Efficient arithmetic operations for rank-structured matrices based on hierarchical low-rank updates

Computing and Visualization in Science

Volumn 16, Issue 6, 2013, Pages 247-258

  Börm, Steffen ^a   Reimer, Knut ^a  

^a UNIVERSITY OF KIEL   (Germany)

Author keywords

[No Author keywords available]

Indexed keywords

FACTORIZATION; INTEGRAL EQUATIONS; INVERSE PROBLEMS; NUMERICAL METHODS;

ALGEBRAIC OPERATIONS; ARITHMETIC OPERATIONS; CHOLESKY FACTORIZATIONS; LINEAR COMPLEXITY; NUMERICAL EXPERIMENTS; OPTIMAL ORDERING; RECURSIVE PROCEDURE; STRUCTURED MATRIXES;

MATRIX ALGEBRA;

EID: 84923564995     PISSN: 14329360     EISSN: 14330369     Source Type: Journal    
DOI: 10.1007/s00791-015-0233-3     Document Type: Article

References (24)

1. Bebendorf, M.: Why finite element discretizations can be factored by triangular hierarchical matrices. SIAM J. Numer. Anal. 45(4), 1472–1494 (2007)
2. Bebendorf, M., Hackbusch, W.: Existence of $${\cal H}$$H-matrix approximants to the inverse FE-matrix of elliptic operators with $$L^{\infty }$$L∞-coefficients. Numer. Math. 95, 1–28 (2003)
3. Börm, S.: $${\cal H}^2$$H2-matrix arithmetics in linear complexity. Computing 77(1), 1–28 (2006)
4. Börm, S.: Adaptive variable-rank approximation of dense matrices. SIAM J. Sci. Comp. 30(1), 148–168 (2007)
5. Börm, S.: Data-sparse approximation of non-local operators by $${\cal H}^2$$H2-matrices. Lin. Algebra Appl. 422, 380–403 (2007)
6. Börm, S.: Approximation of solution operators of elliptic partial differential equations by $${\cal H}$$H- and $${\cal H}^{2}$$H2-matrices. Numer. Math. 115(2), 165–193 (2010)
7. Börm, S.: Efficient Numerical Methods for Non-local Operators: $${\cal H}^{2}$$H2-Matrix Compression, Algorithms and Analysis. EMS Tracts in Mathematics, vol. 14, (2010)
8. Börm, S., Hackbusch, W.: Data-sparse approximation by adaptive $${\cal H}^2$$H2-matrices. Computing 69, 1–35 (2002)
9. Börm, S., Löhndorf, M., Melenk, J.M.: Approximation of integral operators by variable-order interpolation. Numer. Math. 99(4), 605–643 (2005)
10. Chandrasekaran, S., Gu, M., Lyons, W.: A fast adaptive solver for hierarchically semiseparable representations. Calcolo 42, 171–185 (2005)
11. Faustmann, M., Melenk, J.M., Praetorius, D.: Existence of $${\cal H}$$H-matrix approximants to the inverses of BEM matrices: the simple-layer operator. Tech. Rep. 37, Institut für Analysis und Scientific Computing, TU Wien (2013). http://www.asc.tuwien.ac.at/preprint/2013/asc37x2013.pdf
12. Faustmann, M., Melenk, J.M., Praetorius, D.: $${\cal H}$$H-matrix approximability of the inverse of FEM matrices. Tech. Rep. 20, Institut für Analysis und Scientific Computing, TU Wien (2013). http://www.asc.tuwien.ac.at/preprint/2013/asc20x2013.pdf
13. Grasedyck, L., Hackbusch, W.: Construction and arithmetics of $${\cal H}$$H-matrices. Computing 70, 295–334 (2003)
14. Grasedyck, L., Kriemann, R., LeBorne, S.: Domain decomposition based $${\cal H}$$H-LU preconditioning. Numer. Math. 112(4), 565–600 (2009)
15. 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)
16. Hackbusch, W.: Hierarchische Matrizen—Algorithmen und Analysis. Springer, Berlin (2009)
17. Hackbusch, W., Khoromskij, B.N.: A sparse matrix arithmetic based on $${\cal H}$$H-matrices. Part II: application to multi-dimensional problems. Computing 64, 21–47 (2000)
18. Hackbusch, W., Khoromskij, B.N., Sauter, S.A.: On $${\cal H}^{2}$$H2-matrices. In: Bungartz, H., Hoppe, R., Zenger, C. (eds.) Lectures on Applied Mathematics, pp. 9–29. Springer, Berlin (2000)
19. Hestenes, M.R., Stiefel, E.: Methods of conjugate gradients for solving linear systems. J. Res. Nat. B. Stand. 49(6), 409–436 (1952)
20. Lintner, M.: The eigenvalue problem for the 2d Laplacian in $${\cal H}$$H-matrix arithmetic and application to the heat and wave equation. Computing 72, 293–323 (2004)
21. Martinsson, P.G.: A fast direct solver for a class of elliptic partial differential equations. J. Sci. Comput. 38, 316–330 (2008)
22. Martinsson, P.G., Rokhlin, V., Tygert, M.: A fast algorithm for the inversion of general Toeplitz matrices. Comp. Math. Appl. 50, 741–752 (2005)
23. Xia, J., Chandrasekaran, S., Gu, M., Li, X.: Superfast multifrontal method for large structured linear systems of equations. SIAM J. Matrix Anal. Appl. 31(3), 1382–1411 (2009)
24. Xia, J., Chandrasekaran, S., Gu, M., Li, X.S.: Fast algorithms for hierarchically semiseparable matrices. Numer. Lin. Alg. Appl. (2009). doi:10.1002/nla.691