High order approximations of the operator Lyapunov equation have low rank

Luka Grubišić; Harri Hakula

doi:10.1007/s10543-022-00917-z

High order approximations of the operator Lyapunov equation have low rank

Luka Grubišić
, Harri Hakula^*

^*Tämän työn vastaava kirjoittaja

University of Zagreb

Tutkimustuotos: Lehtiartikkeli › Article › Scientific › vertaisarvioitu

89 Lataukset (Pure)

Abstrakti

We 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.

Alkuperäiskieli	Englanti
Sivut	1433-1459
Sivumäärä	27
Julkaisu	BIT Numerical Mathematics
Vuosikerta	62
Numero	4
Varhainen verkossa julkaisun päivämäärä	7 huhtik. 2022
DOI - pysyväislinkit	https://doi.org/10.1007/s10543-022-00917-z
Tila	Julkaistu - jouluk. 2022
OKM-julkaisutyyppi	A1 Alkuperäisartikkeli tieteellisessä aikakauslehdessä

Rahoitus

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.

Pääsy asiakirjaan

10.1007/s10543-022-00917-zLisenssi: CC BY

High order approximations of the operator Lyapunov equation have low rankFinal published version, 2,01 MBLisenssi: CC BY

Muut tiedostot ja linkit

Linkki julkaisuun Scopuksessa

Siteeraa tätä

@article{a516f82b3828490aa4d4dde517eba735,

title = "High order approximations of the operator Lyapunov equation have low rank",

abstract = "We 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.",

keywords = "Exponential decay, hp-finite element methods, Low-rank approximation, Lyapunov equation",

author = "Luka Grubi{\v s}i{\'c} and Harri Hakula",

note = "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: {\textcopyright} 2022, The Author(s).",

year = "2022",

month = dec,

doi = "10.1007/s10543-022-00917-z",

language = "English",

volume = "62",

pages = "1433--1459",

journal = "BIT Numerical Mathematics",

issn = "0006-3835",

publisher = "Springer",

number = "4",

}

TY - JOUR

T1 - High order approximations of the operator Lyapunov equation have low rank

AU - Grubišić, Luka

AU - Hakula, Harri

N1 - 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).

PY - 2022/12

Y1 - 2022/12

N2 - We 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.

AB - We 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.

KW - Exponential decay

KW - hp-finite element methods

KW - Low-rank approximation

KW - Lyapunov equation

UR - https://www.scopus.com/pages/publications/85127679179

U2 - 10.1007/s10543-022-00917-z

DO - 10.1007/s10543-022-00917-z

M3 - Article

AN - SCOPUS:85127679179

SN - 0006-3835

VL - 62

SP - 1433

EP - 1459

JO - BIT Numerical Mathematics

JF - BIT Numerical Mathematics

IS - 4

ER -

High order approximations of the operator Lyapunov equation have low rank

Abstrakti

Rahoitus

Pääsy asiakirjaan

Muut tiedostot ja linkit

Sormenjälki

Siteeraa tätä