A spatial domain decomposition method for parabolic optimal control problems
A spatial domain decomposition method for parabolic optimal control problems
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DOI：
10.1016/j.cam.2006.02.002
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发表时间：
2007-04
影响因子：
2.4
通讯作者：
M. Heinkenschloss;M. Herty
中科院分区：
文献类型：
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作者：
M. Heinkenschloss;M. Herty
我们提出了一种非重叠空间域分解方法来解决线性二次抛物线最优控制问题。空间域被分解为不重叠的子域。原始抛物线最优控制问题被分解为时空圆柱子域上提出的较小问题，其中辅助状态和伴随变量作为时空界面边界上的狄利克雷边界条件施加。子域问题通过 Robin 传输条件耦合。这导致舒尔补方程，其中未知数是时空界面边界上的辅助状态伴随变量。 Schur 补算子是时空子域 Schur 补算子的和。这些子域Schur补算子的应用等价于子域抛物线最优控制问题的求解。子域 Schur 补算子被证明是可逆的，并且它们的逆的应用等价于相关子域抛物线最优控制问题的解。我们为 Schur 补系统引入了一系列新的 Neumann-Neumann 型预处理器，包括几种不同的粗网格校正。我们将预处理器的数值性能与 Benamou 最近引入的替代方法进行了比较。
We present a non-overlapping spatial domain decomposition method for the solution of linear–quadratic parabolic optimal control problems. The spatial domain is decomposed into non-overlapping subdomains. The original parabolic optimal control problem is decomposed into smaller problems posed on space–time cylinder subdomains with auxiliary state and adjoint variables imposed as Dirichlet boundary conditions on the space–time interface boundary. The subdomain problems are coupled through Robin transmission conditions. This leads to a Schur complement equation in which the unknowns are the auxiliary state adjoint variables on the space-time interface boundary. The Schur complement operator is the sum of space–time subdomain Schur complement operators. The application of these subdomain Schur complement operators is equivalent to the solution of an subdomain parabolic optimal control problem. The subdomain Schur complement operators are shown to be invertible and the application of their inverses is equivalent to the solution of a related subdomain parabolic optimal control problem. We introduce a new family of Neumann–Neumann type preconditioners for the Schur complement system including several different coarse grid corrections. We compare the numerical performance of our preconditioners with an alternative approach recently introduced by Benamou.