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A low-rank matrix equation method for solving PDE-constrained optimization problems. (English) Zbl 07418123

Summary: 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.

MSC:

65F10 Iterative numerical methods for linear systems
49M41 PDE constrained optimization (numerical aspects)
65F45 Numerical methods for matrix equations
65F55 Numerical methods for low-rank matrix approximation; matrix compression

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mctoolbox; deal.ii
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References:

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