Efficient scalable algorithms for solving dense linear systems with hierarchically semiseparable structures

SIAM Journal on Scientific Computing

Volumn 35, Issue 6, 2013, Pages

  Wang, Shen ^a   Li, Xiaoye S ^b   Xia, Jianlin ^a   Situ, Yingchong ^c   De Hoop, Maarten V ^a  

^a PURDUE UNIVERSITY   (United States)

^b LAWRENCE BERKELEY NATIONAL LABORATORY   (United States)

^c Purdue University   (United States)

Author keywords

Compression; Direct solver; HSS construction; HSS matrix; Low rank property; Parallel HSS algorithm

Indexed keywords

FACTORIZATION; HELMHOLTZ EQUATION; INVERSE PROBLEMS; ITERATIVE METHODS; LOWER-UPPER DECOMPOSITION; SEISMOLOGY; TREES (MATHEMATICS);

COMPRESSION; DIRECT SOLVERS; FACTORIZATION ALGORITHMS; HELMHOLTZ'S EQUATIONS; HIERARCHICALLY SEMISEPARABLE CONSTRUCTION; HIERARCHICALLY SEMISEPARABLE MATRIXES; LOW-RANK PROPERTIES; PARALLEL HIERARCHICALLY SEMISEPARABLE ALGORITHM; SCALABLE ALGORITHMS; SPARSE LINEAR SYSTEMS;

LINEAR SYSTEMS;

EID: 84892613101     PISSN: 10648275     EISSN: 10957200     Source Type: Journal    
DOI: 10.1137/110848062     Document Type: Article

References (33)

1. T. Bella, Y. Eidelman, V. Gohberg, and I. Olshevsky, Computations with quasiseparable polynomials and matrices, Theoret. Comput. Sci., 409 (2008), pp. 158-179.
2. L. S. Blackford, J. Choi, E. D'Azevedo, J. Demmel, I. Dhillon, J. Dongarra, S. Hammarling, G. Henry, A. Petitet, K. Stanley, D. Walker, and R. C. Whaley, ScaLAPACK Users' Guide, Software Environ. Tools, SIAM, Philadelphia, 1997.
3. BLACS. http://www.netlib.org/blacs.
4. S. Borm, L. Grasedyck, and W. Hackbusch, Introduction to hierarchical matrices with applications, Eng. Anal. Bound. Elem, 27 (2003), pp. 405-422.
5. S. Borm and W. Hackbusch, Data-Sparse Approximation by Adaptive H2-Matrices, Technical report, Max Planck Institute for Mathematics, Leipzig, Germany, 2001.
6. S. Chandrasekaran, P. Dewilde, M. Gu, T. Pals, X. Sun, A.-J. van der Veen, and D. White, Some fast algorithms for sequentially semiseparable representations, SIAM J. Matrix Anal. Appl., 27 (2005), pp. 341-364.
7. S. Chandrasekaran, M. Gu, and T. Pals, A fast ULV decomposition solver for hierarchically semiseparable representations, SIAM J. Matrix Anal. Appl., 28 (2006), pp. 603-622.
8. I. S. Duff and J. K. Reid, The multifrontal solution of indefinite sparse symmetric linear equations, ACM Trans. Math. Softw., 9 (1983), pp. 302-325.
9. Y. Eidelman and I. Gohberg, On a new class of structured matrices, Integral Equations Operator Theory, 34 (1999), pp. 293-324.
10. B. Engquist and L. Ying, Sweeping preconditioner for the Helmholtz equation: Hierarchical matrix representation, Comm. Pure Appl. Math., 64 (2011), pp. 697-735.
11. J. A. George, Nested dissection of a regular finite element mesh, SIAM J. Numer. Anal., 10 (1973), pp. 345-363.
12. G. H. Golub and C. V. Loan, Matrix Computations, 3rd ed., Johns Hopkins University Press, Baltimore, MD, 1996.
13. M. Gu and S. C. Eisenstat, Efficient algorithms for computing a strong rank-revealing qr factorization, SIAM J. Sci. Comput., 17 (1996), pp. 848-869.
14. W. Hackbusch, L. Grasedyck, and S. Borm, An introduction to hierarchical matrices, Math. Bohem., 127 (2002), pp. 229-241.
15. W. Hackbusch and B. N. Khoromskij, A sparse H-matrix arithmetic. Part II: Application to multi-dimensional problems, Computing, 64 (2000), pp. 21-47.
16. W. Hackbusch, B. Khoromskij, and S. A. Sauter, On H2-Matrices, Lectures on Appl. Math., Springer, Berlin, 2000, pp. 9-29.
17. W. Hackbusch, A sparse matrix arithmetic based on H-matrices. Part I: Introduction to H-matrices, Computing, 62 (1999), pp. 89-108.
18. Integrated Performance Monitoring: IPM, http://ipm-hpc.sourceforge.net.
19. LAPACK -Linear Algebra PACKage, http://www.netlib.org/lapack.
20. Message Passing Interface Forum, http://www.mpi-forum.org.
21. ScaLAPACK -Scalable Linear Algebra PACKage, http://www.netlib.org/ scalapack.
22. R. Thakur, R. Rabenseifner, and W. Gropp, Optimization of collective communication operations in MPICH, Int. J. High Performance Comput. Appl., 19 (2005), pp. 49-66.
23. R. Vandebril, M. Van Barel, G. Golub, and N. Mastronardi, A bibliography on semiseparable matrices, Calcolo, 42 (2005), pp. 249-270.
24. S. Wang, M. V. de Hoop, and J. Xia, Seismic inverse scattering via Helmholtz operator factorization and optimization, J. Comput. Phys., 229 (2010), pp. 8445-8462.
25. S. Wang, M. V. de Hoop, and J. Xia, On 3D modeling of seismic wave propagation via a structured massively parallel multifrontal direct Helmholtz solver, Geophys. Prospect., 59 (2011), pp. 857-873.
26. S. Wang, M. V. de Hoop, J. Xia, and X. S. Li, Massively parallel structured multifrontal solver for time-harmonic elastic waves in 3D anisotropic media, Geophys. J. Int., 91 (2012), pp. 346-366.
27. S. Wang, J. Xia, M. V. de Hoop, and X. S. Li, Massively parallel structured direct solver for equations describing time-harmonic qP-polarized waves in TTI media, Geophysics, 77 (2012), pp. T69-T82.
28. J. Xia, S. Chandrasekaran, M. Gu, and X. S. Li, Superfast multifrontal method for large structured linear systems of equations, SIAM J. Matrix Anal. Appl., 31 (2009), pp. 1382-1411.
29. J. Xia, S. Chandrasekaran, M. Gu, and X. S. Li, Fast algorithms for hierarchically semiseparable matrices, Numer. Linear Algebra Appl., 2010 (2010), pp. 953-976.
30. J. Xia and M. Gu, Robust approximate cholesky factorization of rank-structured symmetric positive definite matrices, SIAM J. Matrix Anal. Appl., 31 (2010), pp. 2899-2920.
31. J. Xia, Efficient structured multifrontal factorization for general large sparse matrices, SIAM J. Sci. Comput., 35 (2012), pp. A832-A860.
32. J. Xia, On the complexity of some hierarchical structured matrix algorithms, SIAM J. Matrix Anal. Appl., 33 (2012), pp. 388-410.
33. Y. Xi, J. Xia, S. Cauley, and V. Balakrishnan, Superfast and stable structured solvers for Toeplitz least squares via randomized sampling, SIAM J. Matrix Anal. Appl., submitted.