Robust and efficient multifrontal solver for large discretized PDEs

High-Performance Scientific Computing: Algorithms and Applications

Volumn 9781447124375, Issue , 2012, Pages 199-217

  Xia, Jianlin ^a  

^a PURDUE UNIVERSITY   (United States)

Author keywords

[No Author keywords available]

Indexed keywords

FLEXIBLE STRUCTURES; MATRIX ALGEBRA;

DISCRETIZATIONS; ELLIPTIC EQUATIONS; FACTORIZATION METHODS; LOW-RANK PROPERTIES; MATRIX; MULTIFRONTAL; MULTIFRONTAL METHODS; PERFORMANCE; SPARSE MATRICES; SYMMETRIC POSITIVE DEFINITE;

FACTORIZATION;

EID: 84877338334     PISSN: None     EISSN: None     Source Type: Book    
DOI: 10.1007/978-1-4471-2437-5_10     Document Type: Chapter

References (35)

1. Bebendorf, M.: Efficient inversion of the Galerkin matrix of general second-order elliptic operators with nonsmooth coefficients. Math. Comput. 74(251), 1179-1199 (2005)
2. Bebendorf, M., Hackbusch, W.: Existence of H-matrix approximants to the inverse FE-matrix of elliptic operators with l ∞-coefficients. Numer. Math. 95, 1-28 (2003)
3. Chandrasekaran, S., Dewilde, P., Gu, M., Lyons, W., Pals, T.: A fast solver for HSS representations via sparse matrices. SIAM J. Matrix Anal. Appl. 29, 67-81 (2006)
4. Chandrasekaran, S., Dewilde, P., Gu, M., Somasunderam, N.: On the numerical rank of the off-diagonal blocks of Schur complements of discretized elliptic PDEs. SIAM J. Matrix Anal. Appl. 31, 2261-2290 (2010)
5. Chandrasekaran, S., Gu, M., Pals, T.: A fast ULV decomposition solver for hierarchically semiseparable representations. SIAM J. Matrix Anal. Appl. 28, 603-622 (2006)
6. Chen, L.: iFEM: An innovative finite element methods package in MATLAB. Technical Report (2008). http://math.uci.edu/~chenlong/Papers/iFEMpaper.pdf
7. Davis, T.A., Hu, Y.: The university of Florida sparse matrix collection. ACM Trans. Math. Softw.
8. Demmel, J.W., Gilbert, J.R., Li, X.S.: SuperLU Users' Guide (2003). http://crd.lbl.gov/~xiaoye/SuperLU/
9. Duff, I.S., Reid, J.K.: The multifrontal solution of indefinite sparse symmetric linear. ACM Trans. Math. Softw. 9, 302-325 (1983)
10. Eidelman, Y., Gohberg, I.C.: On a new class of structured matrices. Integral Equ. Oper. Theory 34, 293-324 (1999)
11. Engquist, B., Ying, L.: Sweeping preconditioner for the Helmholtz equation: Hierarchical matrix representation. Commun. Pure Appl. Math. 64, 697-735 (2011)
12. George, A.: Nested dissection of a regular finite element mesh. SIAM J. Numer. Anal. 10, 345-363 (1973)
13. Gilbert, J.R., Teng, S.H.:MESHPART, aMatlabmesh partitioning and graph separator toolbox (2002). http://aton.cerfacs.fr/algor/Softs/MESHPART/
14. Greengard, L., Rokhlin, V.: A fast algorithm for particle simulations. J. Comp. Physiol. 73, 325-348 (1987)
15. Hackbusch, W.: A sparse matrix arithmetic based on H-matrices. Part I: Introduction to Hmatrices. Computer 62, 89-108 (1999)
16. Hackbusch, W., Börm, S.: Data-sparse approximation by adaptive H2-matrices. Computer 69, 1-35 (2002)
17. Hoffman, A.J., Martin, M.S., Rose, D.J.: Complexity bounds for regular finite difference and finite element grids. SIAM J. Numer. Anal. 10, 364-369 (1973)
18. Karypis, G.: METIS: family of multilevel partitioning algorithms (1998). http://glaros.dtc.umn.edu/gkhome/metis/metis/overview
19. Le Borne, S., Grasedyck, L., Kriemann, R.: Domain-decomposition based H-LU preconditioners. Domain Decomposition Methods in Science and Engineering XVI. in O.B. Widlund, D.E. Keyes (Eds) 55, 661-668 (2006)
20. Liu, J.W.H.: The multifrontal method for sparse matrix solution: Theory and practice. SIAM Rev. 34, 82-109 (1992)
21. Lyons, W.: Fast algorithms with applications to PDEs. Ph.D. Thesis, UCSB (2005)
22. Parter, S.: The use of linear graphs in Gauss elimination. SIAM Rev. 3, 119-130 (1961)
23. Polizzi, E., Sameh, A.H.: A parallel hybrid banded system solver: the SPIKE algorithm. Parallel Comput. 32, 177-194 (2006)
24. Sambavaram, S.R., Sarin, V., Sameh, A.H., Grama, A.: Multipole-based preconditioners for large sparse linear systems. Parallel Comput. 29, 1261-1273 (2003)
25. Schmitz, P.G., Ying, L.: A fast direct solver for elliptic problems on general meshes in 2D. J. Comput. Phys. (2011). doi:10.1016/j.jcp.2011.10.013
26. Schmitz, P.G., Ying, L.: A fast direct solver for elliptic problems on Cartesian meshes in 3D (2011, submitted). http://www.math.utexas.edu/users/lexing/publications/direct3d.pdf
27. Tewarson, R.P.: On the product form of inverses of sparse matrices. SIAM Rev. 8 (1966)
28. Vandebril, R., Barel, M.V., Golub, G., Mastronardi, N.: A bibliography on semiseparable matrices. Calcolo 42, 249-270 (2005)
29. Wang, S., Li, X.S., Xia, J., Situ, Y., de Hoop, V.M.: Efficient scalable algorithms for hierarchically semiseparable matrices. Preprint (2011). http://www.math.purdue.edu/~xiaj/work/parhss.pdf
30. Xia, J.: Robust structured multifrontal factorization and preconditioning for discretized PDEs. Preprint (2008). http://www.math.purdue.edu/~xiaj/work/mfprec.pdf
31. Xia, J.: Efficient structured multifrontal factorization for large sparse matrices. Preprint (2010). http://www.math.purdue.edu/~xiaj/work/mfhss.pdf
32. Xia, J.: On the complexity of some hierarchical structured matrices. SIAM J. Matrix Anal. Appl. (2011, submitted). http://www.math.purdue.edu/~xiaj/work/hsscost.pdf
33. Xia, J., Chandrasekaran, S., Gu, M., Li, X.S.: Superfast multifrontal method for large structured linear systems of equations. SIAM J. Matrix Anal. Appl. 31, 1382-1411 (2009)
34. Xia, J., Chandrasekaran, S., Gu, M., Li, X.S.: Fast algorithms for hierarchically semiseparable matrices. Numer. Linear Algebra Appl. 17, 953-976 (2010)
35. Xia, J., Gu, M.: Robust approximate Cholesky factorization of rank-structured symmetric positive definite matrices. SIAM J. Matrix Anal. Appl. 31, 2899-2920 (2010)