H-FAINV: hierarchically factored approximate inverse preconditioners

Computing and Visualization in Science

Volumn 17, Issue 3, 2015, Pages 135-150

  Kriemann, R ^a   Le Borne, S ^b  

^a MAX PLANCK INSTITUTE FOR MATHEMATICS IN THE SCIENCES   (Germany)

^b HAMBURG UNIVERSITY OF TECHNOLOGY   (Germany)

Author keywords

Approximate factored inverse; Hierarchical matrices; Preconditioning

Indexed keywords

APPROXIMATION THEORY; INVERSE PROBLEMS; ITERATIVE METHODS;

APPROXIMATE FACTORED INVERSE; APPROXIMATE INVERSE; APPROXIMATION APPROACH; COMPUTATIONALLY EFFICIENT; HIERARCHICAL MATRICES; LOW RANK APPROXIMATIONS; PRECONDITIONING; SPARSE APPROXIMATE INVERSE;

MATRIX ALGEBRA;

EID: 84955752798     PISSN: 14329360     EISSN: 14330369     Source Type: Journal    
DOI: 10.1007/s00791-015-0254-y     Document Type: Article

References (26)

1. Bangerth, W., Hartmann, R., Kanschat, G.: deal.II—a general purpose object oriented finite element library. ACM Trans. Math. Softw. 33(4), 24/1–24/27 (2007)
2. Bangerth, W., Heister, T., Heltai, L., Kanschat, G., Kronbichler, M., Maier, M., Turcksin, B., Young, T.D.: The deal.ii library, version 8.1. arXiv:1312.2266v4 (preprint) (2013)
3. Bebendorf, M.: Hierarchical matrices. In: A Means to Efficiently Solve Elliptic Boundary Value Problems. Lecture Notes in Computational Science and Engineering, vol. 63. Springer (2008)
4. Benzi, M.: Preconditioning techniques for large linear systems: a survey. J. Comput. Phys. 182, 418–477 (2002)
5. Benzi, M., Meyer, C., Tuma, M.: A sparse approximate inverse preconditioner for the conjugate gradient method. SIAM J. Sci. Comput. 17, 1135–1149 (1996)
6. Benzi, M., Tuma, M.: A sparse approximate inverse preconditioner for nonsymmetric linear systems. SIAM J. Sci. Comput. 19, 968–994 (1998)
7. Benzi, M., Tuma, M.: A comparative study of sparse approximate inverse preconditioners. Appl. Numer. Methods 30, 305–340 (1999)
8. Benzi, M., Tuma, M.: Orderings for factorized sparse approximate inverse preconditioners. SIAM J. Sci. Comput. 21, 1851–1868 (2000)
9. Börm, S.: $${\cal {H}}^2$$H2-matrix arithmetics in linear complexity. Computing 77, 1–28 (2006)
10. Börm, S.: Approximation of solution operators of elliptic partial differential equations by $${{\cal {H}}}$$H- and $${{\cal {H}}}^2$$H2-matrices. Numer. Math. 115, 165–193 (2010)
11. Börm, S., Grasedyck, L.: HLIB version 1.3. www.hlib.org
12. Börm, S., Grasedyck, L., Hackbusch, W.: Hierarchical matrices. Lecture Notes No. 21, Max-Planck-Institute for Mathematics in the Sciences, Leipzig, Germany (2003)
13. Bridson, R., Tang, W.P.: Ordering, anisotropy and factored sparse approximate inverses. SIAM J. Sci. Comput. 21, 867–882 (1999)
14. Bru, R., Marin, J., Mas, J., Tuma, M.: Balanced incomplete factorization. SIAM J. Sci. Comput. 30, 2302–2318 (2008)
15. Byckling, M., Huhtanen, M.: Preconditioning with direct approximate factoring of the inverse. SIAM J. Sci. Comput. 36, A88–A104 (2014)
16. Duff, I.S., Erisman, A.M., Gear, C.W., Reid, J.K.: Sparsity structure and gaussian elimination. SIGNUM Newsl. 23, 2–8 (1988)
17. Faustmann, M., Melenk, J.M., Praetorius, D.: H-matrix approximability of the inverse of FEM matrices. Numer. Math. (2015). doi:10.1007/s00211-015-0706-9
18. Grasedyck, L., Hackbusch, W.: Construction and arithmetics of $$\cal {H}$$H-matrices. Computing 70, 295–334 (2003)
19. Grasedyck, L., Kriemann, R., Le Borne, S.: Domain decomposition based $$\cal {H}$$H-LU preconditioning. Numeri. Math. 112, 565–600 (2009)
20. Hackbusch, W.: A sparse matrix arithmetic based on $$\cal {H}$$H-matrices. Part I: Introduction to $$\cal {H}$$H-matrices. Computing 62, 89–108 (1999)
21. Hackbusch, W.: Hierarchische Matrizen. Springer, Berlin (2009). (in German, English (revised) edition to appear in 2015)
22. Kolotilina, Y., Yeremin, Y.: Factorized sparse approximate inverse preconditionings I. theory. SIAM J. Matrix Anal. Appl. 14, 45–58 (1993)
23. Kriemann, R.: $$\cal {H}$$H-LU factorization on many-core systems. Tech. Rep. MIS-Preprint 5/2014, Max Planck Institute for Mathematics in the Sciences (2014)
24. Saad, Y.: Iterative Methods for Sparse Linear Systems. SIAM, Bangkok (2003)
25. Wang, S., Li, X.S., Xia, J., Situ, Y., de Hoop, M.V.: Efficient scalable algorithms for solving dense linear systems with hierarchically semiseparable structures. SIAM J. Sci. Comput. 35, C519–C544 (2013)
26. Xia, J., Chandrasekaran, S., Gu, M., Li, X.S.: Fast algorithms for hierarchically semiseparable matrices. Numer. Linear Algebra Appl. 17, 953–976 (2010)