A General Algorithm for Reusing Krylov Subspace Information. I. Unsteady Navier-Stokes

A general algorithm is developed that reuses available information to accelerate the iterative convergence of linear systems with multiple right-hand sides A x = b (sup i), which are commonly encountered in steady or unsteady simulations of nonlinear equations. The algorithm is based on the classical GMRES algorithm with eigenvector enrichment but also includes a Galerkin projection preprocessing step and several novel Krylov subspace reuse strategies. The new approach is applied to a set of test problems, including an unsteady turbulent airfoil, and is shown in some cases to provide significant improvement in computational efficiency relative to baseline approaches.

[1]  Misha Elena Kilmer,et al.  Recycling Subspace Information for Diffuse Optical Tomography , 2005, SIAM J. Sci. Comput..

[2]  Henk A. van der Vorst,et al.  Bi-CGSTAB: A Fast and Smoothly Converging Variant of Bi-CG for the Solution of Nonsymmetric Linear Systems , 1992, SIAM J. Sci. Comput..

[3]  Christian Rey,et al.  Iterative accelerating algorithms with Krylov subspaces for the solution to large-scale nonlinear problems , 2004, Numerical Algorithms.

[4]  Cornelis Vuik,et al.  Coarse grid acceleration of a parallel block preconditioner , 2001, Future Gener. Comput. Syst..

[5]  Gene H. Golub,et al.  A Hybrid Approach Combining Chebyshev Filter and Conjugate Gradient for Solving Linear Systems with Multiple Right-Hand Sides , 2007, SIAM J. Matrix Anal. Appl..

[6]  Cornelis Vuik,et al.  Parallelism in ILU-preconditioned GMRES , 1998, Parallel Comput..

[7]  R. Morgan.  Restarted block-GMRES with deflation of eigenvalues , 2005 .

[8]  Ronald B. Morgan,et al.  A Restarted GMRES Method Augmented with Eigenvectors , 1995, SIAM J. Matrix Anal. Appl..

[9]  Yousef Saad,et al.  Hybrid Krylov Methods for Nonlinear Systems of Equations , 1990, SIAM J. Sci. Comput..

[10]  P. Spalart.  A One-Equation Turbulence Model for Aerodynamic Flows , 1992 .

[11]  Kesheng Wu,et al.  Thick-Restart Lanczos Method for Large Symmetric Eigenvalue Problems , 2000, SIAM J. Matrix Anal. Appl..

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

[13]  David E. Keyes,et al.  Nonlinearly Preconditioned Inexact Newton Algorithms , 2002, SIAM J. Sci. Comput..

[14]  A. Segal,et al.  A comparison of various deflation vectors applied to elliptic problems with discontinuous coefficients , 2002 .

[15]  Frédéric Guyomarc'h,et al.  An Augmented Conjugate Gradient Method for Solving Consecutive Symmetric Positive Definite Linear Systems , 2000, SIAM J. Matrix Anal. Appl..

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

[17]  Yousef Saad,et al.  A Flexible Inner-Outer Preconditioned GMRES Algorithm , 1993, SIAM J. Sci. Comput..

[18]  J. Meijerink,et al.  An Efficient Preconditioned CG Method for the Solution of a Class of Layered Problems with Extreme Contrasts in the Coefficients , 1999 .

[19]  Yousef Saad,et al.  Deflated and Augmented Krylov Subspace Techniques , 1997, Numer. Linear Algebra Appl..

[20]  Cornelis Vuik,et al.  The Deflation Accelerated Schwarz Method for CFD , 2005, International Conference on Computational Science.

[21]  Paul Fischer,et al.  PROJECTION TECHNIQUES FOR ITERATIVE SOLUTION OF Ax = b WITH SUCCESSIVE RIGHT-HAND SIDES , 1993 .

[22]  Lois Mansfield.  Damped Jacobi Preconditioning and Coarse Grid Deflation for Conjugate Gradient Iteration on Parallel Computers , 1991, SIAM J. Sci. Comput..

[23]  Gene H. Golub,et al.  Adaptively Preconditioned GMRES Algorithms , 1998, SIAM J. Sci. Comput..

[24]  Kevin Burrage,et al.  On the performance of various adaptive preconditioned GMRES strategies , 1998, Numer. Linear Algebra Appl..

[25]  R. Nicolaides.  Deflation of conjugate gradients with applications to boundary value problems , 1987 .

[26]  E. Sturler,et al.  Truncation Strategies for Optimal Krylov Subspace Methods , 1999 .

[27]  Tony F. Chan,et al.  Analysis of Projection Methods for Solving Linear Systems with Multiple Right-Hand Sides , 1997, SIAM J. Sci. Comput..

[28]  Min Zeng,et al.  Harmonic projection methods for large non-symmetric eigenvalue problems , 1998, Numer. Linear Algebra Appl..

[29]  Paul F. Fischer,et al.  Hybrid Multigrid/Schwarz Algorithms for the Spectral Element Method , 2005, J. Sci. Comput..

[30]  K. Burrage,et al.  Restarted GMRES preconditioned by deflation , 1996 .

[31]  H. V. D. Vorst,et al.  An iterative solution method for solving f ( A ) x = b , using Krylov subspace information obtained for the symmetric positive definite matrix A , 1987 .

[32]  William Gropp,et al.  DOMAIN DECOMPOSITION METHODS IN COMPUTATIONAL FLUID DYNAMICS , 1991 .

[33]  M. Benzi.  Preconditioning techniques for large linear systems: a survey , 2002 .

[34]  Christian Rey,et al.  A Rayleigh–Ritz preconditioner for the iterative solution to large scale nonlinear problems , 1998, Numerical Algorithms.

[35]  Y. Saad.  Analysis of Augmented Krylov Subspace Methods , 1997, SIAM J. Matrix Anal. Appl..

[36]  D. Zingg,et al.  Fast Newton-Krylov method for unstructured grids , 1998 .

[37]  Ronald B. Morgan,et al.  Implicitly Restarted GMRES and Arnoldi Methods for Nonsymmetric Systems of Equations , 2000, SIAM J. Matrix Anal. Appl..

[38]  Efstratios Gallopoulos,et al.  An Iterative Method for Nonsymmetric Systems with Multiple Right-Hand Sides , 1995, SIAM J. Sci. Comput..

[39]  J. Meijerink,et al.  The construction of projection vectors for a deflated ICCG method applied to problems with extreme contrasts in the coefficients , 2001 .

[40]  Danny C. Sorensen,et al.  Implicit Application of Polynomial Filters in a k-Step Arnoldi Method , 1992, SIAM J. Matrix Anal. Appl..

[41]  R. Mittra,et al.  A conjugate gradient algorithm for the treatment of multiple incident electromagnetic fields , 1989 .

[42]  Ronald B. Morgan,et al.  GMRES with Deflated Restarting , 2002, SIAM J. Sci. Comput..

[43]  Cornelis Vuik,et al.  On the Construction of Deflation-Based Preconditioners , 2001, SIAM J. Sci. Comput..

[44]  W. Joubert,et al.  Numerical experiments with parallel orderings for ILU preconditioners. , 1999 .

[45]  Yousef Saad,et al.  Iterative methods for sparse linear systems , 2003 .

[46]  Hester Bijl,et al.  Implicit Time Integration Schemes for the Unsteady Compressible Navier–Stokes Equations: Laminar Flow , 2002 .

[47]  S. A. Kharchenko,et al.  Eigenvalue translation based preconditioners for the GMRES(k) method , 1995, Numer. Linear Algebra Appl..

[48]  Yousef Saad,et al.  High-order ILU preconditioners for CFD problems , 2000 .

[49]  M. Papadrakakis,et al.  A new implementation of the Lanczos method in linear problems , 1990 .

[50]  Eric de Sturler,et al.  Recycling Krylov Subspaces for Sequences of Linear Systems , 2006, SIAM J. Sci. Comput..

[51]  B. Parlett.  The Symmetric Eigenvalue Problem , 1981 .

[52]  Youcef Saad,et al.  A Basic Tool Kit for Sparse Matrix Computations , 1990 .

[53]  Cornelis Vuik,et al.  An Efficient Deflation Method applied on 2-D and 3-D Bubbly Flow Problems , 2006 .

[54]  L. Dutto.  The effect of ordering on preconditioned GMRES algorithm, for solving the compressible Navier-Stokes equations , 1993 .

[55]  Gautam M. Shroff,et al.  Stabilization of unstable procedures: the recursive projection method , 1993 .

[56]  Alan B. Williams,et al.  A Deflation Technique for Linear Systems of Equations , 1998, SIAM J. Sci. Comput..

[57]  Cornelis Vuik,et al.  A Comparison of Deflation and the Balancing Preconditioner , 2005, SIAM J. Sci. Comput..

[58]  D. Keyes,et al.  Jacobian-free Newton-Krylov methods: a survey of approaches and applications , 2004 .

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

[60]  Yuri V. Vassilevski,et al.  Choice of initial guess in iterative solution of series of systems arising in fluid flow simulations , 2006, J. Comput. Phys..

[61]  Hester Bijl,et al.  Fourth-Order Runge–Kutta Schemes for Fluid Mechanics Applications , 2005, J. Sci. Comput..

[62]  G. Golub,et al.  Gmres: a Generalized Minimum Residual Algorithm for Solving , 2022 .