Skip to main content
SpringerLink
Log in

A Matrix-free Two-grid Preconditioner for Solving Boundary Integral Equations in Electromagnetism

  • Published:
Computing Aims and scope Submit manuscript

Abstract

In this paper, we describe a matrix-free iterative algorithm based on the GMRES method for solving electromagnetic scattering problems expressed in an integral formulation. Integral methods are an interesting alternative to differential equation solvers for this problem class since they do not require absorbing boundary conditions and they mesh only the surface of the radiating object giving rise to dense and smaller linear systems of equations. However, in realistic applications the discretized systems can be very large and for some integral formulations, like the popular Electric Field Integral Equation, they become ill-conditioned when the frequency increases. This means that iterative Krylov solvers have to be combined with fast methods for the matrix-vector products and robust preconditioning to be affordable in terms of CPU time. In this work we describe a matrix-free two-grid preconditioner for the GMRES solver combined with the Fast Multipole Method. The preconditioner is an algebraic two-grid cycle built on top of a sparse approximate inverse that is used as smoother, while the grid transfer operators are defined using spectral information of the preconditioned matrix. Experiments on a set of linear systems arising from real radar cross section calculation in industry illustrate the potential of the proposed approach for solving large-scale problems in electromagnetism.

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

  • Alléon, G., Amram, S., Durante, N., Homsi, P., Pogarieloff, D., Farhat, C.: Massively parallel processing boosts the solution of industrial electromagnetic problems: High performance out-of-core solution of complex dense systems. In: Proc. 8th SIAM Conf. on Parallel (M. Heath, V. Torczon, G. Astfalk, P. E. Bjørstad, A. H. Karp, C. H. Koebel, V. Kumar, R. F. Lucas, L. T. Watson, D. E. Womble, eds.), Minneapolis, Minnesota, USA. Philadelphia: SIAM 1997.

  • G. Alléon M. Benzi L. Giraud (1997) ArticleTitleSparse approximate inverse preconditioning for dense linear systems arising in computational electromagnetics Numer. Algorithm 16 1–15 Occurrence Handle10.1023/A:1019170609950 Occurrence Handle1623628

    Article  MathSciNet  Google Scholar 

  • J. Baglama D. Calvetti G. H. Golub L. Reichel (1999) ArticleTitleAdaptively preconditioned GMRES algorithms SIAM J. Sci. Comput. 20 IssueID1 243–269 Occurrence Handle10.1137/S1064827596305258 Occurrence Handle99d:65097

    Article  MathSciNet  Google Scholar 

  • M. Bebendorf (2000) ArticleTitleApproximation of boundary element matrices Numer. Math. 86 IssueID4 565–589 Occurrence Handle10.1007/PL00005410 Occurrence Handle0966.65094 Occurrence Handle2001j:65022

    Article  MATH  MathSciNet  Google Scholar 

  • M. Bebendorf S. Rjasanov (2003) ArticleTitleAdaptive low-rank approximation of collocation matrices Computing 70 IssueID1 1–24 Occurrence Handle10.1007/s00607-002-1469-6 Occurrence Handle2004a:65177

    Article  MathSciNet  Google Scholar 

  • M. Benzi M. Tůma (1998) ArticleTitleA sparse approximate inverse preconditioner for non-symmetric linear systems SIAM J. Sci. Comput. 19 968–994 Occurrence Handle10.1137/S1064827595294691 Occurrence Handle98m:65071

    Article  MathSciNet  Google Scholar 

  • C. Calvez ParticleLe B. Molina (1999) ArticleTitleImplicitly restarted and deflated GMRES Numer. Algorithm 21 261–285 Occurrence Handle10.1023/A:1019113630790 Occurrence Handle2000i:65047

    Article  MathSciNet  Google Scholar 

  • Canning, F. X.: The impedance matrix localization (IML) method for moment-method calculations. IEEE Antennas and Propagation Magazine 1990.

  • B. Carpentieri I. S. Duff L. Giraud (2003) ArticleTitleA class of spectral two-level preconditioners SIAM J. Sci. Comput. 25 IssueID2 749–765 Occurrence Handle10.1137/S1064827502408591 Occurrence Handle2005g:65054

    Article  MathSciNet  Google Scholar 

  • B. Carpentieri I. S. Duff L. Giraud G. Sylvand (2005) ArticleTitleCombining fast multipole techniques and an approximate inverse preconditioner for large parallel electromagnetics calculations SIAM J. Sci. Comput. 27 IssueID3 774–792 Occurrence Handle10.1137/040603917 Occurrence Handle2199907

    Article  MathSciNet  Google Scholar 

  • Carpentieri, B., Giraud, L., Gratton, S.: Additive and multiplicative two-level spectral preconditioning for general linear systems. Technical Report TR/PA/04/38, CERFACS, Toulouse, France, 2004

  • K. Chen (2001) ArticleTitleAn analysis of sparse approximate inverse preconditioners for boundary integral equations SIAM J. Matrix Anal. Appl. 22 IssueID3 1058–1078 Occurrence Handle10.1137/S0895479898348040 Occurrence Handle0985.65038 Occurrence Handle2002a:65188

    Article  MATH  MathSciNet  Google Scholar 

  • W. C. Chew Y. M. Wang (1993) ArticleTitleA recursive t-matrix approach for the solution of electromagnetic scattering by many spheres IEEE Trans. Antennas Propagation 41 IssueID12 1633–1639 Occurrence Handle10.1109/8.273306

    Article  Google Scholar 

  • W. C. Chew K. F. Warnick (2001) ArticleTitleOn the spectrum of the electric field integral equation and the convergence of the moment method Int. J. Numer. Meth. Engng. 51 475–489 Occurrence Handle2002b:78020

    MathSciNet  Google Scholar 

  • E. Chow Y. Saad (1997) ArticleTitleExperimental study of ILU preconditioners for indefinite matrices J. Comput. Appl. Math. 86 387–414 Occurrence Handle10.1016/S0377-0427(97)00171-4 Occurrence Handle98h:65013

    Article  MathSciNet  Google Scholar 

  • E. Darve (2000) ArticleTitleThe fast multipole method: Numerical implementation J. Comp. Phys. 160 IssueID1 195–240 Occurrence Handle10.1006/jcph.2000.6451 Occurrence Handle0974.78012 Occurrence Handle2001a:78050

    Article  MATH  MathSciNet  Google Scholar 

  • Edelman, A.: Records in dense linear algebra [online, cited 27 April 2005]. Available from: http://www-math.mit.edu/edelman/records.html.

  • J. Erhel K. Burrage B. Pohl (1996) ArticleTitleRestarted GMRES preconditioned by deflation J. Comput. Appl. Math. 69 303–318 Occurrence Handle10.1016/0377-0427(95)00047-X Occurrence Handle97a:65035

    Article  MathSciNet  Google Scholar 

  • L. Fournier S. Lanteri (2001) ArticleTitleMultiplicative and additive parallel multigrid algorithms for the acceleration of compressible flow computations on unstructured meshes Appl. Numer. Math. 36 401–426 Occurrence Handle10.1016/S0168-9274(00)00017-9 Occurrence Handle2002a:65166

    Article  MathSciNet  Google Scholar 

  • N. I. M. Gould J. A. Scott (1998) ArticleTitleSparse approximate-inverse preconditioners using norm-minimization techniques SIAM J. Sci. Comput. 19 IssueID2 605–625 Occurrence Handle10.1137/S1064827595288425 Occurrence Handle1618848

    Article  MathSciNet  Google Scholar 

  • L. Greengard W. Gropp (1990) ArticleTitleA parallel version of the fast multipole method Comput. Math. Appl. 20 63–71 Occurrence Handle10.1016/0898-1221(90)90349-O Occurrence Handle91e:78003

    Article  MathSciNet  Google Scholar 

  • L. Greengard V. Rokhlin (1987) ArticleTitleA fast algorithm for particle simulations J. Comput. Phys. 73 325–348 Occurrence Handle10.1016/0021-9991(87)90140-9 Occurrence Handle88k:82007

    Article  MathSciNet  Google Scholar 

  • W. Hackbush (1999) ArticleTitleA sparse matrix arithmetic based on -matrices Computing 62 IssueID2 89–108 Occurrence Handle10.1007/s006070050015 Occurrence Handle2000c:65039

    Article  MathSciNet  Google Scholar 

  • W. Hackbush Z. P. Nowak (1989) ArticleTitleOn the fast matrix multiplication in the boundary-element method by panel clustering Nummath. 54 IssueID4 463–491

    Google Scholar 

  • HSL. A collection of Fortran codes for large scale scientific computation, 2000. http://www.numerical.rl.ac.uk/hsl.

  • S. A. Kharchenko A. Yu (1995) ArticleTitleYeremin. Eigenvalue translation based preconditioners for the GMRES(k) method Numer. Linear Algebra Appl. 2 IssueID1 51–77 Occurrence Handle10.1002/nla.1680020105 Occurrence Handle95i:65069

    Article  MathSciNet  Google Scholar 

  • L.Y. Kolotilina A Yu (1993) ArticleTitleYeremin. Factorized sparse approximate inverse preconditionings. I: Theory SIAM J. Matrix Anal. Appl. 14 45–58 Occurrence Handle10.1137/0614004 Occurrence Handle93m:65061

    Article  MathSciNet  Google Scholar 

  • Lee, J., Lu, C.-C., Zhang, J.: Sparse inverse preconditioning of multilevel fast multipole algorithm for hybrid integral equations in electromagnetics. Technical Report 363-02, Department of Computer Science, University of Kentucky, KY, 2002

  • J. Lee C.-C. Lu J. Zhang (2003) ArticleTitleIncomplete LU preconditioning for large scale dense complex linear systems from electromagnetic wave scattering problems J. Comp. Phys. 185 158–175 Occurrence Handle10.1016/S0021-9991(02)00052-9

    Article  Google Scholar 

  • Morgan, R. B.: Implicitely restarted GMRES and Arnoldi methods for nonsymmetric systems of equations. SIAM J. Matrix Anal. Appl. 21(4), 1112–1135.

  • R. B. Morgan (1995) ArticleTitleA restarted GMRES method augmented with eigenvectors SIAM J. Matrix Anal. Appl. 16 1154–1171 Occurrence Handle10.1137/S0895479893253975 Occurrence Handle0836.65050 Occurrence Handle96f:65037

    Article  MATH  MathSciNet  Google Scholar 

  • R. B. Morgan (2002) ArticleTitleGMRES with deflated restarting SIAM J. Sci. Comput. 24 IssueID1 20–37 Occurrence Handle10.1137/S1064827599364659 Occurrence Handle1018.65042 Occurrence Handle2003h:65040

    Article  MATH  MathSciNet  Google Scholar 

  • Rahola, J.: Experiments on iterative methods and the fast multipole method in electromagnetic scattering calculations. Technical Report TR/PA/98/49, CERFACS, Toulouse, France, 1998

  • S. M. Rao D. R. Wilton A. W. Glisson (1982) ArticleTitleElectromagnetic scattering by surfaces of arbitrary shape IEEE Trans. Antennas Propagat. AP-30 409–418 Occurrence Handle10.1109/TAP.1982.1142818

    Article  Google Scholar 

  • V. Rokhlin (1990) ArticleTitleRapid solution of integral equations of scattering theory in two dimensions J. Comp. Phys. 86 IssueID2 414–439 Occurrence Handle10.1016/0021-9991(90)90107-C Occurrence Handle0686.65079 Occurrence Handle90k:76081

    Article  MATH  MathSciNet  Google Scholar 

  • Y. Saad (1988) ArticleTitleProjection and deflation methods for partial pole assignment in linear state feedback IEEE Trans. Automat. Contr. 33 IssueID3 290–297 Occurrence Handle10.1109/9.406 Occurrence Handle0641.93031 Occurrence Handle927848

    Article  MATH  MathSciNet  Google Scholar 

  • Samant, A. R., Michielssen, E., Saylor, P.:Approximate inverse based preconditioners for 2D dense matrix problems. Technical Report CCEM-11-96, University of Illinois, 1996.

  • K. Sertel J. L. Volakis (2000) ArticleTitleIncomplete LU preconditioner for FMM implementation Micro. Opt. Tech. Lett. 26 IssueID7 265–267 Occurrence Handle10.1002/1098-2760(20000820)26:4<265::AID-MOP18>3.0.CO;2-O

    Article  Google Scholar 

  • J. Song C.-C. Lu W. C. Chew (1997) ArticleTitleMultilevel fast multipole algorithm for electromagnetic scattering by large complex objects IEEE Trans. Antennas Propagation 45 IssueID10 1488–1493 Occurrence Handle10.1109/8.633855

    Article  Google Scholar 

  • Sylvand, G.: La méthode multipôle rapide en electromagnétisme : performances, parallélisation, applications. PhD thesis, Ecole Nationale des Ponts et Chaussées, 2002

  • R. S. Tuminaro (1992) ArticleTitleA highly parallel multigrid-like method for the solution of the Euler equations SIAM J. Sci. Stat. Comput. 13 88–100 Occurrence Handle10.1137/0913005 Occurrence Handle0742.76068 Occurrence Handle92m:76102

    Article  MATH  MathSciNet  Google Scholar 

  • S. A. Vavasis (1992) ArticleTitlePreconditioning for boundary integral equations SIAM J. Matrix Anal. Appl. 13 905–925 Occurrence Handle10.1137/0613055 Occurrence Handle0755.65109 Occurrence Handle93b:65062

    Article  MATH  MathSciNet  Google Scholar 

  • Y. Saad (1993) ArticleTitleAnalysis of augmented Krylov subspace techniques SIAM J. Sci. Comput. 14 461–469 Occurrence Handle10.1137/0914028 Occurrence Handle0780.65022 Occurrence Handle1204241

    Article  MATH  MathSciNet  Google Scholar 

Download references

Author information

Authors and Affiliations

  1. Institute for Mathematics and Scientific Computing, Karl-Franzens University, Heinrichstraße 36, 8010, Graz, Austria

    B. Carpentieri

Authors
  1. B. Carpentieri
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to B. Carpentieri.

Rights and permissions

Reprints and permissions

About this article

Cite this article

Carpentieri, B. A Matrix-free Two-grid Preconditioner for Solving Boundary Integral Equations in Electromagnetism. Computing 77, 275–296 (2006). https://doi.org/10.1007/s00607-006-0161-7

Download citation

  • Received:

  • Revised:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s00607-006-0161-7

AMS Subject Classifications

Keywords

Search

Navigation