2020/05/29 by Alexandra Bünger, Valeria Simoncini, Bünger, Alexandra +3
Computer Science · Engineering · Mathematics · #15A69 #65F10 #65F50 #93C20 #FOS: Mathematics #Matrix Theory and Algorithms #Numerical Analysis (math.NA) #Sparse and Compressive Sensing Techniques #Statistical and numerical algorithms
paper · pdf · doi:10.48550/arxiv.2005.14499
openalex publication_date 2020/05/29 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
PDE-constrained optimization problems arise in a broad number of applications\nsuch as hyperthermia cancer treatment or blood flow simulation. Discretization\nof the optimization problem and using a Lagrangian approach result in a\nlarge-scale saddle-point system, which is challenging to solve, and acquiring a\nfull space-time solution is often infeasible. We present a new framework to\nefficiently compute a low-rank approximation to the solution by reformulating\nthe KKT system into a Sylvester-like matrix equation. This matrix equation is\nsubsequently projected onto a small subspace via an iterative rational Krylov\nmethod and we obtain a reduced problem by imposing a Galerkin condition on its\nresidual. In our work we discuss implementation details and dependence on the\nvarious problem parameters. Numerical experiments illustrate the performance of\nthe new strategy also when compared to other low-rank approaches.\n