Abstract
In aggregation based algebraic multigrids, the current schemes are to construct the grid hierarchy from bottom to top, where several nodes on the finer level are clustered into a node on the coarser level step by step. Therefore this kind of scheme is mainly based on local information. In this paper, we present a new aggregation scheme, where the grid hierarchy is formed from top to bottom in a natural way. The adjacent graph of the original coefficient matrix is partitioned first, and then each part is recursively partitioned until some limitations are met for a certain level. Then the grid hierarchy is formed based on the global information, which is completely different from the classical ones. When partitioning graphs, any kind of method can be used, including those based on coordinate information and those based on the element of the matrix only, such as the methods provided in the software package METIS. Finally, the new scheme is validated from the solution of some discrete two-dimensional systems with preconditioned conjugate gradient iterations.
This is a preview of subscription content, log in via an institution to check access.
Access this chapter
Tax calculation will be finalised at checkout
Purchases are for personal use only
References
Notay, Y.: Aggregation-based algebraic multilevel preconditioning. SIAM J. Matrix Anal. Appl. 27(4), 998–1018 (2006)
Kim, H., Xu, J., Zikatanov, L.: A multigrid method based on graph matching for convection-diffusion equations. Numer. Linear Algebra Appl. 10, 181–195 (2003)
Notay, Y.: Aggregation-based algebraic multigrid for convection-diffusion equations. SIAM J. Sci. Comput. 34(4), A2288–A2316 (2012)
D’Ambra, P., Buttari, A., di Serafino, D., Filippone, S., Gentile, S., Ucar, B.: A novel aggregation method based on graph matching for algebraic multigrid preconditioning of sparse linear systems. In: International Conference on Preconditioning Techniques for Scientific & Industrial Applications, May 2011, Bordeaux, France (2011)
Dendy Jr., J.E., Moulton, J.D.: Black Box Multigrid with coarsening by a factor of three. Numer. Linear Algebra Appl. 17(2–3), 577–598 (2010)
Vanek, P., Mandel, J., Brezina, M.: Algebraic multigrid by smoothed aggregation for second order and fourth order elliptic problems. Computing 56, 179–196 (1996)
Kumar, P.: Aggregation based on graph matching and inexact coarse grid solve for algebraic two grid. Int. J. Comput. Math. 91(5), 1061–1081 (2014)
Wu, J.P., Song, J.Q., Zhang, W.M., Ma, H.F.: Coarse grid correction to domain decomposition based preconditioners for meso-scale simulation of concrete. Appl. Mech. Mater. 204–208, 4683–4687 (2012)
Chen, M.H., Greenbaum, A.: Analysis of an aggregation-based algebraic two-grid method for a rotated anisotropic diffusion problem. Numer. Linear Algebra Appl. 22(4), 681–701 (2015)
Braess, D.: Towards algebraic multigrid for elliptic problems of second order. Computing 55, 379–393 (1995)
Deng, L.J., Huang, T.Z., Zhao, X.L., Zhao, L., Wang, S.: An economical aggregation algorithm for algebraic multigrid. (AMG). J. Comput. Anal. Appl. 16(1), 181–198 (2014)
Wu, J.P., Yin, F.K., Peng, J., Yang, J.H.: Research on two-point aggregated algebraic multigrid preconditioning methods. In: International Conference on Computer Engineering and Information System [CEIS 2016], Shanghai, China (2016)
Saad, Y.: Iterative methods for Sparse Linear Systems. PWS Pub. Co., Boston (1996)
Wagner, C.: Introduction to algebraic multigrid, Course Notes, University of Heidelberg (1998/1999). http://www.iwr.uni-heidelberg.de/~Christian.Wagner/
Karypis, G., Kumar, G.: MeTiS – a software package for partitioning unstructured graphs, partitioning meshes, and computing fill-reducing orderings of sparse matrices – Version 4.0, Technical report, University of Minnesota, September 1998
Acknowledgment
This work is funded by NSFC(61379022).
Author information
Authors and Affiliations
Corresponding authors
Editor information
Editors and Affiliations
Rights and permissions
Copyright information
© 2018 Springer Nature Singapore Pte Ltd.
About this paper
Cite this paper
Wu, J., Yin, F., Peng, J., Yang, J. (2018). An Algebraic Multigrid Preconditioner Based on Aggregation from Top to Bottom. In: Yuan, H., Geng, J., Liu, C., Bian, F., Surapunt, T. (eds) Geo-Spatial Knowledge and Intelligence. GSKI 2017. Communications in Computer and Information Science, vol 849. Springer, Singapore. https://doi.org/10.1007/978-981-13-0896-3_20
Download citation
DOI: https://doi.org/10.1007/978-981-13-0896-3_20
Published:
Publisher Name: Springer, Singapore
Print ISBN: 978-981-13-0895-6
Online ISBN: 978-981-13-0896-3
eBook Packages: Computer ScienceComputer Science (R0)