Journal of applied mathematics & informatics

The Korean Society for Computational and Applied Mathematics (한국전산응용수학회)
권호 표지

DATA MINING AND PREDICTION OF SAI TYPE MATRIX PRECONDITIONER

  • Kim, Sang-Bae (Department of Mathematics, Hannam University) ;
  • Xu, Shuting (Department of Computer Information Systems, Virginia State University) ;
  • Zhang, Jun (Department of Computer Science, University of Kentucky)
  • Published : 2010.01.30
Download PDF
⟨ Previous Next ⟩

Abstract

The solution of large sparse linear systems is one of the most important problems in large scale scientific computing. Among the many methods developed, the preconditioned Krylov subspace methods are considered the preferred methods. Selecting a suitable preconditioner with appropriate parameters for a specific sparse linear system presents a challenging task for many application scientists and engineers who have little knowledge of preconditioned iterative methods. The prediction of ILU type preconditioners was considered in [27] where support vector machine(SVM), as a data mining technique, is used to classify large sparse linear systems and predict best preconditioners. In this paper, we apply the data mining approach to the sparse approximate inverse(SAI) type preconditioners to find some parameters with which the preconditioned Krylov subspace method on the linear systems shows best performance.

Keywords

References

  1. M. W. Benson and P. O. Frederickson. Iterative solution of large sparse linear systems arising in certain multidimensional approximation problems. Utilitas Math., 22:127-140, 1982.
  2. M. W. Benson, J. Krettmann, and M. Wright. Parallel algorithms for the solution of certain large sparse linear systems. Int. J. Comput. Math, 16:245-260, 1984. https://doi.org/10.1080/00207168408803441
  3. E. Chow. A priori sparsity patterns for parallel sparse approximate inverse preconditioners. SIAM J. Sci. Comput., 21(5):1804-1822, 2000.
  4. E. Chow and Y. Saad. Approximate inverse preconditioners via sparse-sparse iterations. SIAM J. Sci. Comput., 19(3):995-1023, 1998. https://doi.org/10.1137/S1064827594270415
  5. J. D. F. Cosgrove, J. C. Diaz, and A. Griewank. Approximate inverse preconditionings for sparse linear systems. Int. J. Comput. Math., 44:91-110, 1992. https://doi.org/10.1080/00207169208804097
  6. T. Davis. University of Florida sparse matrix collection. NA Digest, 97(23), June 1997.
  7. I.S. Duff and J.K. Reid A.M. Erisman. Direct Methods for Sparse Matrices. Clarendon Press, New York, NY, 1986.
  8. M. Grote and T. Huckle. Parallel preconditioning with sparse approximate inverses. SIAM J. Sci. Comput., 18:838-853, 1997. https://doi.org/10.1137/S1064827594276552
  9. M. Grote and H. D. Simon. Parallel preconditioning and approximate inverse on the Connection machines. In R. F. Sincovec, D. E. Keyes, M. R. Leuze, L. R. Petzold, and D. A. Reed, editors, Proceedings of the Sixth SIAM Conference on Parallel Proceessing for Scientific Computing, pages 519-523, Philadelphia, PA, 1993.
  10. T. Joachims. Making large-scale svm learning practical. In C. Burges B. Scholkopf and A. Smola, editors, Advances in Kernel Methods - Support Vector Learning. MIT-Press, 1999.
  11. N. Kang, E. S. Carlson, and J. Zhang. Parallel simulation of anisotropic diffusion with human brain dt-mri data. Computers and Structures, 82(28):2389-2399, 2004. https://doi.org/10.1016/j.compstruc.2004.04.011
  12. S. Karaa, C. C. Douglas, and J. Zhang. Preconditioned multigrid simulation of an axisymmetric laminar diffusion flame. Math. Comput. Model., 38:269-279, 2003. https://doi.org/10.1016/S0895-7177(03)90086-0
  13. L. Y. Kolotilina. Explicit preconditioning of systems of linear algebraic equations with dense matrices. J. Soviet Math., 43:2566-2573, 1988. https://doi.org/10.1007/BF01374987
  14. J. Lee, C. Lu, and J. Zhang. Performance of preconditioned Krylov iterative methods for solving hybrid integral equations in electromagnetics. J. of Applied Comput. Electromagnetics Society, 18:54-61, 2003.
  15. http://math.nist.gov / MatrixMarket.
  16. Y. Saad. Iterative Methods for Sparse Linear Systems. PWS Publishing, New York, NY, 1996.
  17. V. N. Vapnik. Tshe Nature of Statistical Learning Theory. Springer, New York, 1995.
  18. V. N. Vapnik. Statistical Learning Theory. John Wiley and Sons, New York, 1998.
  19. K. Wang, S.-B. Kim, and J. Zhang. A comparative study on dynamic and static sparsity patterns in parallel sparse approximate inverse preconditioning. Journal of Math. Model. Alg., 2(3): 203-215, 2003.
  20. S. Xu. Study and Design of An Intelligent Preconditioner Recommendation System. PhD thesis, Department of Computer Science, University of Kentucky, Lexington, KY, 2005.
  21. S. Xu, E. Lee, and J. Zhang. Designing and building an intelligent preconditioning recommendation system. In Abstracts of the 2003 International Conference of Preconditioning Techniques for Large Sparse Matrix Problems in Scientific and Industrial Applications, Napa, CA, 2003.
  22. S. Xu, E. Lee, and J. Zhang. An interim analysis report on preconditioners and matrices. Technical Report 388-03, Department of Computer Science, University of Kentucky, Lexington, KY, 2003.
  23. S. Xu and J. Zhang. A new data mining approach to predicting matrix condition numbers. Commun.Inform. Systems, 4(4):325-340, 2004.
  24. S. Xu and J. Zhang. A data mining approach to matrix preconditioning problem. In Proceedings of the Eighth Workshop on Mining Scientific and Engineering Datasets (MSD '05), in conjunction with the Fifth SIAM International Confererence on Data Mining, pages 49-57, Newport Beach, CA, 2005.
  25. S. Xu and J. Zhang. Matrix condition number prediction with SVM regression and feature selection. In Proceedings of the Fifth SIAM International Conference on Data Mining, Newport Beach, CA, 2005.
  26. S. Xu and J. Zhang. Solvability prediction of sparse matrices with matrix structure-based preconditioners. In Abstracts of the 2005 International Conference on Preconditioning Techniques for Large Sparse Matrix Problems in Scientific and Industrial Applications, Atlanta, GA, 2005.
  27. S. Xu and J. Zhang. SVM classification for predicting sparse matrix solvability with parameterized matrix preconditioners. Technical Report 405-06, Department of Computer Science, University of Kentucky, Lexington, KY, 2006.
  28. J. Zhang. Preconditioned Krylov subspace methods for solving nonsymmetric matrices from CFD applications. Comput. Methods Appl. Mech. Engrg., 189(3):825-840, 2000. https://doi.org/10.1016/S0045-7825(99)00345-X
  29. J. Zhang. Performance of ILU preconditioners for stationary 3D Navier-Stokes simulation and the matrix mining project. In proceedings of the 2001 International Conference on Preconditioning Techniques for Large Sparse Matrix Problems in Scientific and Industrial Applications, pages 89-90, Tahoe City, CA, 2001.