We consider a PDE-constrained optimization problem of tracking type with parabolic state equation. The solution to the problem is characterized by the Karush-Kuhn-Tucker (KKT) system, which we formulate using a strong variational formulation of the state equation and a super weak formulation of the adjoined state equation. This allows us to propose a preconditioner that is robust both in the regularization and the diffusion parameter. In order to discretize the problem, we use Isogeometric Analysis since it allows the construction of sufficiently smooth basis functions effortlessly. To realize the preconditioner, one has to solve a problem over the whole space time cylinder that is elliptic with respect to certain non-standard norms. Using a fast diagonalization approach in time, we reformulate the problem as a collection of elliptic problems in space only. These problems are not only smaller, but our approach also allows to solve them in a time-parallel way. We show the efficiency of the preconditioner by rigorous analysis and illustrate it with numerical experiments.

翻译：我们考虑一类具有抛物型状态方程的跟踪型偏微分方程约束优化问题。该问题的解由Karush-Kuhn-Tucker（KKT）系统表征，我们通过状态方程的强变分形式与伴随状态方程的超弱形式构建该系统。这使得我们能够提出一种在正则化参数和扩散参数下均具有鲁棒性的预处理器。为离散化该问题，我们采用等几何分析方法，因其可便捷地构造充分光滑的基函数。为实现该预处理器，需要在整个时空柱域上求解一个在特定非标准范数下呈椭圆型的问题。通过时间维度的快速对角化方法，我们将该问题重构为一系列仅含空间维度的椭圆型问题。这些问题不仅规模更小，而且我们的方法支持以时间并行的方式求解。我们通过严格的理论分析证明了该预处理器的有效性，并通过数值实验加以验证。