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Parallelization of the Rational Arnoldi Algorithm

Rational Krylov methods are applicable to a wide range of scientific computing problems, and the rational Arnoldi algorithm is a commonly used procedure for computing an orthonormal basis of a rational Krylov space. Typically, the computationally most expensive component of this algorithm is the solution of a large linear system of equations at each iteration. We explore the option of solving several linear systems simultaneously, thus constructing the rational Krylov basis in parallel. If this is not done carefully, the basis being orthogonalized may become badly conditioned, leading to numerical instabilities in the orthogonalization process. We introduce the new concept of continuation pairs, which gives rise to a near-optimal parallelization strategy that allows us to control the growth of the condition number of this nonorthogonal basis. As a consequence we obtain a significantly more accurate and reliable parallel rational Arnoldi algorithm. The computational benefits are illustrated using several numerical examples from different application areas.

  • 1.  Benchmark Collection, Oberwolfach Model Reduction Benchmark Collection, data collection, 2003, https://portal.uni-freiburg.de/imteksimulation/downloads/benchmark. Google Scholar

  • 2.  Z. Bai, J. Demmel, J. Dongarra, A. Ruhe, and H. van der Vorst, Templates for the Solution of Algebraic Eigenvalue Problems: A Practical Guide, Software Environ. Tools 11, SIAM, Philadelphia, 2000. Google Scholar

  • 3.  P. Benner and  J. Saak , Numerical solution of large and sparse continuous time algebraic matrix Riccati and Lyapunov equations: A state of the art survey , GAMM-Mitt. Ges. Angew. Math. Mech. , 36 ( 2013 ), pp. 32 -- 52 . CrossrefGoogle Scholar

  • 4.  M. Berljafa and  S. Güttel , A Rational Krylov Toolbox for MATLAB , MIMS EPrint 2014 . 56 , Manchester Institute for Mathematical Sciences, The University of Manchester, Manchester, UK, 2014, software available for download at http://guettel.com/rktoolbox/. Google Scholar

  • 5.  M. Berljafa and  S. Güttel , Generalized rational Krylov decompositions with an application to rational approximation , SIAM J. Matrix Anal. Appl. , 36 ( 2015 ), pp. 894 -- 916 , https://doi.org/10.1137/140998081. LinkISIGoogle Scholar

  • 6.  M. Berljafa and  S. Güttel , The RKFIT algorithm for nonlinear rational approximation , MIMS EPrint 2015 . 38 , Manchester Institute for Mathematical Sciences, The University of Manchester, Manchester, UK, 2015. Google Scholar

  • 7.  R.-U. Börner O. G. Ernst and  S. Güttel , Three-dimensional transient electromagnetic modelling using rational Krylov methods , Geophys. J. Int. , 202 ( 2015 ), pp. 2025 -- 2043 . CrossrefISIGoogle Scholar

  • 8.  T. A. Davis and  Y. Hu , The University of Florida Sparse Matrix Collection , ACM Trans. Math. Software , 38 ( 2011 ), pp. 1 -- 25 . ISIGoogle Scholar

  • 9.  V. Druskin L. Knizhnerman and  V. Simoncini , Analysis of the rational Krylov subspace and ADI methods for solving the Lyapunov equation , SIAM J. Numer. Anal. , 49 ( 2011 ), pp. 1875 -- 1898 , https://doi.org/10.1137/100813257. LinkISIGoogle Scholar

  • 10.  V. Druskin L. Knizhnerman and  M. Zaslavsky , Solution of large scale evolutionary problems using rational Krylov subspaces with optimized shifts , SIAM J. Sci. Comput. , 31 ( 2009 ), pp. 3760 -- 3780 , https://doi.org/10.1137/080742403. LinkISIGoogle Scholar

  • 11.  K. Gallivan and  E. Grimme . Van Dooren, A rational Lanczos algorithm for model reduction , Numer. Algorithms , 12 ( 1996 ), pp. 33 -- 63 . CrossrefISIGoogle Scholar

  • 12.  L. Giraud and  J. Langou , When modified Gram--Schmidt generates a well-conditioned set of vectors , IMA J. Numer. Anal. , 22 ( 2002 ), pp. 521 -- 528 . CrossrefISIGoogle Scholar

  • 13.  T. Göckler and  V. Grimm , Uniform approximation of $\varphi$-functions in exponential integrators by a rational Krylov subspace method with simple poles , SIAM J. Matrix Anal. Appl. , 35 ( 2014 ), pp. 1467 -- 1489 , https://doi.org/10.1137/140964655. LinkISIGoogle Scholar

  • 14.  G. H. Golub and C. F. Van Loan, Matrix Computations, 4th ed., Johns Hopkins University Press, Baltimore, MD, 2013. Google Scholar

  • 15.  L. Grasedyck , Existence of a low rank or ${\mathcal H}$-matrix approximant to the solution of a Sylvester equation , Numer. Linear Algebra Appl. , 11 ( 2004 ), pp. 371 -- 389 . CrossrefISIGoogle Scholar

  • 16.  E. Grimme, Krylov Projection Methods for Model Reduction, Ph.D. thesis, University of Illinois at Urbana-Champaign, Champaign, IL, 1997. Google Scholar

  • 17.  S. Gugercin A. C. Antoulas and  C. Beattie , ${\mathcal H}_2$ model reduction for large-scale linear dynamical systems , SIAM J. Matrix Anal. Appl. , 30 ( 2008 ), pp. 609 -- 638 , https://doi.org/10.1137/060666123. LinkISIGoogle Scholar

  • 18.  S. Güttel, Rational Krylov Methods for Operator Functions, Ph.D. thesis, Institut für Numerische Mathematik und Optimierung der Technischen Universität Bergakademie Freiberg, Freiberg, Germany, 2010. Google Scholar

  • 19.  S. Güttel Krylov approximation of matrix functions: Numerical methods and optimal pole selection , GAMM-Mitt. Ges. Angew. Math. Mech. , 36 ( 2013 ), pp. 8 -- 31 . CrossrefGoogle Scholar

  • 20.  S. Güttel R. V. Beeumen K. Meerbergen and  W. Michiels , NLEIGS: A class of fully rational Krylov methods for nonlinear eigenvalue problems , SIAM J. Sci. Comput. , 36 ( 2014 ), pp. A2842 -- A2864 , https://doi.org/10.1137/130935045. LinkISIGoogle Scholar

  • 21.  E. Jarlebring and  H. Voss , Rational Krylov for nonlinear eigenproblems, An iterative projection method , Appl. Math. , 50 ( 2005 ), pp. 543 -- 554 . CrossrefGoogle Scholar

  • 22.  G. Lassaux and  K. Willcox , Model reduction for active control design using multiple-point Arnoldi methods , AIAA Paper , 616 ( 2003 ), pp. 1 -- 11 . Google Scholar

  • 23.  R. B. Lehoucq and  K. Meerbergen , Using generalized Cayley transformations within an inexact rational Krylov sequence method , SIAM J. Matrix Anal. Appl. , 20 ( 1998 ), pp. 131 -- 148 , https://doi.org/10.1137/S0895479896311220. LinkISIGoogle Scholar

  • 24.  J.-R. Li and  J. White , Low rank solution of Lyapunov equations , SIAM J. Matrix Anal. Appl. , 24 ( 2002 ), pp. 260 -- 280 , https://doi.org/10.1137/S0895479801384937. LinkISIGoogle Scholar

  • 25.  C. C. Paige B. N. Parlett and  H. . van der Vorst, Approximate solutions and eigenvalue bounds from Krylov subspaces , Numer. Linear Algebra Appl. , 2 ( 1995 ), pp. 115 -- 133 . CrossrefISIGoogle Scholar

  • 26.  A. Ruhe Krylov sequence methods for eigenvalue computation , Linear Algebra Appl. , 58 ( 1984 ), pp. 391 -- 405 . CrossrefISIGoogle Scholar

  • 27.  A. Ruhe and  Rational Krylov : A practical algorithm for large sparse nonsymmetric matrix pencils , SIAM J. Sci. Comput. , 19 ( 1998 ), pp. 1535 -- 1551 , https://doi.org/10.1137/S1064827595285597. LinkISIGoogle Scholar

  • 28.  Y. Saad, Iterative Methods for Sparse Linear Systems, 2nd ed., SIAM, Philadelphia, 2003. Google Scholar

  • 29.  D. Skoogh, A parallel rational Krylov algorithm for eigenvalue computations, in Applied Parallel Computing Large Scale Scientific and Industrial Problems, B. K\ragström, J. Dongarra, E. Elmroth, and J. Waśniewski, eds., Lecture Notes in Comput. Sci. 1541, Springer, Berlin, 1998, pp. 521--526. Google Scholar

  • 30.  G. W. Stewart , Backward error bounds for approximate Krylov subspaces , Linear Algebra Appl. , 340 ( 2002 ), pp. 81 -- 86 . CrossrefISIGoogle Scholar

  • 31.  R. Van Beeumen K. Meerbergen and  W. Michiels , Compact rational Krylov methods for nonlinear eigenvalue problems , SIAM J. Matrix Anal. Appl. , 36 ( 2015 ), pp. 820 -- 838 , https://doi.org/10.1137/140976698. LinkISIGoogle Scholar

  • 32.  H. A. Van der Vorst , Computational methods for large eigenvalue problems, in Handb. Numer. Anal., VIII, North--Holland , Amsterdam , 2002 , pp. 3 -- 179 . Google Scholar