Advertisement

A Parallel Adaptive Block FSAI Preconditioner for Finite Element Geomechanical Models

  • Carlo Janna
  • Massimiliano Ferronato
  • Giuseppe Gambolati
Part of the Advances in Intelligent and Soft Computing book series (AINSC, volume 145)

Abstract

Efficient ad hoc preconditioners are a key factor for a successful implementation of linear solvers in a parallel computing environment. The class of Factorized Sparse Approximate Inverses (FSAI), although originally developed for scalar machines, has proven extremely promising in multicore hardware. A recent evolution of FSAI is Block FSAI (BFSAI) which clusters the largest coefficients of the preconditionedmatrix in a number of diagonal blocks defined in advance. A further improvement of BFSAI is the adaptive BFSAI (labelled ABF) where the non zero pattern of the BFSAI preconditioner is not prescribed a priori but computed automatically and adaptively by a suitable algorithm. Numerical results from large finite element (FE) geomechanical models show that ABF coupled with an incomplete Cholesky factorization of each individual diagonal block, i.e. ABF-IC, may outperform BFSAI-IC by up to a factor 4 while exhibiting an excellent degree of parallelization on any multiprocessor computer.

Keywords

Frobenius Norm Diagonal Block Approximate Inverse Geomechanical Model Preconditioned Matrix 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. 1.
    Agullo, E., Guermouche, A., L’Excellent, J.Y.: A parallel out-of-core multifrontal method: storage of factors on disk and analysis of models for an out-of-core active memory. Parall. Comp. 34, 296–317 (2008)MathSciNetCrossRefGoogle Scholar
  2. 2.
    Benzi, M.: Preconditioning techniques for large linear systems: A survey. J. Comp. Phys. 182, 418–477 (2002)MathSciNetzbMATHCrossRefGoogle Scholar
  3. 3.
    Chow, E.: A priori sparsity patterns for parallel sparse approximate inverse preconditioners. SIAM J. Sci. Comput. 21, 1804–1822 (2000)MathSciNetzbMATHCrossRefGoogle Scholar
  4. 4.
    Chow, E., Saad, Y.: Approximate inverse preconditioners via sparse-sparse iterations. SIAM J. Sci. Comput. 19, 995–1023 (1998)MathSciNetzbMATHCrossRefGoogle Scholar
  5. 5.
    Couturier, R., Denis, C., Jezequel, F.: GREMLINS: A large sparse linear solver for grid environment. Par. Comput. 34, 380–391 (2008)MathSciNetCrossRefGoogle Scholar
  6. 6.
    Erhel, J.: Some properties of Krylov projection methods for large linear systems. In: Ivany, P., Topping, B.H.V. (eds.) Computational Technology Reviews, vol. III, pp. 41–70. Saxe-Coburg Publications, Stirlingshire (2011)Google Scholar
  7. 7.
    Holland, R.M., Wathen, A.J., Shaw, G.J.: Sparse approximate inverses and target matrices. SIAM J. Sci. Comput. 26, 1000–1011 (2005)MathSciNetzbMATHCrossRefGoogle Scholar
  8. 8.
    Janna, C., Ferronato, M., Gambolati, G.: A block FSAI-ILU parallel preconditioner for symmetric positive definite linear systems. SIAM J. Sci. Comput. 32, 2468–2484 (2010)MathSciNetzbMATHCrossRefGoogle Scholar
  9. 9.
    Janna, C., Ferronato, M.: Adaptive pattern research for block FSAI preconditioning. SIAM J. Sci. Comput. (to appear)Google Scholar
  10. 10.
    Kaporin, I.E.: New convergence results and preconditioning strategies for the conjugate gradient method. Numer. Lin. Alg. App. 1, 179–210 (1994)MathSciNetzbMATHCrossRefGoogle Scholar
  11. 11.
    Kolotilina, L.Y., Yeremin, A.Y.: Factorized sparse approximate inverse preconditionings, 1. Theory. SIAM J. Matrix Anal. Appl. 14, 45–58 (1993)MathSciNetzbMATHCrossRefGoogle Scholar
  12. 12.
    Kolotilina, L.Y., Yeremin, A.Y.: Factorized sparse approximate inverse preconditionings, 4. Sipmple approaches to rising efficiency. Numer. Lin. Alg. Appl. 6, 515–531 (1999)MathSciNetzbMATHGoogle Scholar
  13. 13.
    Li, X.Y.S.: An overview of SuperLU: algorithms, implementation and user interface. ACM Trans. on Math. Software 31, 302–325 (2005)zbMATHCrossRefGoogle Scholar
  14. 14.
    Resh, M.M.: High performance computer simulation for engineering: A review. In: Ivany, P., Topping, B.H.V. (eds.) Trends in parallel, distributed, grid and cloud computing for engineering. Computer Science, Engineering and Technology Series, vol. 27, pp. 177–186. Saxe-Coburg Publications, UK (2011)CrossRefGoogle Scholar
  15. 15.
    Saad, Y.: Iterative methods for sparse linear systems. SIAM, Philadelphia (2003)Google Scholar

Copyright information

© Springer-Verlag GmbH Berlin Heidelberg 2012

Authors and Affiliations

  • Carlo Janna
    • 1
  • Massimiliano Ferronato
    • 1
  • Giuseppe Gambolati
    • 1
  1. 1.Department of Mathematical Methods and Models for Scientific ApplicationsUniversity of PadovaPaduaItaly

Personalised recommendations