Deflation techniques are typically used to shift isolated clusters of small eigenvalues in order to obtain a tighter distribution and a smaller condition number. Such changes induce a positive effect in the convergence behavior of Krylov subspace methods, which are among the most popular iterative solvers for large sparse linear systems. We develop a deflation strategy for symmetric saddle point matrices by taking advantage of their underlying block structure. The vectors used for deflation come from an elliptic singular value decomposition relying on the generalized Golub–Kahan bidiagonalization process. The block targeted by deflation is the off-diagonal one since it features a problematic singular value distribution for certain applications. One example is the Stokes flow in elongated channels, where the off-diagonal block has several small, isolated singular values, depending on the length of the channel. Applying deflation to specific parts of the saddle point system is important when using solvers such as CRAIG, which operates on individual blocks rather than the whole system. The theory is developed by extending the existing framework for deflating square matrices before applying a Krylov subspace method such as MINRES. Numerical experiments confirm the merits of our strategy and lead to interesting questions about using approximate vectors for deflation.


  1. Golub–Kahan bidiagonalization
  2. eigenvalue deflation
  3. singular value decomposition
  4. saddle point problems
  5. Stokes equation

MSC codes

  1. 15A18
  2. 35P15
  3. 65F10
  4. 65F15
  5. 65N22

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K. Aishima, Global convergence of the restarted Lanczos and Jacobi–Davidson methods for symmetric eigenvalue problems, Numer. Math., 131 (2015), pp. 405–423.
M. Arioli, Generalized Golub–Kahan bidiagonalization and stopping criteria, SIAM J. Matrix Anal. Appl., 34 (2013), pp. 571–592, https://doi.org/10.1137/120866543.
O. Axelsson and J. Karátson, Reaching the superlinear convergence phase of the CG method, J. Comput. Appl. Math., 260 (2014), pp. 244–257.
J. Baglama and L. Reichel, Augmented implicitly restarted Lanczos bidiagonalization methods, SIAM J. Sci. Comput., 27 (2005), pp. 19–42, https://doi.org/10.1137/04060593X.
J. Baglama, L. Reichel, and D. Richmond, An augmented LSQR method, Numer. Algorithms, 64 (2013), pp. 263–293.
J. L. Barlow, Reorthogonalization for the Golub–Kahan–Lanczos bidiagonal reduction, Numer. Math., 124 (2013), pp. 237–278.
R. Barrett, M. Berry, T. F. Chan, J. Demmel, J. Donato, J. Dongarra, V. Eijkhout, R. Pozo, C. Romine, and H. van der Vorst, Templates for the Solution of Linear Systems: Building Blocks for Iterative Methods, SIAM, 1994, https://doi.org/10.1137/1.9781611971538.
M. Benzi, G. H. Golub, and J. Liesen, Numerical solution of saddle point problems, Acta Numer., 14 (2005), pp. 1–137.
E. V. Chizhonkov and M. A. Olshanskii, On the domain geometry dependence of the LBB condition, ESAIM Math. Model. Numer. Anal., 34 (2000), pp. 935–951.
O. Coulaud, L. Giraud, P. Ramet, and X. Vasseur, Deflation and Augmentation Techniques in Krylov Subspace Methods for the Solution of Linear Systems, arXiv preprint, arXiv:1303.5692, 2013.
H. A. Daas, L. Grigori, P. Hénon, and P. Ricoux, Recycling Krylov subspaces and truncating deflation subspaces for solving sequence of linear systems, ACM Trans. Math. Software, 47 (2021), pp. 1–30.
V. Darrigrand, A. Dumitrasc, C. Kruse, and U. Rüde, Inexact inner–outer Golub–Kahan bidiagonalization method: A relaxation strategy, Numer. Linear Algebra Appl., (2022), e2484.
H. C. Elman, D. J. Silvester, and A. J. Wathen, Finite Elements and Fast Iterative Solvers: With Applications in Incompressible Fluid Dynamics, 2nd ed., Numer. Math. Sci. Comput., Oxford Academic, 2014.
B. Fischer, Polynomial Based Iteration Methods for Symmetric Linear Systems, Classics Appl. Math. 68, SIAM, 2011, https://doi.org/10.1137/1.9781611971927.
B. Fischer, A. Ramage, D. J. Silvester, and A. J. Wathen, Minimum residual methods for augmented systems, BIT, 38 (1998), pp. 527–543.
A. Gaul, M. H. Gutknecht, J. Liesen, and R. Nabben, Deflated and Augmented Krylov Subspace Methods: Basic Facts and a Breakdown-Free Deflated MINRES, MATHEON preprint 759, Technical University Berlin, 2011.
A. Gaul, M. H. Gutknecht, J. Liesen, and R. Nabben, A framework for deflated and augmented Krylov subspace methods, SIAM J. Matrix Anal. Appl., 34 (2013), pp. 495–518, https://doi.org/10.1137/110820713.
L. Giraud and S. Gratton, On the sensitivity of some spectral preconditioners, SIAM J. Matrix Anal. Appl., 27 (2006), pp. 1089–1105, https://doi.org/10.1137/040617546.
L. Giraud, J. Langou, and M. Rozložník, The loss of orthogonality in the Gram-Schmidt orthogonalization process, Comput. Math. Appl., 50 (2005), pp. 1069–1075.
G. H. Golub and R. Underwood, The block Lanczos method for computing eigenvalues, in Mathematical Software, Elsevier, 1977, pp. 361–377.
G. H. Golub and C. F. Van Loan, Matrix Computations, Johns Hopkins Stud. Math. Sci., Johns Hopkins University Press, 1996.
G. H. Golub, Z. Zhang, and H. Zha, Large sparse symmetric eigenvalue problems with homogeneous linear constraints: The Lanczos process with inner–outer iterations, Linear Algebra Appl., 309 (2000), pp. 289–306.
M. Griebel, T. Dornseifer, and T. Neunhoeffer, Numerical Simulation in Fluid Dynamics: A Practical Introduction, Math. Model. Comput. 3, SIAM, 1998, https://doi.org/10.1137/1.9780898719703.
M. H. Gutknecht, Spectral deflation in Krylov solvers: A theory of coordinate space based methods, Electron. Trans. Numer. Anal., 39 (2012), pp. 156–185.
F. H. Harlow and J. E. Welch, Numerical calculation of time-dependent viscous incompressible flow of fluid with free surface, Phys. Fluids, 8 (1965), pp. 2182–2189.
N. Kohl and U. Rüde, Textbook Efficiency: Massively Parallel Matrix-Free Multigrid for the Stokes System, arXiv preprint, arXiv:2010.13513, 2020.
C. Kruse, V. Darrigrand, N. Tardieu, M. Arioli, and U. Rüde, Application of an iterative Golub-Kahan algorithm to structural mechanics problems with multi-point constraints, Adv. Model. Simul. Eng. Sci., 7 (2020), 45.
C. Kruse, M. Sosonkina, M. Arioli, N. Tardieu, and U. Rüde, Parallel performance of an iterative solver based on the Golub-Kahan bidiagonalization, in Parallel Processing and Applied Mathematics (PPAM 2019), Lect. Notes Comp. Sci., 12043, R. Wyrzykowski, E. Deelman, J. Dongarra, and K. Karczewski, eds., Springer, 2020, pp. 104–116.
C. Kruse, M. Sosonkina, M. Arioli, N. Tardieu, and U. Rüde, Parallel solution of saddle point systems with nested iterative solvers based on the Golub-Kahan bidiagonalization, Concurr. Comput., 33 (2021), e5914.
J. Liesen and P. Tichỳ, Convergence analysis of Krylov subspace methods, GAMM-Mitt., 27 (2004), pp. 153–173.
L. A. M. Mello, E. De Sturler, G. H. Paulino, and E. C. N. Silva, Recycling Krylov subspaces for efficient large-scale electrical impedance tomography, Comput. Methods Appl. Mech. Engrg., 199 (2010), pp. 3101–3110.
S. Mercier, S. Gratton, N. Tardieu, and X. Vasseur, A new preconditioner update strategy for the solution of sequences of linear systems in structural mechanics: Application to saddle point problems in elasticity, Comput. Mech., 60 (2017), pp. 969–982.
R. Nabben and C. Vuik, A comparison of deflation and coarse grid correction applied to porous media flow, SIAM J. Numer. Anal., 42 (2004), pp. 1631–1647, https://doi.org/10.1137/S0036142903430451.
M. A. Olshanskii and V. Simoncini, Acquired clustering properties and solution of certain saddle point systems, SIAM J. Matrix Anal. Appl., 31 (2010), pp. 2754–2768, https://doi.org/10.1137/100792652.
D. Orban and M. Arioli, Iterative Solution of Symmetric Quasi-definite Linear Systems, SIAM Spotlights 3, SIAM, 2017, https://doi.org/10.1137/1.9781611974737.
C. C. Paige, Accuracy and effectiveness of the Lanczos algorithm for the symmetric eigenproblem, Linear Algebra Appl., 34 (1980), pp. 235–258.
A. Ramage, D. Ruiz, A. Sartenaer, and C. Tannier, Using partial spectral information for block diagonal preconditioning of saddle-point systems, Comput. Optim. Appl., 78 (2021), pp. 353–375.
M. Rozložník, M. Tůma, A. Smoktunowicz, and J. Kopal, Numerical stability of orthogonalization methods with a non-standard inner product, BIT, 52 (2012), pp. 1035–1058.
Y. Saad, Iterative Methods for Sparse Linear Systems, 2nd ed., SIAM, 2003, https://doi.org/10.1137/1.9780898718003.
M. A. Saunders, Solution of sparse rectangular systems using LSQR and CRAIG, BIT, 35 (1995), pp. 588–604.
J. A. Sifuentes, M. Embree, and R. B. Morgan, GMRES convergence for perturbed coefficient matrices, with application to approximate deflation preconditioning, SIAM J. Matrix Anal. Appl., 34 (2013), pp. 1066–1088, https://doi.org/10.1137/120884328.
K. M. Soodhalter, E. de Sturler, and M. E. Kilmer, A survey of subspace recycling iterative methods, GAMM-Mitt., 43 (2020), e202000016.
A. Stathopoulos and K. Orginos, Computing and deflating eigenvalues while solving multiple right-hand side linear systems with an application to quantum chromodynamics, SIAM J. Sci. Comput., 32 (2010), pp. 439–462, https://doi.org/10.1137/080725532.
H. A. Van der Vorst, Iterative Krylov Methods for Large Linear Systems, Cambridge Monogr. Appl. Comput. Math. 13, Cambridge University Press, 2003.
E. van’t Wout, M. B. van Gijzen, A. Ditzel, A. van der Ploeg, and C. Vuik, The deflated relaxed incomplete Cholesky CG method for use in a real-time ship simulator, Proc. Comput. Sci., 1 (2010), pp. 249–257.
N. Venkovic, P. Mycek, L. Giraud, and O. Le Maitre, Comparative Study of Harmonic and Rayleigh-Ritz Procedures with Applications to Deflated Conjugate Gradients, Technical report TR/PA/20-3, CERFACS, 2020.

Information & Authors


Published In

cover image SIAM Journal on Matrix Analysis and Applications
SIAM Journal on Matrix Analysis and Applications
Pages: 203 - 231
ISSN (online): 1095-7162


Submitted: 6 December 2022
Accepted: 2 October 2023
Published online: 17 January 2024


  1. Golub–Kahan bidiagonalization
  2. eigenvalue deflation
  3. singular value decomposition
  4. saddle point problems
  5. Stokes equation

MSC codes

  1. 15A18
  2. 35P15
  3. 65F10
  4. 65F15
  5. 65N22



Chair for Computer Science 10 - System Simulation, Friedrich-Alexander University Erlangen-Nuremberg, 91058, Erlangen, Germany.
CERFACS, 31057, Toulouse, France.
Ulrich Rüde
Chair for Computer Science 10 - System Simulation, Friedrich-Alexander University Erlangen-Nuremberg, 91058, Erlangen, Germany.

Funding Information

Bavarian Academic Center for Central, Eastern and Southeastern Europe (BAYHOST)
Funding: This research was supported by the Bavarian Academic Center for Central, Eastern and Southeastern Europe (BAYHOST).

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