Parallel Solution Methods and Preconditioners for Evolution Equations

0495421 - ÚGN 2019 RIV LT eng J - Journal Article
Axelsson, Owe - Neytcheva, M. - Liang, Z. Z.
Parallel Solution Methods and Preconditioners for Evolution Equations.
Mathematical Modeling and Analysis. Roč. 23, č. 2 (2018), s. 287-308. ISSN 1392-6292. E-ISSN 1648-3510
R&D Projects: GA MŠMT LD15105; GA MŠMT LQ1602
Institutional support: RVO:68145535
Keywords : parallel solution * evolution equation * preconditioning * PDE-constrained optimization
OECD category: Applied mathematics
Impact factor: 1.038, year: 2018
https://www.mla.vgtu.lt/index.php/MMA/article/view/1424/1134

The recent development of the high performance computer platforms shows a clear trend towards heterogeneity and hierarchy. In order to utilize the computational power, particular attention must be paid to finding new algorithms or adjust existing ones so that they better match the HPC computer architecture. In this work we consider an alternative to classical time-stepping methods based on use of time-harmonic properties and discuss solution approaches that allow efficient utilization of modern HPC resources. The method in focus is based on a truncated Fourier expansion of the solution of an evolutionary problem. The analysis is done for linear equations and it is remarked on the possibility to use two- or multilevel mesh methods for nonlinear problems, which can enable further, even higher degree of parallelization.
The arising block matrix system to be solved admits a two-by-two block form with square blocks, for which a very efficient preconditioner exists. It leads to tight eigenvalue bounds for the preconditioned matrix and, hence, to a very fast convergence of a preconditioned Krylov subspace or iterative refinement method. The analytical background is shown as well as some illustrating numerical examples.
Permanent Link: http://hdl.handle.net/11104/0288410