Abstract
The GMRES method is an iterative method that provides better solutions when dealing with large linear systems of equations with a non-symmetric coefficient matrix. The GMRES method generates a Krylov subspace for the solution, and the augmented GMRES method allows augmentation of the Krylov subspaces by a user supplied subspace which represents certain known features of the desired solution. The augmented GMRES method performs well with suitable augmentation, but performs poorly with unsuitable augmentation. The adaptive augmented GMRES method automatically selects a suitable subspace from a set of candidates supplied by the user. This study shows that this method maintains the performance level of augmented GMRES and lightens the burden it puts on its users. Numerical experiments compare robustness as well as the efficiency of various heuristic strategies. References M. L. Baart. The use of auto-correlation for pseudo-rank determination in noisy ill-conditioned linear least-squares problems. IMA Journal of Numerical Analysis, 2:241--247, 1982. doi:10.1093/imanum/2.2.241 J. Baglama and L. Reichel. Augmented GMRES-type method. Numerical Linear Algebra with Applications, 14:337--350, 2007. doi:10.1002/nla.518 D. Calvetti, B. Lewis, and L. Reichel. GMRES-type methods for inconsistent systems. Linear Algebra and its Applications, 316:157--169, 2000. doi:10.1016/S0024-3795(00)00064-1 D. Calvetti, B. Lewis, and L. Reichel. On the choice of subspace for iterative methods for linear discrete ill-posed problems. Int. J. Appl. Math. Comput. Sci, 11:1069--1092, 2001. http://www.math.kent.edu/ reichel/publications/subspaceselect.pdf G. H. Golub and C. F. Van Loan. Matrix Computations. Johns Hopkins University Press, 2nd edition, 1989. P. C. Hansen. Regularization tools: A matlab package for analysis and solution of discrete ill-posed problems. Numerical Algorithms, 6:1--35, 1994. doi:10.1007/BF02149761 P. C. Hansen. Rank Deficient and Discrete Ill-Posed Problems. SIAM, Philadelphia, 1998. Y. Saad. and M. H. Schultz. GMRES: A generalized minimal residual method for solving nonsymmetric linear systems. SIAM J. Sci. Stat. Comput., 7:856--869, 1986. http://www.stanford.edu/class/cme324/saad-schultz.pdf
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