High order approximations of the operator Lyapunov equation have low rank
No Thumbnail Available
A1 Alkuperäisartikkeli tieteellisessä aikakauslehdessä
BIT Numerical Mathematics, Volume 62, issue 4
AbstractWe present a low-rank greedily adapted hp-finite element algorithm for computing an approximation to the solution of the Lyapunov operator equation. We show that there is a hidden regularity in eigenfunctions of the solution of the Lyapunov equation which can be utilized to justify the use of high order finite element spaces. Our numerical experiments indicate that we achieve eight figures of accuracy for computing the trace of the solution of the Lyapunov equation posed in a dumbbell-domain using a finite element space of dimension of only 10 4 degrees of freedom. Even more surprising is the observation that hp-refinement has an effect of reducing the rank of the approximation of the solution.
Funding Information: This research was funded by the Hrvatska Zaklada za Znanost (Croatian Science Foundation) under the Grant IP-2019-04-6268 - Randomized low-rank algorithms and applications to parameter dependent problems. Publisher Copyright: © 2022, The Author(s).
Exponential decay, hp-finite element methods, Low-rank approximation, Lyapunov equation
Grubišić , L & Hakula , H 2022 , ' High order approximations of the operator Lyapunov equation have low rank ' , BIT Numerical Mathematics , vol. 62 , no. 4 , pp. 1433-1459 . https://doi.org/10.1007/s10543-022-00917-z