Scalable parallel implementations of the GMRES algorithm via Householder reflections

Applications involving large sparse nonsymmetric linear systems encourage parallel implementations of robust iterative solution methods, such as GMRES(k). One variation of GMRES(k) is to adapt the restart value k for any given problem and use Householder reflections in the orthogonalization phase to achieve high accuracy. The Householder transformations can be performed without global communications and modified to use an arbitrary row distribution of the coefficient matrix. The effect of this modification on the GMRES(k) performance is discussed here. This paper compares the abilities of various parallel GMRES(k) implementations to maintain fixed efficiency with increase in problem size and number of processors.

[1]  Vipin Kumar,et al.  Isoefficiency: measuring the scalability of parallel algorithms and architectures , 1993, IEEE Parallel & Distributed Technology: Systems & Applications.

[2]  Anoop Gupta,et al.  Scaling parallel programs for multiprocessors: methodology and examples , 1993, Computer.

[3]  Roger B. Sidje,et al.  Alternatives for parallel Krylov subspace basis computation , 1997, Numer. Linear Algebra Appl..

[4]  Sosonkina Maria,et al.  A New Adaptive GMRES Algorithm for Achieving High Accuracy , 1996 .

[5]  H. Walker.  Implementation of the GMRES method using householder transformations , 1988 .

[6]  Vipin Kumar,et al.  Performance and Scalability of Preconditioned Conjugate Gradient Methods on Parallel Computers , 1995, IEEE Trans. Parallel Distributed Syst..

[7]  Maria Sosonkina Driver,et al.  Parallel Sparse Linear Algebra for Homotopy Methods , 1997 .

[8]  Yousef Saad,et al.  Iterative solution of general sparse linear systems on clusters of workstations , 1996 .

[9]  Gene H. Golub,et al.  Matrix computations , 1983 .

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

[11]  C. Bischof,et al.  Robust incremental condition estimation , 1991 .

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

[13]  H. V. D. Vorst,et al.  The superlinear convergence behaviour of GMRES , 1993 .

[14]  Å. Björck.  Solving linear least squares problems by Gram-Schmidt orthogonalization , 1967 .