Iterative Solution Methods for Large Linear Discrete Ill-Posed Problems

1999 ◽  
pp. 313-367 ◽  
Author(s):  
Daniela Calvetti ◽  
Lothar Reichel ◽  
Qin Zhang
Keyword(s):  
Iterative Solution ◽  
Solution Methods ◽  
Ill Posed ◽  
Iterative Solution Methods

Iterative solution methods for ill-posed problems

1995 ◽  
Author(s):  
Daniela Calvetti ◽  
Lothar Reichel ◽  
Q. Zhang
Keyword(s):  
Iterative Solution ◽  
Solution Methods ◽  
Ill Posed ◽  
Iterative Solution Methods

scholarly journals Efficient Preconditioners for Large Scale Binary Cahn-Hilliard Models

2012 ◽  
Vol 12 (1) ◽  
pp. 1-22 ◽  
Author(s):  
Petia Boyanova ◽  
Minh Do-Quang ◽  
Maya Neytcheva
Keyword(s):  
Large Scale ◽  
Iterative Solution ◽  
Flow Problems ◽  
Hilliard Equation ◽  
Finite Element Discretizations ◽  
Solution Methods ◽  
Iterative Solution Methods ◽  
Preconditioned Iterative ◽  
Cahn Hilliard Equation ◽  
Element Discretizations

AbstractIn this work we consider preconditioned iterative solution methods for numerical simulations of multiphase flow problems, modelled by the Cahn-Hilliard equation. We focus on diphasic flows and the construction and efficiency of a preconditioner for the algebraic systems arising from finite element discretizations in space and the θ-method in time. The preconditioner utilises to a full extent the algebraic structure of the underlying matrices and exhibits optimal convergence and computational complexity properties. Various numerical experiments, including large scale examples, are presented as well as performance comparisons with other solution methods.

scholarly journals Superior properties of the PRESB preconditioner for operators on two-by-two block form with square blocks

2020 ◽  
Vol 146 (2) ◽  
pp. 335-368
Author(s):  
Owe Axelsson ◽  
János Karátson
Keyword(s):  
Optimal Control ◽  
Rate Of Convergence ◽  
Large Scale ◽  
Optimal Control Problems ◽  
Iterative Solution ◽  
Control Problems ◽  
Block Form ◽  
Solution Methods ◽  
Preconditioned Matrix ◽  
Iterative Solution Methods

Abstract Matrices or operators in two-by-two block form with square blocks arise in numerous important applications, such as in optimal control problems for PDEs. The problems are normally of very large scale so iterative solution methods must be used. Thereby the choice of an efficient and robust preconditioner is of crucial importance. Since some time a very efficient preconditioner, the preconditioned square block, PRESB method has been used by the authors and coauthors in various applications, in particular for optimal control problems for PDEs. It has been shown to have excellent properties, such as a very fast and robust rate of convergence that outperforms other methods. In this paper the fundamental and most important properties of the method are stressed and presented with new and extended proofs. Under certain conditions, the condition number of the preconditioned matrix is bounded by 2 or even smaller. Furthermore, under certain assumptions the rate of convergence is superlinear.

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