| 1. |
- Axelsson, Owe, et al.
(författare)
-
A black-box generalized conjugate gradient minimum residual method based on variable preconditioners and local element approximations
- 2007
-
Rapport (övrigt vetenskapligt/konstnärligt)abstract
- In order to control the accuracy of a preconditioner for an outer iterative process one often involves variable preconditioners. The variability may for instance be due to the use of inner iterations in the construction of the preconditioner. Both the outer and inner iterations may be based on some conjugate gradient type of method, e.g. generalized minimum residual methods.A background for such methods, including results about their computational complexity and rate of convergence, is given. It is then applied for a variable preconditioner arising for matrices partitioned in two-by-two block form. The matrices can be unsymmetric and also indefinite. The aim is to provide a black-box solver, applicable for all ranges of problem parameters such as coefficient jumps and anisotropy.When applying this approach for elliptic boundary value problems, in order to achieve the latter aim, it turns out to be efficient to use local element approximations of arising block matrices as preconditioners for the inner iterations.It is illustrated by numerical examples how the convergence rate of the inner-outer iteration method approaches that for the more expensive fixed preconditioner when the accuracies of the inner iterations increase
|
|
| 2. |
- Axelsson, Owe, et al.
(författare)
-
A Boundary Optimal Control Identification Problem
- 2020
-
Rapport (övrigt vetenskapligt/konstnärligt)abstract
- Optimal control methods are applied in various problems and can be efficient also for solving inverse problems, such as parameter identification and boundary control which arise in many important applications. For boundary optimal control methods one can identify conditions on an inaccessible part of the boundary by letting them play the role of a control variable function and by overimposing boundary conditions at another part of the boundary of the given domain. The paper shows a mathematical formulation of the problem, the arising (regularized) Karush-Kuhn-Tucker (KKT) system and introduces a preconditioner for this system. The spectral analysis of the preconditioner and numerical tests with preconditioning are included.
|
|
| 3. |
|
|
| 4. |
- Axelsson, Owe, et al.
(författare)
-
A general approach to analyse preconditioners for two-by-two block matrices
- 2010
-
Rapport (övrigt vetenskapligt/konstnärligt)abstract
- Two-by-two block matrices arise in various applications, such as in domain decomposition methods or, more generally, when solving boundary value problems discretized by finite elements from the separation of the node set of the mesh into 'fine' and 'coarse' nodes. Matrices with such a structure, in saddle point form arise also in mixed variable finite element methods and in constrained optimization problems.A general algebraic approach to construct, analyse and control the accuracy of preconditioners for matrices in two-by-two block form is presented. This includes both symmetric and nonsymmetric matrices, as well as indefinite matrices. The action of the preconditioners can involve element-by-element approximations and/or geometric or algebraic multigrid/multilevel methods.
|
|
| 5. |
|
|
| 6. |
|
|
| 7. |
|
|
| 8. |
- Axelsson, Owe, et al.
(författare)
-
An Efficient Preconditioning Method for State Box-Constrained Optimal Control Problems
- 2017
-
Rapport (övrigt vetenskapligt/konstnärligt)abstract
- An efficient preconditioning technique used earlier for two-by-two block matrix systems with square matrices is shown to be applicable also for a state variable box-constrained optimal control problem. The problem is penalized by a standard regularization term for the control variable and for the box-constraint, using a Moreau-Yosida penalization method. It is shown that there arises very few nonlinear iteration steps and also few iterations to solve the arising linearized equations on the fine mesh. This holds for a wide range of the penalization and discretization parameters. The arising nonlinearity can be handled with a hybrid nonlinear-linear procedure that raises the computational efficiency of the overall solution method.
|
|
| 9. |
|
|
| 10. |
|
|
