Hierarchical matrix techniques for low- and high-frequency Helmholtz problems

IMA Journal of Numerical Analysis

Volumn 28, Issue 1, 2008, Pages 46-79

  Banjai, Lehel ^a   Hackbusch, Wolfgang ^b  

^a UNIVERSITY OF ZURICH   (Switzerland)

^b MAX PLANCK INSTITUTE FOR MATHEMATICS IN THE SCIENCES   (Germany)

Author keywords

Boundary element method; Helmholtz equation; Hierarchical matrices

Indexed keywords

BOUNDARY ELEMENT METHOD; GALERKIN METHODS; HELMHOLTZ EQUATION; ITERATIVE METHODS; LINEAR SYSTEMS; NUMERICAL METHODS; SAILING VESSELS;

APPROXIMATE LU DECOMPOSITION; ARITHMETICAL OPERATIONS; GALERKIN BOUNDARY ELEMENT METHOD; HIERARCHICAL MATRICES; LOW AND HIGH FREQUENCIES; MATRIX-VECTOR PRODUCTS; NUMERICAL INSTABILITY; PRECONDITIONED ITERATIVE METHODS;

MATRIX ALGEBRA;

EID: 37849033831     PISSN: 02724979     EISSN: 14643642     Source Type: Journal    
DOI: 10.1093/imanum/drm001     Document Type: Article

References (46)

1. ABRAMOWITZ, M. & STEGUN, I. A. (eds) (1992) Handbook of Mathematical Functions with Formulas, Graphs, and Mathematical Tables. New York: Dover Publications Inc.
2. AMINI, S. & MAINES, N. D. (1998) Preconditioned Krylov subspace methods for boundary element solution of the Helmholtz equation. Int. J. Numer. Methods Eng., 41, 875-898.
3. AMINI, S. & PROFIT, A. (2000) Analysis of the truncation errors in the fast multipole method for scattering problems. Proceedings of the 8th International Congress on Computational and Applied Mathematics, ICCAM-98 (Leuven), vol. 115, pp. 23-33.
4. AMINI, S. & PROFIT, A. (2003) Multi-level fast multipole solution of the scattering problem.. Eng. Anal. Bound. Elem., 27, 547-564.
5. AMINI, S. & PROFIT, A. T. J. (1999) Analysis of a diagonal form of the fast multipole algorithm for scattering theory. BIT, 39, 585-602.
6. BANJAI, L. & SAUTER, S. (2007) A refined Galerkin error and stability analysis for highly indefinite variational problems. SIAM J. Numer. Anal., 45, 37-53.
7. BEBENDORF, M. (2000) Approximation of boundary element matrices. Numer. Math., 86, 565-589.
8. BEBENDORF, M. (2005) Hierarchical LU decomposition-based preconditioners for BEM. Computing, 74, 225-247.
9. 2-matrices - multilevel methods for the approximation of integral operators. Comput. Vis. Sci., 7, 173-181.
10. 2-matrix arithmetics in linear complexity. Computing, 77, 1-28.
11. BÖRM, S. & GRASEDYCK, L. (2004) Low-rank approximation of integral operators by interpolation. Computing, 72, 325-332.
12. BÖRM, S. & GRASEDYCK, L. (2005) Hybrid cross approximation of integral operators. Numer. Math., 101, 221-249.
13. BRAKHAGE, H. & WERNER, P. (1965) Über das Dirichletsche Außenraumproblem für die Helmholtzsche Schwingungsgleichung. Arch Math., 16, 325-329.
14. CHEN, G. & ZHOU, J. (1992) Boundary Element Methods. Computational Mathematics and Applications. London: Academic Press Ltd.
15. CHEW, W. C., JIN, J.-M., MICHIELSSEN, E. & SONG, J. M. (2001) Fast and Efficient Algorithms in Computational Electromagnetics. Boston, MA: Artech House.
16. COLTON, D. & KRESS, R. (1983) integral Equation Methods in Scattering Theory. New York: John Wiley.
17. COLTON, D. & KRESS, R. (1998) Inverse Acoustic and Electromagnetic Scattering Theory, 2nd edn. Berlin, Germany: Springer.
18. DAHMEN, W. (1997) Wavelet and multiscale methods for operator equations. Acta numerica, 1997 (A. Iserles ed.), vol. 6. Cambridge: Cambridge University Press, pp. 55-228.
19. DARVE, E. (2000) The fast multipole method: numerical implementation. J. Comput. Phys., 160, 195-240.
20. DARVE, E. & HAVÉ, P. (2004a) A fast multipole method for Maxwell equations stable at all frequencies. Philos. Trans. R. Soc. Lond. Ser. A Math Phys. Eng. Sci., 362, 603-628.
21. DARVE, E. & HAVÉ, P. (2004b) Efficient fast multipole method for low-frequency scattering. J. Comput. Phys., 197, 341-363.
22. DAVIS, P. J. & RABINOWITZ, P. (1984) Methods of Numerical Integration, 2nd edn. Orlando, FL: Academic Press Inc.
23. 3. Ph.D. Thesis, Universität Karlsruhe, Karlsruhe, Germany.
24. GRADSHTEYN, I. S. & RYZHIK, I. M. (2000) Table of Integrals, Series, and Products. San Diego, CA: Academic Press Inc.
25. GRASEDYCK, L. (2005) Adaptive recompression of H-matrices for BEM. Computing, 74, 205-223.
26. GRASEDYCK, L. & HACKBUSCH, W. (2003) Construction and arithmetics of H-matrices. Computing, 70, 295-334.
27. GREENGARD, L., HUANG, J., ROKHLIN, V. & WANDZURA, S. (1998) Accelerating fast multipole methods for the Helmholtz equation at low frequencies. IEEE Comput. Sci. Eng., 32, 32-38.
28. GREENGARD, L. & ROKHLIN, V. (1987) A fast algorithm for particle simulations. J. Comput. Phys., 73, 325-348.
29. HACKBUSCH, W. (1995) Integral Equations. Basel: Birkhäuser.
30. HACKBUSCH, W. (1999) A sparse matrix arithmetic based on H-matrices. I. Introduction to H-matrices. Computing, 62, 89-108.
31. 2-matrices. Computing, 69, 1-35.
32. HACKBUSCH, W. & KHOROMSKIJ, B. N. (2000) A sparse H-matrix arithmetic. II. Application to multi-dimensional problems. Computing, 64, 21-47.
33. HACKBUSCH, W. & NOWAK, Z. P. (1989) On the fast matrix multiplication in the boundary element method by panel clustering. Numer. Math., 54, 463-491.
34. HARRIS, P. J. & CHEN, K. (2003) On efficient preconditioned for iterative solution of a Galerkin boundary element equation for the three-dimensional exterior Helmholtz problem. J. Comput. Appl. Math., 156, 303-318.
35. HENRICI, P. (1986) Applied and Computational Complex Analysis, vol. 3. New York: John Wiley.
36. HRYCAK, T. & ROKHLIN, V. (1998) An improved fast multipole algorithm for potential fields. SIAM J. Sci. Comput., 19, 1804-1826.
37. LU, C. & CHEW, W. C. (1994) A multilevel algorithm for solving a boundary integral equation of wave scattering. Microw. Opt. Technol. Lett., 7, 466-470.
38. NÉDÉLEC, J.-C. (2001) Acoustic and Electromagnetic Equations. New York: Springer.
39. OHNUKI, S. & CHEW, W. C. (2003/2004) Truncation error analysis of multipole expansion. SIAM J. Sci. Comput., 25, 1293-1306.
40. ROKHLIN, V. (1990) Rapid solution of integral equations of scattering theory in two dimensions. J. Comput. Phys., 86, 414-439.
41. ROKHLIN, V. (1993) Diagonal forms of translation operators for the Helmholtz equation in three dimensions. Appl. Comput. Harmon. Anal., 1, 82-93.
42. SAUTER, S. & SCHWAB, C. (2004) Randelementmethoden. Leipzig, Germany: Teubner.
43. STOLPER, M. (2004) Computing and compression of the boundary element matrices for the Helmholtz equation. J. Numer. Math., 12, 55-75.
44. TREFETHEN, L. N. & BAU III, D. (1997) Numerical Linear Algebra. Philadelphia, PA: Society for Industrial, and Applied Mathematics (SIAM).
45. WATSON, G. N. (1944) A Treatise on the Theory of Bessel Functions. Cambridge, England: Cambridge University Press.
46. ZHAO, J.-S. &CHEW, C. C. (1999) MLFMA for solving integral equations of 2-D electromagnetic problems from static to electrodynamic. Microw. Opt. Technol. Lett., 20, 306-311.