Skip to main content
SpringerLink
Log in

Toward Memory-Efficient Linear Solvers

  • Conference paper
  • First Online:
High Performance Computing for Computational Science — VECPAR 2002 (VECPAR 2002)

Part of the book series: Lecture Notes in Computer Science ((LNCS,volume 2565))

  • 700 Accesses

Abstract

We describe a new technique for solvinga sparse linear system Ax = b as a block system AX = B, where multiple startingv ectors and right-hand sides are chosen so as to accelerate convergence. Efficiency is gained by reusing the matrix A in block operations with X and B. Techniques for reducingthe cost of the extra matrix-vector operations are presented.

This is a preview of subscription content, log in via an institution to check access.

Access this chapter

Log in via an institution
Chapter
USD 29.95
Price excludes VAT (USA)
  • Available as PDF
  • Read on any device
  • Instant download
  • Own it forever
eBook
USD 84.99
Price excludes VAT (USA)
  • Available as PDF
  • Read on any device
  • Instant download
  • Own it forever
Softcover Book
USD 109.99
Price excludes VAT (USA)
  • Compact, lightweight edition
  • Dispatched in 3 to 5 business days
  • Free shipping worldwide - see info

Tax calculation will be finalised at checkout

Purchases are for personal use only

Institutional subscriptions

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. Field, M.: Optimizing a parallel conjugate gradient solver. SIAM J. Sci. Stat. Comput. 19 (1998) 27–37 315

    Article  MATH  MathSciNet  Google Scholar 

  2. Simon, H., Yeremin, A.: A new approach to construction of efficient iterative schemes for massively parallel applications: variable block CG and BiCG methods and variable block Arnoldi procedure. In R. Sincovec et al., ed.: Parallel Processingfor Scientific Computing. (1993) 57–60 315

    Google Scholar 

  3. Anderson, W.K., Gropp, W.D., Kaushik, D., Keyes, D. E., Smith, B. F.: Achieving high sustained performance in an unstructured mesh CFD application. In: Proceedings of Supercomputing 99. (1999) Also published as Mathematics and Computer Science Division, Argonne National Laboratory, Technical Report ANL/MCS-P776-0899 315

    Google Scholar 

  4. Farhat, C., Macedo, A., Lesoinne, M.: A two-level domain decomposition method for the iterative solution of high frequency exterior Helmholtz problems. Numerische Mathematik 85 (2000) 283–308 315

    Article  MATH  MathSciNet  Google Scholar 

  5. Dongarra, J., Hammarling, S., Sorensen, D.: Block reduction of matrices to condensed form for eigenvalue computations. J. Comp. Appl. Math. 27 (1989) 215–227 315

    Article  MathSciNet  MATH  Google Scholar 

  6. Dongarra, J.,, DuCroz, J., Hammarling, S., Hanson, R.: An extended set of Fortran Basic Linear Algebra Subprograms. ACM Trans. Math. Software 14 (1988) 1–17 315

    Article  MATH  Google Scholar 

  7. Dongarra, J., DuCroz, J., Du., I., Hammarling, S.: A set of level 3 Basic Linear Algebra Subprograms. ACM TOMS 16 (1990) 1. 315

    Article  MATH  Google Scholar 

  8. Patterson, D., Anderson, T., Cardwell, N., Fromm, R., Keeton, K., Kozyrakis, C., Thomas, R., Yelick, K.: A case for intelligent RAM. IEEE Micro March/April (1997) 34–44 315

    Google Scholar 

  9. Gropp, W., Kaushik, D., Keyes, D., Smith, B.: Toward realistic performance bounds for implicit CFD codes. In the Proceedings of the International Conference on Parallel CFD (1999) 315

    Google Scholar 

  10. Kaushik, D., Keyes, D.: Efficient parallelization of an unstructured grid solver: A memory-centric approach. In the Proceedings of the International Conference on Parallel CFD (1999) 315

    Google Scholar 

  11. Behling, S., Bell, R., Farrell, P., Holtho., H., O'Connell, F., Weir, W.: The POWER4 Processor Introduction and Tuning Guide. IBM Redbooks (2001) 316, 323

    Google Scholar 

  12. Gropp, W., et al.: PETSc 2.0 for MPI. http://www.mcs.anl.gov/petsc/ (1999) 316

  13. Basic Linear Algebra Subprograms Technical (BLAST) Forum: Document for the Basic Linear Algebra Subprograms (BLAS) standard. http://www.netlib.org/utk/papers/blast-forum.html (1998) 316

  14. Hestenes, M., Stiefel, E.: Methods of conjugate gradients for solving linear systems. J. Res. Nat. Bur. Stds. 49 (1952) 409–436 316

    MATH  MathSciNet  Google Scholar 

  15. Nachtigal, N.M., Reddy, S.C., Trefethen, L.N.: How fast are nonsymmetric matrix iterations? SIAM Journal on Matrix Analysis Applications 13 (1992) 778–795 316

    Article  MATH  MathSciNet  Google Scholar 

  16. Saad, Y., Schultz, M.: GMRES: A generalized minimal residual algorithm for solvingn onsymmetric linear systems. SIAM Journal on Scientific and Statistical Computing 7 (1986) 856–869 317

    Article  MATH  MathSciNet  Google Scholar 

  17. Saad, Y.: Iterative Methods for Sparse Linear Systems. PWS Publishing Company (1996) 317, 325

    Google Scholar 

  18. Chapman, A., Saad, Y.: Deflated and augmented Krylov subspace techniques. Linear Algebra with Applications 4 (1997) 43–66 317, 318

    Article  MATH  MathSciNet  Google Scholar 

  19. Morgan, R.B.: A restarted GMRES method augmented with eigenvectors. SIAM Journal on Matrix Analysis and Applications 16 (1995) 1154–1171 317, 318, 320

    Article  MATH  MathSciNet  Google Scholar 

  20. Morgan, R.B.: Implicitly restarted GMRES and Arnoldi methods for nonsymmetric systems of equations. SIAM Journal of Matrix Analysis and Applications 21 (2000) 1112–1135 317

    Article  MATH  Google Scholar 

  21. Saad, Y.: Analysis of augmented Krylov subspace methods. SIAM Journal on Matrix Analysis and Applications 18 (1997) 435–449 317

    Article  MATH  MathSciNet  Google Scholar 

  22. Baglama, J., Calvetti, D., Golub, G., Reichel, L.: Adaptively preconditioned GMRES algorithms. SIAM Journal on Scientific Computing 20 (1998) 243–269 318

    Article  MathSciNet  Google Scholar 

  23. Erhel, J., Burrage, K., Pohl, B.: Restarted GMRES preconditioned by deflation. Journal of Computational Applied Mathematics 69 (1996) 303–318 318

    Article  MATH  MathSciNet  Google Scholar 

  24. Eiermann, M., Ernst, O.G., Schneider, O.: Analysis of acceleration strategiesfor restarted minimum residual methods. Journal of Computational and Applied Mathematics 123 (2000) 261–292 318, 320

    Article  MATH  MathSciNet  Google Scholar 

  25. van der Vorst, H. A., Vuik, C.: GMRESR: a family of nested GMRES methods. Numerical Linear Algebra with Applications 1 (1994) 369–386 318, 319, 320

    Article  MATH  MathSciNet  Google Scholar 

  26. de Sturler, E.: Nested Krylov methods based on GCR. Journal of Computational and Applied Mathematics 67 (1996) 15–41 318, 319, 320

    Article  MATH  MathSciNet  Google Scholar 

  27. de Sturler, E., Fokkema, D.: Nested Krylov methods and preservingort hogonality. In Melson, N., Manteuffel, T., McCormick, S., eds.: Sixth Copper Mountain Conference on Multigrid Methods. Part 1 of NASA conference Publication 3324, NASA (1993) 111–126 318

    Google Scholar 

  28. de Sturler, E.: Truncation strategies for optimal Krylov subspace methods. SIAM Journal on Numerical Analysis 36 (1999) 864–889 318

    Article  MathSciNet  Google Scholar 

  29. Nachitgal, N.M., Reichel, L., Trefethen, L.N.: A hybrid GMRES algorithm for nonsymmetric linear systems. SIAM Journal of Matrix Analysis Applications 13 (1992) 796–825 319

    Article  Google Scholar 

  30. Manteuffel, T. A., Starke, G.: On hybrid iterative methods for nonsymmetric systems of linear equations. Numerical Mathematics 73 (1996) 489–506 319

    Article  MATH  MathSciNet  Google Scholar 

  31. Joubert, W.: A robust GMRES-base adaptive polynomial preconditioningalg orithm for nonsymmetric linear systems. SIAM Journal on Scientific Computing 15 (1994) 427–439 319

    Article  MATH  MathSciNet  Google Scholar 

  32. National Institute of Standards and Technology, Mathematical and Computational Sciences Division: Matrix Market. http://math.nist.gov/MatrixMarket (2002) 321

  33. S. Naffziger, Hammond, G.: The implementation of the next generation 64bitanium microprocessor. In: Proceedings of the IEEE International Solid-State Circuits Conference. (2002) 323

    Google Scholar 

  34. Kessler, R.E., McLellan, E. J., Webb, D.A.: The alpha 21264 microprocessor architecture (2002) http://www.compaq.com/alphaserver/download/ev6chip.pdf. 323

  35. DeGelas, J.: Alphalinux: The penguin drives a Ferrari (2000) http://www.aceshardware.com/Spades 323

  36. Hennessey, J., Patterson, D.: Computer Architecture: A Quantitative Approach. 2nd edn. Morgan Kaufmann (1996) 324

    Google Scholar 

  37. Dongarra, J., Bunch, J., Moler, C., Stewart, G.: LINPACK Users’ Guide. SIAM Publications (1979) 324

    Google Scholar 

  38. Gropp, W.D., Kaushik, D.K., Keyes, D. E., B.F. Smith: Toward realistic performance bounds for implicit CFD codes. In A. Ecer et al., ed.: Proceedings of Parallel CFD’99, Elsevier (1999) 324

    Google Scholar 

Download references

Author information

Authors and Affiliations

  1. University of Colorado, 80302, Boulder, CO, USA

    Allison Baker, John Dennis & Elizabeth R. Jessup

Authors
  1. Allison Baker
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. John Dennis
    View author publications

    You can also search for this author in PubMed Google Scholar

  3. Elizabeth R. Jessup
    View author publications

    You can also search for this author in PubMed Google Scholar

Editor information

Editors and Affiliations

  1. Faculdade de Engenharia da, Universidade do Porto, Rua Dr. Roberto Frias, 4200-465, Porto, Portugal

    José M. L. M. Palma  & A. Augusto Sousa  & 

  2. Department of Computer Science, University of Tennessee, 37996-1301, Knoxville, TN, USA

    Jack Dongarra

  3. Departamento de Sistemas Informáticos y Computación, Universidad Politécnica de Valencia, Camino de Vera, s/n, Apartado 22012, 46020, Valencia, Spain

    Vicente Hernández

Rights and permissions

Reprints and permissions

Copyright information

© 2003 Springer-Verlag Berlin Heidelberg

About this paper

Cite this paper

Baker, A., Dennis, J., Jessup, E.R. (2003). Toward Memory-Efficient Linear Solvers. In: Palma, J.M.L.M., Sousa, A.A., Dongarra, J., Hernández, V. (eds) High Performance Computing for Computational Science — VECPAR 2002. VECPAR 2002. Lecture Notes in Computer Science, vol 2565. Springer, Berlin, Heidelberg. https://doi.org/10.1007/3-540-36569-9_20

Download citation

  • DOI: https://doi.org/10.1007/3-540-36569-9_20

  • Published:

  • Publisher Name: Springer, Berlin, Heidelberg

  • Print ISBN: 978-3-540-00852-1

  • Online ISBN: 978-3-540-36569-3

  • eBook Packages: Springer Book Archive

Publish with us

Policies and ethics

Search

Navigation