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Towards H—Matrix Approximation of Linear Complexity

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Problems and Methods in Mathematical Physics

Part of the book series: Operator Theory: Advances and Applications ((OT,volume 121))

Abstract

In preceding papers [10]-[15], a class of matrices (H-matrices) has been analysed which are data-sparse and allow an approximate matrix arithmetic of almost linear complexity. Several types of H-matrices were shown to provide good approximation of nonlocal (integral) operators in FEM and BEM applications.

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References

  1. J.H. Bramble and J.T. King: A robust finite element method for nonhomogeneous Dirichlet problems in domains with curved boundaries. Math. Comp. 63(207), 1994, 1–17.

    Article  MathSciNet  MATH  Google Scholar 

  2. W. Dahmen, S. Prößdorf and R. Schneider: Wavelet approximation methods for pseudodifferential equations II: Matrix compression and fast solution, Adv. Comput. Math. 1 (1993) 259–335.

    Article  MathSciNet  MATH  Google Scholar 

  3. W. Dahmen, S. Prößdorf and R. Schneider, Wavelet approximation methods for pseudodifferential equations I: Stability and convergence, Math. Z. 215 (1994) 583–620.

    Article  MathSciNet  MATH  Google Scholar 

  4. K. Giebermann: A New Version of Panel Clustering for the Boundary Element Method. Preprint University of Bonn, January 1999.

    Google Scholar 

  5. M. Gnewuch and S. Sauter: Boundary Integral Equations for Second Order Elliptic BVPs, Preprin MPI, 56/1999, Leipzig, 1999.

    Google Scholar 

  6. L. Greengard and V. Rokhlin: A new version of the fast multipole method for the Laplace equation in three dimensions. Acta Numerica, 1997, 229–269.

    Google Scholar 

  7. W. Hackbusch: The panel clustering algorithm. MAFELAP 1990 (J. R. Whiteman, ed.). Academic Press, London, 1990, 339–348.

    Google Scholar 

  8. W. Hackbusch: Integral Equations. Theory and Numerical Treatment. ISNM 128. Birkhäuser, Basel, 1995.

    MATH  Google Scholar 

  9. W. Hackbusch: Elliptic Differential Equations: Theory and Numerical Treatment. Springer-Verlag, Berlin, 1992.

    MATH  Google Scholar 

  10. W. Hackbusch: A Sparse Matrix Arithmetic based on H-Matrices. Part I: Introduction to H-Matrices. Computing 62 (1999), 89–108.

    Article  MathSciNet  MATH  Google Scholar 

  11. W. Hackbusch and B. N. Khoromskij: A Sparse H-Matrix Arithmetic. Part II. Application to Multi-Dimensional Problems. Computing 64 (2000), 1, 21–47.

    MathSciNet  MATH  Google Scholar 

  12. W. Hackbusch and B. N. Khoromskij: A sparse H-matrix arithmetic: General complexity estimates. Preprint No. 66, MPI Leipzig, 1999; In: Numerical Analysis in the 20th Century, vol. 6: Ordinary Differential and Integral Equations (J. Pryce, G. Van den Berghe, C.T.H. Baker, G. Monegato, eds.), J. Comp. Appl. Math., Elsevier, 2000.

    Google Scholar 

  13. W. Hackbusch and B. N. Khoromskij: H-Matrix Approximation on Graded Meshes. Preprint No. 54, MPI Leipzig, 1999; Proc. of MAFELAP 1999, (J.R. Whiteman, ed.). Academic Press, London, to appear.

    Google Scholar 

  14. W. Hackbusch and B. N. Khoromskij: Towards H-Matrix Approximation of Linear Complexity. Preprint No. 15, MPI Leipzig, 2000.

    Google Scholar 

  15. W. Hackbusch, B. N. Khoromskij and S. Sauter: On H2-matrices. Preprint MPI, Leipzig, No. 50, 1999; In: Lectures on Applied Mathematics (H.-J. Bungartz, R. Hoppe, C. Zenger, eds.), Springer-Verlag, 2000, 9-30.

    Google Scholar 

  16. W. Hackbusch and Z. P. Nowak: On the fast matrix multiplication in the boundary element method by panel clustering. Numer. Math. 54 (1989) 463–491.

    Article  MathSciNet  MATH  Google Scholar 

  17. B.N. Khoromskij: Lectures on Multilevel Schur-Complement Methods for Elliptic Differential Equations. Preprint 98/4, ICA3, University of Stuttgart, 1998.

    Google Scholar 

  18. B.N. Khoromskij and S. Prößdorf: Multilevel preconditioning on the refined interface and optimal boundary solvers for the Laplace equation. Advances in Comp. Math. 4 (1995), 331–355.

    Article  MATH  Google Scholar 

  19. B. N. Khoromskij and S. Prößdorf: Fast computations with harmonic Poincaré-Steklov operators on nested refined meshes. Advances in Comp. Math., 8 (1998), 111–135.

    Article  MATH  Google Scholar 

  20. C. Lage and C. Schwab: Wavelet Galerkin algorithms for Boundary Integral Equations, Preprint ETH Zürich, October, 1997.

    Google Scholar 

  21. J.M. Melenk: Operator adapted spectral element methods I: Harmonic and generalized harmonic polynomials. Research Report 97-07, ETH Zürich, 1997.

    Google Scholar 

  22. T. von Petersdorff and C. Schwab: Wavelet approximations for first kind boundary integral equations, Numer. Math. 74 (1996) 479–516.

    Article  MathSciNet  MATH  Google Scholar 

  23. E. Tadmor: The exponential accuracy of Fourier and Chebyshev differencing methods. SIAM J. Numer. Anal., v. 23, 1986, 1–23.

    Google Scholar 

  24. V. M. Tikhomirov: Approximation Theory. In Analysis II, Encyclopaedia of Mathematical Sciences, v. 14, (R.V. Gamkrelidze, ed.) Springer-Verlag, 1990.

    Google Scholar 

  25. E. E. Tyrtyshnikov: Mosaic-skeleton approximations. Calcolo 33 (1996) 47–57.

    Article  MathSciNet  MATH  Google Scholar 

  26. E. P. Zhidkov and B. N. Khoromski J. Boundary integral equations on special surfaces and their applications. Sov. J. Numerical Anal. Math. Modelling, North Holland, Antwerpen, 2, 6, 1988, 463–488.

    Google Scholar 

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Author information

Authors and Affiliations

  1. Max-Planck-Institut für Mathematik in den Naturwissenschaften, Inselstr. 22-26, D-04103, Leipzig, Germany

    Wolfgang Hackbusch & Boris N. Khoromskij

Authors
  1. Wolfgang Hackbusch
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  2. Boris N. Khoromskij
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Editor information

Editors and Affiliations

  1. Weierstrass Institute for Applied Analysis and Stochastics, Mohrenstrasse 39, 10117, Berlin, Germany

    Johannes Elschner

  2. Department of Mathematical Sciences, Raymond and Beverly Sackler Faculty of Exact Sciences, Tel Aviv University, 69978, Ramat Aviv, Israel

    I. Gohberg

  3. Faculty of Mathematics, Technical University of Chemnitz, 09107, Chemnitz, Germany

    Bernd Silbermann

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© 2001 Springer Basel AG

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Hackbusch, W., Khoromskij, B.N. (2001). Towards H—Matrix Approximation of Linear Complexity. In: Elschner, J., Gohberg, I., Silbermann, B. (eds) Problems and Methods in Mathematical Physics. Operator Theory: Advances and Applications, vol 121. Birkhäuser, Basel. https://doi.org/10.1007/978-3-0348-8276-7_13

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  • DOI: https://doi.org/10.1007/978-3-0348-8276-7_13

  • Publisher Name: Birkhäuser, Basel

  • Print ISBN: 978-3-0348-9500-2

  • Online ISBN: 978-3-0348-8276-7

  • eBook Packages: Springer Book Archive

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