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Updating incomplete factorization preconditioners for model order reduction

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Abstract

When solving a sequence of related linear systems by iterative methods, it is common to reuse the preconditioner for several systems, and then to recompute the preconditioner when the matrix has changed significantly. Rather than recomputing the preconditioner from scratch, it is potentially more efficient to update the previous preconditioner. Unfortunately, it is not always known how to update a preconditioner, for example, when the preconditioner is an incomplete factorization. A recently proposed iterative algorithm for computing incomplete factorizations, however, is able to exploit an initial guess, unlike existing algorithms for incomplete factorizations. By treating a previous factorization as an initial guess to this algorithm, an incomplete factorization may thus be updated. We use a sequence of problems from model order reduction. Experimental results using an optimized GPU implementation show that updating a previous factorization can be inexpensive and effective, making solving sequences of linear systems a potential niche problem for the iterative incomplete factorization algorithm.

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References

  1. Ahmad, M.I., Szyld, D.B., van Gijzen, M.B.: Preconditioned multishift biCG for H2-optimal model reduction. Research Report 12-06-15. Department of Mathematics, Temple University (2012)

  2. Alvarado, F.L., Schreiber, R.: Optimal parallel solution of sparse triangular systems. SIAM J. Sci. Comput. 14, 446–460 (1993)

    Article  MathSciNet  MATH  Google Scholar 

  3. Antoulas, A.C.: Approximation of large-scale dynamical systems. SIAM publications, philadelphia (2005)

    Book  MATH  Google Scholar 

  4. Anzt, H.: Asynchronous and multiprecision linear solvers - scalable and fault-tolerant numerics for energy efficient high performance computing. Ph.D. thesis, Karlsruhe Institute of Technology Institute for Applied and Numerical Mathematics (2012)

  5. Anzt, H., Chow, E., Dongarra, J.: Iterative sparse triangular solves for preconditioning. In: Träff, J.L., Hunold, S., Versaci, F. (eds.) Euro-Par 2015: parallel processing, lecture notes in computer science, vol. 9233, pp 650–661. Springer, Berlin (2015), 10.1007/978-3-662-48096-0_50

  6. Badía, J.M., Benner, P., Mayo, R., Quintana-Ortí, E.S.: Parallel algorithms for balanced truncation model reduction of sparse systems. In: Dongarra, J.J., Madsen, K., Wasniewski, J. (eds.) Applied parallel computing: 7th international conference, PARA 2004. no. 3732 in Lecture Notes in Comput. Sci., pp 267–275. Springer, Berlin (2006)

  7. Badía, J.M., Benner, P., Mayo, R., Quintana-Ortí, E.S., Quintana-Ortí, G., Remón, A.: Balanced truncation model reduction of large and sparse generalized linear systems. Technical report Chemnitz Scientific Computing Preprints 06-04. Fakultät für Mathematik, TU Chemnitz (2006)

  8. Badía, J.M., Benner, P., Mayo, R., Quintana-Ortí, E.S., Quintana-Ortí, G., Saak, J.: Parallel order reduction via balanced truncation for optimal cooling of steel profiles. In: Cunha, J.C., Medeiros, P.D. (eds.) Euro-Par 2005 parallel processing, Lecture Notes in Comput. Sci., vol. 3648, pp 857–866. Springer, Berlin (2005), 10.1007/11549468∖_93

  9. Baumann, M., van Gijzen, M.B.: Nested Krylov methods for shifted linear systems. Report 14-01, Delft Institute of Applied Mathematics, Delft University of Technology (2014)

  10. Beattie, C.A., Gugercin, S., Wyatt, S.: Inexact solves in interpolatory model reduction. Linear Algebra Appl. 436(8), 2916–2943 (2012). doi:10.1016/j.laa.2011.07.015

    Article  MathSciNet  MATH  Google Scholar 

  11. Bellavia, S., De Simone, V., di Serafino, D., Morini, B.: Efficient preconditioner updates for shifted linear systems. SIAM J. Sci. Comput. 33(4), 1785–1809 (2011). doi:10.1137/100803419

    Article  MathSciNet  MATH  Google Scholar 

  12. Benner, P., Ezzatti, P., Kressner, D., Quintana-Ortí, E.S., Remón, A.: Accelerating model reduction of large linear systems with graphics processors. In: Jónasson, K. (ed.) Applied Parallel and Scientific Computing, Lecture Notes in Comput. Sci., vol. 7134, pp 88–97. Springer, Berlin (2012), 10.1007/978-3-642-28145-7∖_9

  13. Benner, P., Ezzatti, P., Quintana-Ortí, E.S., Remón, A.: Using hybrid CPU-GPU platforms to accelerate the computation of the matrix sign function. In: Lin, H.X., Alexander, M., Forsell, M., Knüpfer, A., Prodan, R., Sousa, L., Streit, A. (eds.) Euro-Par 2009 – parallel processing workshops, lecture notes in Comput. Sci., vol. 6043, pp 132–139. Springer, Berlin (2010), 10.1007/978-3-642-14122-5_17

  14. Benner, P., Li, J.R., Penzl, T.: Numerical solution of large Lyapunov equations, Riccati equations, and linear-quadratic control problems. Numer. Lin. Alg. Appl. 15(9), 755–777 (2008)

    Article  MathSciNet  MATH  Google Scholar 

  15. Benner, P., Saak, J.: A semi-discretized heat transfer model for optimal cooling of steel profiles. In: Benner, P., Mehrmann, V., Sorensen, D. (eds.) Dimension reduction of large-scale systems, lect. Notes Comput. Sci. Eng., vol. 45, pp 353–356. Springer, Berlin (2005)

  16. Benzi, M., Joubert, W., Mateescu, G.: Numerical experiments with parallel orderings for ILU preconditioners. Electron. Trans. Numer. Anal. 8, 88–114 (1999)

    MathSciNet  MATH  Google Scholar 

  17. Bergman, K., et al.: Exascale computing study: technology challenges in achieving exascale systems. DARPA IPTO ExaScale Computing Study (2008)

  18. Bertaccini, D.: Efficient preconditioning for sequences of parametric complex symmetric linear systems. Electron. Trans. Numer. Anal. 18, 49–64 (2004)

    MathSciNet  MATH  Google Scholar 

  19. Calgaro, C., Chehab, J.P., Saad, Y.: Incremental incomplete LU factorizations with applications. Numerical Linear Algebra with Applications 17(5), 811–837 (2010). doi:10.1002/nla.756

    Article  MathSciNet  MATH  Google Scholar 

  20. Chow, E., Anzt, H., Dongarra, J.: Asynchronous iterative algorithm for computing incomplete factorizations on GPUs. In: Lecture Notes in Comput. Sci., vol. 9137, pp 1–16 (2015)

  21. Chow, E., Patel, A.: Fine-grained parallel incomplete LU factorization. SIAM J. Sci. Comput. 37, C169–C193 (2015)

    Article  MathSciNet  MATH  Google Scholar 

  22. Doi, S.: On parallelism and convergence of incomplete LU factorizations. Appl. Numer. Math. 7(5), 417–436 (1991). doi:10.1016/0168-9274(91)90011-N

    Article  MathSciNet  MATH  Google Scholar 

  23. Druskin, V., Knizhnerman, L., Simoncini, V.: Analysis of the rational Krylov subspace and ADI methods for solving the Lyapunov equation. SIAM J. Numer. Anal. 49, 1875–1898 (2011)

    Article  MathSciNet  MATH  Google Scholar 

  24. Duff, I.S., Meurant, G.A.: The effect of ordering on preconditioned conjugate gradients. BIT 29(4), 635–657 (1989)

    Article  MathSciNet  MATH  Google Scholar 

  25. Duintjer Tebbens, J., Tüma, M.: Efficient preconditioning of sequences of nonsymmetric linear systems. SIAM J. Sci. Comput. 29(5), 1918–1941 (2007)

    Article  MathSciNet  MATH  Google Scholar 

  26. Frommer, A., Szyld, D.B.: On asynchronous iterations. J. Comput. Appl. Math. 123, 201–216 (2000)

    Article  MathSciNet  MATH  Google Scholar 

  27. Gugercin, S., Antoulas, A.C., Beattie, C.: \(\mathcal {H}_2\) model reduction for large-scale dynamical systems. SIAM J. Matrix Anal. Appl. 30 (2), 609–638 (2008)

    Article  MathSciNet  MATH  Google Scholar 

  28. Innovative Computing Lab: Software distribution of MAGMA version 1.6. http://icl.cs.utk.edu/magma/ (2014)

  29. Köhler, M., Saak, J.: Efficiency improving implementation techniques for large scale matrix equation solvers. Chemnitz scientific computing prep. CSC 09-10 TU chemnitz (2009)

  30. Köhler, M., Saak, J.: A shared memory parallel implementation of the IRKA algorithm for \(\mathcal {H}_{2}\) model order reduction. In: Manninen, P., Öster, P. (eds.) Applied Parallel and Scientific Computing, Lecture Notes in Comput. Sci., vol. 7782, pp 541–544. Springer, Berlin (2013), 10.1007/978-3-642-36803-5_42

  31. Laub, A.J., Heath, M.T., Paige, C.C., Ward, R.C.: Computation of system balancing transformations and other applications of simultaneous diagonalization algorithms. IEEE Trans. Automat. Control 32(2), 115–122 (1987)

    Article  MATH  Google Scholar 

  32. Logg, A., Mardal, K.A., Wells, G.N., et al.: Automated solution of differential equations by the finite element method. Springer (2012)

  33. Lukarski, D.: Parallel sparse linear algebra for multi-core and many-core platforms - parallel solvers and preconditioners. Ph.D. thesis, Karlsruhe Institute of Technology (KIT), Germany (2012)

    Google Scholar 

  34. Naumov, M.: Parallel solution of sparse triangular linear systems in the preconditioned iterative methods on the GPU. Tech. Rep. NVR-2011-001 NVIDIA (2011)

  35. Corporation, N V I D I A: NVIDIA CUDA Compute Unified Device Architecture Programming Guide 2.3.1 Edn (2009)

  36. NVIDIA Corporation: NVIDIA CUDA TOOLKIT V6.5 (2014)

  37. NVIDIA Corporation: CUBLAS library user guide Du-06702-001_v6.5 edn (2014)

  38. NVIDIA Corporation: CUSPARSE LIBRARY (2014)

  39. Poole, E.L., Ortega, J.M.: Multicolor ICCG methods for vector computers. SIAM J. Numer. Anal. 24, 1394–1417 (1987)

    Article  MathSciNet  MATH  Google Scholar 

  40. Pothen, A., Alvarado, F.: A fast reordering algorithm for parallel sparse triangular solution. SIAM J. Sci. Stat. Comput. 13(2), 645–653 (1992). doi:10.1137/0913036

    Article  MathSciNet  MATH  Google Scholar 

  41. Saad, Y.: Iterative methods for sparse linear systems. Society for Industrial and Applied Mathematics, Philadelphia (2003)

    Book  MATH  Google Scholar 

  42. Tombs, M.S., Postlethwaite, I.: Truncated balanced realization of a stable nonminimal state-space system. Int. J. Control 46(4), 1319–1330 (1987)

    Article  MathSciNet  MATH  Google Scholar 

  43. Wachspress, E.L.: The ADI model problem. Springer, New York (2013)

    Book  MATH  Google Scholar 

  44. Wolf, M., Heroux, M., Boman, E.: Factors impacting performance of multithreaded sparse triangular solve. In: Palma, J., Daydé, M., Marques, O., Lopes, J. (eds.) High Performance Computing for Computational Science – VECPAR 2010, Lecture Notes in Comput. Sci., vol. 6449, pp 32–44. Springer, Berlin (2011), 10.1007/978-3-642-19328-6_6

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Authors and Affiliations

  1. Innovative Computing Lab, University of Tennessee, Knoxville, TN, USA

    Hartwig Anzt & Jack Dongarra

  2. School of Computational Science and Engineering, Georgia Institute of Technology, Atlanta, GA, USA

    Edmond Chow

  3. Max Planck Institute for Dynamics of Complex Technical Systems, Magdeburg, Germany

    Jens Saak

  4. Oak Ridge National Laboratory, Oak Ridge, TN, 37831, USA

    Jack Dongarra

  5. University of Manchester, Manchester, UK

    Jack Dongarra

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  1. Hartwig Anzt
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  2. Edmond Chow
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  3. Jens Saak
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  4. Jack Dongarra
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Correspondence to Hartwig Anzt.

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Anzt, H., Chow, E., Saak, J. et al. Updating incomplete factorization preconditioners for model order reduction. Numer Algor 73, 611–630 (2016). https://doi.org/10.1007/s11075-016-0110-2

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