Additive Schwarz solvers for hp-FEM discretizations of PDE-constrained optimzation problems
Additive Schwarz solvers for hp-FEM discretizations of PDE-constrained optimzation problems
Dokument öffnen (545.5KB)Datum der Veröffentlichung
10.2016Verlag / herausgebende Einrichtung
Institut für Numerische Simulation (INS)Verlagsort
BonnMetadaten
Zur Langanzeige
Zitierbarer Link (Handle URI) 
https://hdl.handle.net/20.500.11811/11862
Inhalt
In this paper, we investigate the minimization of a quadratic functional subject to a boundary value problem of a second order linear elliptic partial differential equation. There are no inequality constraints for state and control. This problem is discretized by hp-finite elements. The main focus of this talk is the development of efficient solution methods for the corresponding system of linear algebraic equations. From the literature it is known that this system is symmetric and indefinite. This paper considers two different iterative solvers
• a conjugate gradient (CG) method in a special inner product, following Schöberl/Zulehner, and
• the minimal residual method.
In both methods, efficient preconditioners for finite element mass and stiffness matrix acclerate the convergence speed of the underlying iterative method. This contribution presents overlapping hp- FEM preconditioners for mass and stiffness matrix. Optimal condition number estimates are proved. Robustness with respect to the regularization parameter can be shown for the CG-method. Finally several numerical examples show the efficiency of the presented algorithm.
• a conjugate gradient (CG) method in a special inner product, following Schöberl/Zulehner, and
• the minimal residual method.
In both methods, efficient preconditioners for finite element mass and stiffness matrix acclerate the convergence speed of the underlying iterative method. This contribution presents overlapping hp- FEM preconditioners for mass and stiffness matrix. Optimal condition number estimates are proved. Robustness with respect to the regularization parameter can be shown for the CG-method. Finally several numerical examples show the efficiency of the presented algorithm.
Klassifikation (DDC)
510 Mathematik 518 Numerische AnalysisErstpublikation
https://ins.uni-bonn.de/publication/preprintsReihe, Nummer
INS Preprints ; 1625Beuchler, Sven; Hofer, Katharina: Additive Schwarz solvers for hp-FEM discretizations of PDE-constrained optimzation problems. In: INS Preprints, 1625.
Online-Ausgabe in bonndoc: https://hdl.handle.net/20.500.11811/11862
Online-Ausgabe in bonndoc: https://hdl.handle.net/20.500.11811/11862
@unpublished{handle:20.500.11811/11862,
author = {{Sven Beuchler} and {Katharina Hofer}},
title = {Additive Schwarz solvers for hp-FEM discretizations of PDE-constrained optimzation problems},
publisher = {Institut für Numerische Simulation (INS)},
year = 2016,
month = oct,
INS Preprints},
volume = 1625,
note = {In this paper, we investigate the minimization of a quadratic functional subject to a boundary value problem of a second order linear elliptic partial differential equation. There are no inequality constraints for state and control. This problem is discretized by hp-finite elements. The main focus of this talk is the development of efficient solution methods for the corresponding system of linear algebraic equations. From the literature it is known that this system is symmetric and indefinite. This paper considers two different iterative solvers
• a conjugate gradient (CG) method in a special inner product, following Schöberl/Zulehner, and
• the minimal residual method.
In both methods, efficient preconditioners for finite element mass and stiffness matrix acclerate the convergence speed of the underlying iterative method. This contribution presents overlapping hp- FEM preconditioners for mass and stiffness matrix. Optimal condition number estimates are proved. Robustness with respect to the regularization parameter can be shown for the CG-method. Finally several numerical examples show the efficiency of the presented algorithm.},
url = {https://hdl.handle.net/20.500.11811/11862}
}
author = {{Sven Beuchler} and {Katharina Hofer}},
title = {Additive Schwarz solvers for hp-FEM discretizations of PDE-constrained optimzation problems},
publisher = {Institut für Numerische Simulation (INS)},
year = 2016,
month = oct,
INS Preprints},
volume = 1625,
note = {In this paper, we investigate the minimization of a quadratic functional subject to a boundary value problem of a second order linear elliptic partial differential equation. There are no inequality constraints for state and control. This problem is discretized by hp-finite elements. The main focus of this talk is the development of efficient solution methods for the corresponding system of linear algebraic equations. From the literature it is known that this system is symmetric and indefinite. This paper considers two different iterative solvers
• a conjugate gradient (CG) method in a special inner product, following Schöberl/Zulehner, and
• the minimal residual method.
In both methods, efficient preconditioners for finite element mass and stiffness matrix acclerate the convergence speed of the underlying iterative method. This contribution presents overlapping hp- FEM preconditioners for mass and stiffness matrix. Optimal condition number estimates are proved. Robustness with respect to the regularization parameter can be shown for the CG-method. Finally several numerical examples show the efficiency of the presented algorithm.},
url = {https://hdl.handle.net/20.500.11811/11862}
}
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