Preconditioning of two-by-two block matrix systems with square matrix blocks, with applications

Axelsson, Owe. "Preconditioning of two-by-two block matrix systems with square matrix blocks, with applications." Applications of Mathematics 62.6 (2017): 537-559. <http://eudml.org/doc/294273>.

@article{Axelsson2017,
abstract = {Two-by-two block matrices of special form with square matrix blocks arise in important applications, such as in optimal control of partial differential equations and in high order time integration methods. Two solution methods involving very efficient preconditioned matrices, one based on a Schur complement reduction of the given system and one based on a transformation matrix with a perturbation of one of the given matrix blocks are presented. The first method involves an additional inner solution with the pivot matrix block but gives a very tight condition number bound when applied for a time integration method. The second method does not involve this matrix block but only inner solutions with a linear combination of the pivot block and the off-diagonal matrix blocks. Both the methods give small condition number bounds that hold uniformly in all parameters involved in the problem, i.e. are fully robust. The paper presents shorter proofs, extended and new results compared to earlier publications.},
author = {Axelsson, Owe},
journal = {Applications of Mathematics},
keywords = {preconditioning; Schur complement; transformation; optimal control; implicit time integration},
language = {eng},
number = {6},
pages = {537-559},
publisher = {Institute of Mathematics, Academy of Sciences of the Czech Republic},
title = {Preconditioning of two-by-two block matrix systems with square matrix blocks, with applications},
url = {http://eudml.org/doc/294273},
volume = {62},
year = {2017},
}

TY - JOUR
AU - Axelsson, Owe
TI - Preconditioning of two-by-two block matrix systems with square matrix blocks, with applications
JO - Applications of Mathematics
PY - 2017
PB - Institute of Mathematics, Academy of Sciences of the Czech Republic
VL - 62
IS - 6
SP - 537
EP - 559
AB - Two-by-two block matrices of special form with square matrix blocks arise in important applications, such as in optimal control of partial differential equations and in high order time integration methods. Two solution methods involving very efficient preconditioned matrices, one based on a Schur complement reduction of the given system and one based on a transformation matrix with a perturbation of one of the given matrix blocks are presented. The first method involves an additional inner solution with the pivot matrix block but gives a very tight condition number bound when applied for a time integration method. The second method does not involve this matrix block but only inner solutions with a linear combination of the pivot block and the off-diagonal matrix blocks. Both the methods give small condition number bounds that hold uniformly in all parameters involved in the problem, i.e. are fully robust. The paper presents shorter proofs, extended and new results compared to earlier publications.
LA - eng
KW - preconditioning; Schur complement; transformation; optimal control; implicit time integration
UR - http://eudml.org/doc/294273
ER -