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A parallel subdomain by subdomain implementation of the implicitly restarted Arnoldi/Lanczos method

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Abstract

This work presents a parallel implementation of the implicitly restarted Arnoldi/Lanczos method for the solution of eigenproblems approximated by the finite element method. The implicitly restarted Arnoldi/Lanczos uses a restart scheme in order to improve the convergence of the desired portion of the spectrum, addressing issues such as memory requirements and computational costs related to the generation and storage of the Krylov basis. The presented implementation is suitable for distributed memory architectures, especially PC clusters. In the parallel solution, a subdomain by subdomain approach was implemented and overlapping and non-overlapping mesh partitions were tested. Compressed data structures in the formats CSRC and CSRC/CSR were used to store the coefficient matrices. The parallelization of numerical linear algebra operations present in both Krylov and implicitly restarted methods are discussed. Numerical examples are shown, in order to point out the efficiency and applicability of the proposed method.

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References

  1. P_ARPACK (1996) An efficient portable large scale eigenvalue package for distributed memory parallel architectures

  2. Arbenz P, Becka M, Geus R, Hetmaniuk U, Mengotti T (2006) On a parallel multilevel preconditioned maxwell eigensolver. Parallel Comput 32: 157–165

    Article  MathSciNet  Google Scholar 

  3. Arbenz P, Hetmaniuk U, Lehoucq RB, Tuminaro R (2005) A comparison of eigensolvers for large-scale 3D modal analysis using AMG-preconditioned iterative solvers. Int J Numer Methods Eng 1: 1–21

    Google Scholar 

  4. Baker CG, Hetmaniuk UL, Lehoucq RB, Tornquist HK (2008) Anasazi software for the numerical solution of scale eigenvalue problems. ACM Trans Math Softw 5: 1–22

    Google Scholar 

  5. Balay S, Gropp WD, McInnes LC, Smith BF (1997) Efficient management of parallelism in object oriented numerical software libraries. In: Arge E, Bruaset AM, Langtangen HP (eds) Modern software tools in scientific computing. Birkhäuser, Boston, pp 163–202

    Chapter  Google Scholar 

  6. Betcke T (2006) A GSVD formulation of a domain decomposition method for planar eigenvalue problems. IMA J Numer Anal 27: 451–478

    Article  MathSciNet  Google Scholar 

  7. Bitzarakis S, Papadrakis M, Kotsopulos A (1997) Parallel solution techniques in computational structural mechanics. Comput Methods Appl Mech Eng 148(1): 75–104

    Article  MATH  Google Scholar 

  8. Daniel JW, Gragg WB, Kaufman L, Stewart GW (1976) Reorthogonalization and stable algorithms for updating the Gram-Schmidt QR factorization. Math Comput 30: 772–795

    MathSciNet  MATH  Google Scholar 

  9. Emad N, Petiton S, Edjlali G (2005) Multiple explicitly restarted Arnoldi method for solving large eigenproblems. SIAM J Sci Comput 27(1): 253–277

    Article  MathSciNet  MATH  Google Scholar 

  10. Ferreira IA (2008) Parallel solution of a thermo-chemo-mechanical model for early age concrete. PhD thesis, Universidade Federal do Rio de Janeiro, COPPE (in Portuguese)

  11. Francis JGF (1961) The QR transformation—part 1. Comput J 4: 265–271

    Article  MathSciNet  Google Scholar 

  12. Francis JGF (1962) The QR transformation—part 2. Comput J 4: 332–345

    Article  MathSciNet  Google Scholar 

  13. Gensenberger M (2010) Improving the parallel performance of a domain decomposition preconditioning technique in the Jacobi-Davidson method for large scale eigenvalue problems. J Appl Numer Math 60(11): 1083–1099

    Article  Google Scholar 

  14. Golub GH, Loan CFV (1996) Matrix computations, 3rd edn. The Jonh Hopkins University Press, Baltimore

    Google Scholar 

  15. Gropp W (1996) A high-performance, portable implementation of the mpi message passing interface standard. Parallel Comput 22(6): 789–828. doi:10.1016/0167-8191(96)00024-5

    Article  MATH  Google Scholar 

  16. Knyazev AV, Skorokhodov L (1994) The preconditioned gradient-type iterative methods in a subspace for partial generalized symmetric eigenvalue problem. SIAM J Numer Anal 31: 451–478

    Article  MathSciNet  Google Scholar 

  17. Hernandez V, Roman JE, Tomas A (2007) Parallel Arnoldi eigensolvers with enhanced scalability via global communications rearrangement. Parallel Comput 33(7–8): 521–540

    Article  MathSciNet  Google Scholar 

  18. Hernández V, Román JE, Tomás A, Vidal V (2007) SLEPc users manual. Scalable library for eigenvalue problem computations. Universidad Politecnica de Valencia

  19. Hestenes M, Stiefel E (1952) Methods of conjugate gradients for solving linear systems. J Res Natl Bur Stand 49(6): 409–436

    MathSciNet  MATH  Google Scholar 

  20. Lehoucq RB (1995) Analysis and implementation of an implicitly restarted Arnoldi iteration. PhD thesis, Rice University, Houston, TX

  21. Lehoucq RB (2001) Implicitly restarted Arnoldi method and subspace iteration. SIAM J Matrix Anal Appl 23(2): 551–562

    Article  MathSciNet  MATH  Google Scholar 

  22. Lehoucq RB (2005) On the convergence of an implicitly restarted Arnoldi method. Technical report, Sandia National Laboratories Albuquerque, NM

  23. Manteuffel TA (1978) Adaptive procedure for estimating parameters for the nonsymmetric Tchebychev iteration. Numer Math 31: 183–208

    Article  MathSciNet  MATH  Google Scholar 

  24. Meerbergen K, Spence A (1997) Implicitly restarted Arnoldi and purification for the shift-invert transformation. Math Comput 66: 667–689

    Article  MathSciNet  MATH  Google Scholar 

  25. Nour-Omid B, Parlett BN, Ericsson T, Jensen PS (1987) How to implement the spectral transformation. Math Comput 48(178): 663–673

    Article  MathSciNet  MATH  Google Scholar 

  26. Paige CC (1970) Pratical use of the symmetric Lanczos process with re-orthogonalization. BIT 10: 183–195

    Article  MathSciNet  MATH  Google Scholar 

  27. Papadrakis M, Bitzarakis S (1996) Domain decomposition methods for serial and parallel processing. Adv Eng Softw 25: 291–307

    Article  Google Scholar 

  28. Rao ARM (2005) MPI-based parallel finite element approaches for implicit nonlinear dynamic analysis employing sparse PCG solvers. Adv Eng Softw 36: 181–198

    Article  MATH  Google Scholar 

  29. Rebollo TC, Vera EC (2004) Study of a non-overlapping domain decomposition method: Poisson and Stokes problems. Appl Numer Math 48: 169–194

    Article  MathSciNet  MATH  Google Scholar 

  30. Ribeiro FLB, Coutinho ALGA (2005) Comparison between element, edge and compressed storage schemes for iterative solutions in finite element analyses. Int J Numer Methods Eng 63: 569–588

    Article  MATH  Google Scholar 

  31. Ribeiro FLB, Ferreira IA (2007) Parallel implementation of the finite element method using compressed data structures. Comput Mech 1: 187–204

    Google Scholar 

  32. Saad Y (1984) Chebyshev acceleration techniques for solving nonsymmetric eigenvalue problems. Math Comput 42: 567–588

    Article  MathSciNet  MATH  Google Scholar 

  33. Saad Y (1994) Sparsekit: a basic tool kit for sparse matrix computations. Technical report, Computer Science Department, University of Minessota

  34. Saad Y (1996) Iterative methods for sparse linear systems. PWS Publishing Company, Boston

    MATH  Google Scholar 

  35. Saad Y, Schultz M (1986) Gmres: a generalized minimal residual algorithm for solving nonsymmetric linear systems. SIAM J Sci Stat Comput 7: 856–869

    Article  MathSciNet  MATH  Google Scholar 

  36. Shahnaz R, Usman A, Chughtai I (2006) Implementation and evaluation of parallel sparse matrix-vector products on distributed memory parallel computers. In: IEEE international conference on cluster computing, 1–6. http://doi.ieeecomputersociety.org/10.1109/CLUSTR.2006.311878

  37. Sorensen DC (1992) Implicit application of polynomial filters in a k-step Arnoldi method. SIAM J Matrix Anal Appl 13(1): 357–385

    Article  MathSciNet  MATH  Google Scholar 

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Authors and Affiliations

  1. Programa de Engenharia Civil, COPPE/Universidade Federal do Rio de Janeiro, Caixa Postal 68506, Rio de Janeiro, RJ, 21945-970, Brazil

    G. O. Ainsworth Jr., F. L. B. Ribeiro & C. Magluta

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  1. G. O. Ainsworth Jr.
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  2. F. L. B. Ribeiro
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  3. C. Magluta
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Correspondence to F. L. B. Ribeiro.

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Ainsworth, G.O., Ribeiro, F.L.B. & Magluta, C. A parallel subdomain by subdomain implementation of the implicitly restarted Arnoldi/Lanczos method. Comput Mech 48, 563–577 (2011). https://doi.org/10.1007/s00466-011-0607-4

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  • DOI: https://doi.org/10.1007/s00466-011-0607-4

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