Analysis on inexact block diagonal preconditioners for elliptic PDE-constrained optimization problems
type: Item
instance of: scholarly article
title: Analysis on inexact block diagonal preconditioners for elliptic PDE-constrained optimization problems
zbMATH Open document ID: 1402.65145
author: Na Huang
author: Changfeng Ma
publication date: 2018-08-30T00:00:00Z
review text: The authors consider an efficient preconditioning technique to a linear elliptic distributed control problem in two-dimensional spaces. By following the discretize-then-optimize approach, a Galerkin finite element method is used to transform the control problem into a linear system of saddle point form. A class of inexact block diagonal preconditioners is introduced and the lower and upper bounds of positive and negative eigenvalues of the preconditioned matrices are estimated. Based on the Cholesky decomposition of the known matrices, a lower triangular preconditioner to accelerate the minimal residual method for the reduced linear system is also analyzed. The real and complex eigenvalues of the reduced linear system are discussed. It is shown that these bounds do not rely on the regularization parameter and the eigenvalues of the matrices in the discrete system. Some numerical experiments are presented to demonstrate the effectiveness and robustness of the two new preconditioners.
reviewed by: Abdallah Bradji
Mathematics Subject Classification ID: 65N22
Mathematics Subject Classification ID: 65F08
Mathematics Subject Classification ID: 93C20
Mathematics Subject Classification ID: 65N30
Mathematics Subject Classification ID: 49J20
Mathematics Subject Classification ID: 49K20
Mathematics Subject Classification ID: 90C25
Mathematics Subject Classification ID: 15A18
Mathematics Subject Classification ID: 65F05
zbMATH DE Number: 6929639
zbMATH Keywords: PDE-constrained optimization
zbMATH Keywords: saddle point matrices
zbMATH Keywords: preconditioner
zbMATH Keywords: Cholesky decomposition
zbMATH Keywords: spectral bound
describes a project that uses: IFISS
MaRDI profile type: Publication
full work available at URL: j.camwa.2017.07.018
OpenAlex ID: W2739751763
cites work: Parallel Lagrange--Newton--Krylov--Schur Methods for PDE-Constrained Optimization. Part I: The Krylov--Schur Solver
cites work: Parallel Lagrange--Newton--Krylov--Schur Methods for PDE-Constrained Optimization. Part II: The Lagrange--Newton Solver and Its Application to Optimal Control of Steady Viscous Flows
cites work: A certified reduced basis method for parametrized elliptic optimal control problems
cites work: Reduced basis method for parametrized elliptic optimal control problems
cites work: Symmetric Indefinite Preconditioners for Saddle Point Problems with Applications to PDE-Constrained Optimization Problems
cites work: Preconditioned Conjugate Gradient Method for Optimal Control Problems with Control and State Constraints
cites work: Optimal Solvers for PDE-Constrained Optimization
cites work: Block preconditioners for elliptic PDE-constrained optimization problems
cites work: Primal-Dual Strategy for Constrained Optimal Control Problems
cites work: Primal-dual strategy for state-constrained optimal control problems
cites work: Finite elements and fast iterative solvers. With applications in incompressible fluid dynamics
cites work: Eigenvalue estimates for preconditioned saddle point matrices
cites work: Accurate projection methods for the incompressible Navier-Stokes equations
cites work: A dimensional split preconditioner for Stokes and linearized Navier-Stokes equations
cites work: Block Triangular and Skew-Hermitian Splitting Methods for Positive-Definite Linear Systems
cites work: On the eigenvalues of a class of saddle point matrices
cites work: Preconditioning for the Steady-State Navier--Stokes Equations with Low Viscosity
cites work: Algorithm 866
cites work: Fast Nonsymmetric Iterations and Preconditioning for Navier–Stokes Equations
cites work: Performance and analysis of saddle point preconditioners for the discrete steady-state Navier-Stokes equations
cites work: An inexact relaxed DPSS preconditioner for saddle point problem
cites work: A new approximation of the Schur complement in preconditioners for PDE-constrained optimization.
cites work: Preconditioners for reduced saddle point systems arising in elliptic PDE-constrained optimization problems
cites work: Sharp error bounds of some Krylov subspace methods for non-Hermitian linear systems
cites work: Solution of Sparse Indefinite Systems of Linear Equations
cites work: Parallel iterative methods for sparse linear systems
cites work: GMRES: A Generalized Minimal Residual Algorithm for Solving Nonsymmetric Linear Systems
cites work: Q5639278
DOI: 10.1016/J.CAMWA.2017.07.018
published in: Computers & Mathematics with Applications
description: scientific article; zbMATH DE number 6929639