论文标题
一种用于求解PDE受限优化问题的低级矩阵方程方法
A low-rank matrix equation method for solving PDE-constrained optimization problems
论文作者
论文摘要
PDE约束的优化问题在大量应用中（例如高温癌治疗或血流模拟）出现。优化问题和使用拉格朗日方法的离散化会导致大规模的鞍点系统，该系统难以解决，并且获得完整的时空解决方案通常是不可行的。我们提出了一个新的框架，通过将KKT系统重新定义为Sylvester样矩阵方程，以有效地计算出对解决方案的低级近似值。该基质方程随后通过迭代有理Krylov方法投射到一个小子空间上，我们通过在其残留物上施加盖尔金条件来减少问题。在我们的工作中，我们讨论了实施细节以及对各种问题参数的依赖。与其他低级方法相比，数值实验说明了新策略的性能。
PDE-constrained optimization problems arise in a broad number of applications such as hyperthermia cancer treatment or 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.
