SIAM Journal on Scientific Computing, Ahead of Print.
PDE-constrained optimization problems arise in a broad number of applications such as hyperthermia cancer treatment and blood flow simulation. Discretization of the optimization problem and using a Lagrangian approach result in a large-scale saddle-point system, which is challenging to solve, and acquiring a full space-time solution is often infeasible. We present a new framework to efficiently compute a low-rank approximation to the solution by reformulating the KKT system into a Sylvester-like matrix equation. This matrix equation is subsequently projected onto a small subspace via an iterative rational Krylov method, and we obtain a reduced problem by imposing a Galerkin condition on its residual. In our work we discuss implementation details and dependence on the various problem parameters. Numerical experiments illustrate the performance of the new strategy also when compared to other low-rank approaches.

中文翻译：

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一种求解偏微分方程约束优化问题的低秩矩阵方程方法

SIAM 科学计算杂志，提前印刷。
PDE 约束优化问题出现在大量应用中，例如热疗癌症治疗和血流模拟。优化问题的离散化和使用拉格朗日方法会导致大规模的鞍点系统，这很难解决，并且获得完整的时空解通常是不可行的。我们提出了一个新的框架，通过将 KKT 系统重新构造为类似 Sylvester 的矩阵方程来有效地计算解的低秩近似。该矩阵方程随后通过迭代有理 Krylov 方法投影到一个小的子空间，我们通过对其残差施加伽辽金条件来获得简化的问题。在我们的工作中，我们讨论了实现细节和对各种问题参数的依赖。