Skip to main content
SpringerLink
Log in

A direct solver with O(N) complexity for integral equations on one-dimensional domains

  • Research Article
  • Published:
Frontiers of Mathematics in China Aims and scope Submit manuscript

Abstract

An algorithm for the direct inversion of the linear systems arising from Nyström discretization of integral equations on one-dimensional domains is described. The method typically has O(N) complexity when applied to boundary integral equations (BIEs) in the plane with non-oscillatory kernels such as those associated with the Laplace and Stokes’ equations. The scaling coefficient suppressed by the “big-O” notation depends logarithmically on the requested accuracy. The method can also be applied to BIEs with oscillatory kernels such as those associated with the Helmholtz and time-harmonic Maxwell equations; it is efficient at long and intermediate wave-lengths, but will eventually become prohibitively slow as the wave-length decreases. To achieve linear complexity, rank deficiencies in the off-diagonal blocks of the coefficient matrix are exploited. The technique is conceptually related to the ℋ- and ℋ 2-matrix arithmetic of Hackbusch and co-workers, and is closely related to previous work on Hierarchically Semi-Separable matrices.

This is a preview of subscription content, log in via an institution to check access.

Access this article

Log in via an institution

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

Similar content being viewed by others

References

  1. Atkinson K E. The Numerical Solution of Integral Equations of the Second Kind. Cambridge: Cambridge University Press, 1997

    Book  MATH  Google Scholar 

  2. Barnes J, Hut P. A hierarchical O(n log n) force-calculation algorithm. Nature, 1986, 324(4): 446–449

    Article  Google Scholar 

  3. Beylkin G, Coifman R, Rokhlin V. Wavelets in numerical analysis. In: Wavelets and Their Applications. Boston: Jones and Bartlett, 1992, 181–210

    Google Scholar 

  4. Börm S. Efficient Numerical Methods for Non-local Operators: ℋ 2-matrix Compression, Algorithms and Analysis. European Mathematics Society, 2010

  5. Bremer J, Rokhlin V. Efficient discretization of Laplace boundary integral equations on polygonal domains. J Comput Phys, 2010, 229: 2507–2525

    Article  MathSciNet  MATH  Google Scholar 

  6. Chandrasekaran S, Gu M. A divide-and-conquer algorithm for the eigendecomposition of symmetric block-diagonal plus semiseparable matrices. Numer Math, 2004, 96(4): 723–731

    Article  MathSciNet  MATH  Google Scholar 

  7. Chandrasekaran S, Gu M, Li X S, Xia J. Superfast multifrontal method for large structured linear systems of equations. SIAM J Matrix Anal Appl, 2009, 31: 1382–1411

    MathSciNet  Google Scholar 

  8. Chandrasekaran S, Gu M, Li X S, Xia J. Fast algorithms for hierarchically semiseparable matrices. Numer Linear Algebra Appl, 2010, 17: 953–976

    Article  MathSciNet  MATH  Google Scholar 

  9. Cheng H, Gimbutas Z, Martinsson P G, Rokhlin V. On the compression of low rank matrices. SIAM J Sci Comput, 2005, 26(4): 1389–1404

    Article  MathSciNet  MATH  Google Scholar 

  10. Gillman A. Fast direct solvers for elliptic partial differential equations. Ph D Thesis. Boulder: University of Colorado at Boulder, 2011

    Google Scholar 

  11. Golub G H, Van Loan C F. Matrix Computations. 3rd ed. Johns Hopkins Studies in the Mathematical Sciences. Baltimore: Johns Hopkins University Press, 1996

    MATH  Google Scholar 

  12. Grasedyck L, Kriemann R, Le Borne S. Domain decomposition based H-LU preconditioning. Numer Math, 2009, 112: 565–600

    Article  MathSciNet  MATH  Google Scholar 

  13. Greengard L, Gueyffier D, Martinsson P G, Rokhlin V. Fast direct solvers for integral equations in complex three-dimensional domains. Acta Numer, 2009, 18: 243–275

    Article  MathSciNet  MATH  Google Scholar 

  14. Greengard L, Rokhlin V. A fast algorithm for particle simulations. J Comput Phys, 1987, 73(2): 325–348

    Article  MathSciNet  MATH  Google Scholar 

  15. Gu M, Eisenstat S C. Efficient algorithms for computing a strong rank-revealing QR factorization. SIAM J Sci Comput, 1996, 17(4): 848–869

    Article  MathSciNet  MATH  Google Scholar 

  16. Hackbusch W. The panel clustering technique for the boundary element method (invited contribution). In: Boundary Elements IX, Vol 1 (Stuttgart, 1987), Comput Mech Southampton. 1987, 463–474

  17. Hackbusch W. A sparse matrix arithmetic based on H-matrices; Part I: Introduction to H-matrices. Computing, 1999, 62: 89–108

    Article  MathSciNet  MATH  Google Scholar 

  18. Helsing J, Ojala R. Corner singularities for elliptic problems: Integral equations, graded meshes, quadrature, and compressed inverse preconditioning. J Comput Phys, 2008, 227: 8820–8840

    Article  MathSciNet  MATH  Google Scholar 

  19. Kapur S, Rokhlin V. High-order corrected trapezoidal quadrature rules for singular functions. SIAM J Numer Anal, 1997, 34: 1331–1356

    Article  MathSciNet  MATH  Google Scholar 

  20. Martinsson P G. A fast randomized algorithm for computing a hierarchically semiseparable representation of a matrix. SIAM J Matrix Anal Appl, 2011, 32(4): 1251–1274

    Article  MathSciNet  MATH  Google Scholar 

  21. Martinsson P G, Rokhlin V. A fast direct solver for boundary integral equations in two dimensions. J Comp Phys, 2005, 205(1): 1–23

    Article  MathSciNet  MATH  Google Scholar 

  22. Michielssen E, Boag A, Chew W C. Scattering from elongated objects: direct solution in O(N log2 N) operations. IEE Proc Microw Antennas Propag, 1996, 143(4): 277–283

    Article  Google Scholar 

  23. O’Donnell S T, Rokhlin V. A fast algorithm for the numerical evaluation of conformal mappings. SIAM J Sci Stat Comput, 1989, 10: 475–487

    Article  MathSciNet  MATH  Google Scholar 

  24. Schmitz P, Ying L. A fast direct solver for elliptic problems on general meshes in 2d. 2010

  25. Sheng Z, Dewilde P, Chandrasekaran S. Algorithms to solve hierarchically semiseparable systems. In: System Theory, the Schur Algorithm and Multidimensional Analysis. Oper Theory Adv Appl, Vol 176. Basel: Birkhäuser, 2007, 255–294

    Chapter  Google Scholar 

  26. Starr P, Rokhlin V. On the numerical solution of two-point boundary value problems. II. Comm Pure Appl Math, 1994, 47(8): 1117–1159

    Article  MathSciNet  MATH  Google Scholar 

  27. Xiao H, Rokhlin V, Yarvin N. Prolate spheroidal wavefunctions, quadrature and interpolation. Inverse Problems, 2001, 17(4): 805–838

    Article  MathSciNet  MATH  Google Scholar 

  28. Young P, Hao S, Martinsson P G. A high-order Nyström discretization scheme for boundary integral equations defined on rotationally symmetric surfaces. J Comput Phys (to appear)

Download references

Author information

Authors and Affiliations

  1. Department of Applied Mathematics, University of Colorado at Boulder, Boulder, CO, 80309-0526, USA

    Adrianna Gillman, Patrick M. Young & Per-Gunnar Martinsson

Authors
  1. Adrianna Gillman
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Patrick M. Young
    View author publications

    You can also search for this author in PubMed Google Scholar

  3. Per-Gunnar Martinsson
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Per-Gunnar Martinsson.

Rights and permissions

Reprints and permissions

About this article

Cite this article

Gillman, A., Young, P.M. & Martinsson, PG. A direct solver with O(N) complexity for integral equations on one-dimensional domains. Front. Math. China 7, 217–247 (2012). https://doi.org/10.1007/s11464-012-0188-3

Download citation

  • Received:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s11464-012-0188-3

Keywords

MSC

Search

Navigation