A New Analysis of Block Preconditioners for Saddle Point Problems

We consider symmetric saddle point matrices. We analyze block preconditioners based on the knowledge of a good approximation for both the top left block and the Schur complement resulting from its elimination. We obtain bounds on the eigenvalues of the preconditioned matrix that depend only of the quality of these approximations, as measured by the related condition numbers. Our analysis applies to indefinite block diagonal preconditioners, block triangular preconditioners, inexact Uzawa preconditioners, block approximate factorization preconditioners, and a further enhancement of these preconditioners based on symmetric block Gauss--Seidel-type iterations. The analysis is unified and allows the comparison of these different approaches. In particular, it reveals that block triangular and inexact Uzawa preconditioners lead to identical eigenvalue distributions. These theoretical results are illustrated on the discrete Stokes problem. It turns out that the provided bounds allow one to localize accurately bo...

[1]  A. Wathen,et al.  Fast iterative solution of stabilised Stokes systems part II: using general block preconditioners , 1994 .

[2]  R. Bank,et al.  A class of iterative methods for solving saddle point problems , 1989 .

[3]  Barry Lee,et al.  Finite elements and fast iterative solvers: with applications in incompressible fluid dynamics , 2006, Math. Comput..

[4]  Owe Axelsson,et al.  Preconditioning methods for linear systems arising in constrained optimization problems , 2003, Numer. Linear Algebra Appl..

[5]  S. Eisenstat,et al.  Variational Iterative Methods for Nonsymmetric Systems of Linear Equations , 1983 .

[6]  Gene H. Golub,et al.  Numerical solution of saddle point problems , 2005, Acta Numerica.

[7]  Cornelis Vuik,et al.  GMRESR: a family of nested GMRES methods , 1994, Numer. Linear Algebra Appl..

[8]  Artem Napov,et al.  Algebraic Multigrid for Moderate Order Finite Elements , 2014, SIAM J. Sci. Comput..

[9]  P. Strevens.  Iii , 1985 .

[10]  Andrew J. Wathen,et al.  Combination preconditioning of saddle point systems for positive definiteness , 2013, Numer. Linear Algebra Appl..

[11]  John N. Shadid,et al.  A taxonomy and comparison of parallel block multi-level preconditioners for the incompressible Navier-Stokes equations , 2008, J. Comput. Phys..

[12]  Michele Benzi,et al.  On the eigenvalues of a class of saddle point matrices , 2006, Numerische Mathematik.

[13]  M. Saunders,et al.  Solution of Sparse Indefinite Systems of Linear Equations , 1975 .

[14]  Yvan Notay.  Flexible Conjugate Gradients , 2000, SIAM J. Sci. Comput..

[15]  D. Spalding,et al.  A calculation procedure for heat, mass and momentum transfer in three-dimensional parabolic flows , 1972 .

[16]  J. Pasciak,et al.  A preconditioning technique for indefinite systems resulting from mixed approximations of elliptic problems , 1988 .

[17]  NapovArtem,et al.  An Algebraic Multigrid Method with Guaranteed Convergence Rate , 2012 .

[18]  S. SIAMJ.,et al.  AGGREGATION-BASED ALGEBRAIC MULTIGRID FOR CONVECTION-DIFFUSION EQUATIONS∗ , 2012 .

[19]  Walter Zulehner,et al.  Analysis of iterative methods for saddle point problems: a unified approach , 2002, Math. Comput..

[20]  Valeria Simoncini,et al.  Block triangular preconditioners for symmetric saddle-point problems , 2004 .

[21]  Ilaria Perugia,et al.  Block-diagonal and indefinite symmetric preconditioners for mixed finite element formulations , 2000, Numer. Linear Algebra Appl..

[22]  Yvan Notay,et al.  Algebraic multigrid and algebraic multilevel methods: a theoretical comparison , 2005, Numer. Linear Algebra Appl..

[23]  V. Simoncini,et al.  Block--diagonal and indefinite symmetric preconditioners for mixed finite element formulations , 1999 .

[24]  Luca Bergamaschi,et al.  On eigenvalue distribution of constraint‐preconditioned symmetric saddle point matrices , 2012, Numer. Linear Algebra Appl..

[25]  L. Trefethen.  Spectra and pseudospectra , 2005 .

[26]  H. Walker,et al.  GMRES On (Nearly) Singular Systems , 1997, SIAM J. Matrix Anal. Appl..

[27]  G. Golub,et al.  Inexact and preconditioned Uzawa algorithms for saddle point problems , 1994 .

[28]  Barbara Kaltenbacher,et al.  Iterative Solution Methods , 2015, Handbook of Mathematical Methods in Imaging.

[29]  Howard C. Elman,et al.  Finite Elements and Fast Iterative Solvers: with Applications in Incompressible Fluid Dynamics , 2014 .

[30]  Owe Axelsson,et al.  Eigenvalue estimates for preconditioned saddle point matrices , 2006, Numer. Linear Algebra Appl..

[31]  E. Sturler,et al.  Block-diagonal and constraint preconditioners for nonsymmetric indefinite linear systems , 2006 .

[32]  Masaaki Sugihara,et al.  A geometric view of Krylov subspace methods on singular systems , 2011, Numer. Linear Algebra Appl..

[33]  A. Wathen,et al.  Minimum residual methods for augmented systems , 1998 .

[34]  Apostol T. Vassilev,et al.  Analysis of the Inexact Uzawa Algorithm for Saddle Point Problems , 1997 .

[35]  Y. Notay.  An aggregation-based algebraic multigrid method , 2010 .

[36]  Nicholas I. M. Gould,et al.  Preconditioning Saddle-Point Systems with Applications in Optimization , 2010, SIAM J. Sci. Comput..

[37]  Luca Bergamaschi,et al.  A note on eigenvalue distribution of constraint‐preconditioned symmetric saddle point matrices , 2014, Numer. Linear Algebra Appl..

[38]  StübenKlaus.  Algebraic multigrid (AMG) , 1983 .

[39]  Gene H. Golub,et al.  A Note on Preconditioning for Indefinite Linear Systems , 1999, SIAM J. Sci. Comput..

[40]  Yvan Notay,et al.  Algebraic analysis of two‐grid methods: The nonsymmetric case , 2010, Numer. Linear Algebra Appl..

[41]  L. Trefethen,et al.  Numerical linear algebra , 1997 .

[42]  Nicholas I. M. Gould,et al.  Constraint Preconditioning for Indefinite Linear Systems , 2000, SIAM J. Matrix Anal. Appl..

[43]  G. P. Boerstoel,et al.  The Krylov accelerated SIMPLE(R) method for flow problems in industrial furnaces , 2000 .

[44]  D. Bartuschat.  Algebraic Multigrid , 2007 .

[45]  Y. Saad,et al.  GMRES: a generalized minimal residual algorithm for solving nonsymmetric linear systems , 1986 .

[46]  T. Rees,et al.  Block‐triangular preconditioners for PDE‐constrained optimization , 2010, Numer. Linear Algebra Appl..

[47]  Artem Napov,et al.  An Algebraic Multigrid Method with Guaranteed Convergence Rate , 2012, SIAM J. Sci. Comput..

[48]  Valeria Simoncini,et al.  Spectral analysis of inexact constraint preconditioning for symmetric saddle point matrices , 2013 .

[49]  Eric de Sturler,et al.  Block-Diagonal and Constraint Preconditioners for Nonsymmetric Indefinite Linear Systems. Part I: Theory , 2005, SIAM J. Sci. Comput..