In this paper we will consider distributed Linear-Quadratic Optimal Control Problems dealing with Advection-Diffusion PDEs for high values of the P\'eclet number. In this situation, computational instabilities occur, both for steady and unsteady cases. A Streamline Upwind Petrov-Galerkin technique is used in the optimality system to overcome these unpleasant effects. We will apply a finite element method discretization in a optimize-then-discretize approach. For the parabolic case, a space-time framework will be considered and stabilization will also occur in the bilinear forms involving time derivatives. Then we will build Reduced Order Models on this discretization procedure and two possible settings can be analyzed: whether or not stabilization is needed in the online phase, too. In order to build the reduced bases for state, control, and adjoint variables we will consider a Proper Orthogonal Decomposition algorithm in a partitioned approach. The discussion is supported by computational experiments, where relative errors between the FEM and ROM solutions are studied together with the respective computational times.
翻译:在本文中, 我们将考虑分布式线性- 二次曲线最佳控制问题, 涉及对流- 扩散 PDE 的分布式线性- 二次曲线优化控制问题, 其数值为 P\'eclet 编号的高值 。 在这种情况下, 计算不稳定性会发生, 无论是稳定型还是不稳定型。 在优化系统中, 使用精简式Upliste Petrov- Galerkin 技术来克服这些不愉快的影响 。 我们将采用一种最优化的、 时分解式的方法, 应用一种有限的元素分解法 。 对于parboli 案例, 将考虑一个空间- 时间框架, 并在涉及时间衍生物的双线形式中实现稳定化 。 然后我们将在这个离散化程序上建立减序模型, 并且可以分析两种可能的设置 : 是否也需要在在线阶段稳定化 。 为了在分解式方法中建立缩小的状态、 和连接变量基础, 我们将考虑一种适当的分解算法 。 讨论得到计算实验的支持, 在这种实验中, FEM 和 ROM 解决方案 之间的相对错误会与相应的计算与各自的计算时间一起研究 。