Inner product free iterative solution and elimination methods for linear systems of a three-by-three block matrix form

0534467 - ÚGN 2021 RIV NL eng J - Článek v odborném periodiku
Axelsson, Owe - Liang, Z.-Z. - Kružík, Jakub - Horák, David
Inner product free iterative solution and elimination methods for linear systems of a three-by-three block matrix form.
Journal of Computational and Applied Mathematics. Roč. 383, February 2021 (2021), č. článku 113117. ISSN 0377-0427. E-ISSN 1879-1778
Grant CEP: GA MŠMT LQ1602
Institucionální podpora: RVO:68145535
Klíčová slova: PDE-constrained optimization * iterative solution * preconditioning * global communication * inner product free * parallel efficiency
Obor OECD: Applied mathematics
Impakt faktor: 2.872, rok: 2021
Způsob publikování: Omezený přístup
https://www.sciencedirect.com/science/article/pii/S0377042720304088

Large scale systems of algebraic equations are frequently solved by iterative solution methods, such as the conjugate gradient method for symmetric or a generalized conjugate gradient or generalized minimum residual method for nonsymmetric linear systems. In practice, to get an acceptable elapsed computing time when solving large scale problems, one shall use parallel computer platforms. However, such methods involve orthogonalization of search vectors which requires computation of many inner products and, hence, needs global communication of data, which will be costly in computer times. In this paper, we propose various inner product free methods, such as the Chebyshev acceleration method. We study the solution of linear systems arising from optimal control problems for PDEs, such as the edge element discretization of the time-periodic eddy current optimal control problem. Following a discretize-then-optimize scheme, the resulting linear system is of a three-by-three block matrix form. Various solution methods based on an approximate Schur complement and inner product free iterative solution methods for this linear system are analyzed and compared with an earlier used method for two-by-two block matrices with square blocks. The convergence properties and implementation details of the proposed methods are analyzed to show their effectiveness and practicality. Both serial and parallel numerical experiments are presented to further investigate the performance of the proposed methods compared with some other existing methods.
Trvalý link: http://hdl.handle.net/11104/0312661