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PDE-constrained optimization problems arise in a broad number of applicationssuch as hyperthermia cancer treatment or blood flow simulation. Discretizationof the optimization problem and using a Lagrangian approach result in alarge-scale saddle-point system, which is challenging to solve, and acquiring afull space-time solution is often infeasible. We present a new framework toefficiently compute a low-rank approximation to the solution by reformulatingthe KKT system into a Sylvester-like matrix equation. This matrix equation issubsequently projected onto a small subspace via an iterative rational Krylovmethod and we obtain a reduced problem by imposing a Galerkin condition on itsresidual. In our work we discuss implementation details and dependence on thevarious problem parameters. Numerical experiments illustrate the performance ofthe new strategy also when compared to other low-rank approaches.

A low-rank matrix equation method for solving PDE-constrained optimization problems