次问题集。对于这类问题，我们提出了两种不同的简化基近似方法和相应的误差估计方法。在第一种方法中，我们直接考虑所得到的最优性系统，对状态乘子、伴随乘子、控制乘子和拉格朗日乘子引入适当的约化基近似，并使用投影方法来估计约化最优控制中的误差。对于第二种方法，我们首先用松弛变量重新描述最优控制问题，然后通过适当地限制解空间来建立松弛问题的降维基近似，最后推导出基于松弛的最优控制的误差界。我们讨论了B和B的优点和缺点。
In this paper, we employ the reduced basis method for the efficient and reliable solution of parametrized optimal control problems governed by scalar coercive elliptic partial differential equations. We consider the standard linear-quadratic problem setting with distributed control and unilateral control constraints. For this problem class, we propose two different reduced basis approximations and associated error estimation procedures. In our first approach, we directly consider the resulting optimality system, introduce suitable reduced basis approximations for the state, adjoint, control, and Lagrange multipliers, and use a projection approach to bound the error in the reduced optimal control. For our second approach, we first reformulate the optimal control problem using a slack variable, then develop a reduced basis approximation for the slack problem by suitably restricting the solution space, and finally derive error bounds for the slack based optimal control. We discuss benefits and drawbacks of b...