Size of Krylov Subspace

Size of Krylov subspace = <integer>

Description / Usage

This optional card allows the user to specify the dimension (size) of the Krylov subspace for the gmres option of the Solution Algorithm card, where

<integer>

m, specifies the number of orthogonalization directions and can be any positive integer less than or equal to the order of the matrix.

If the Size of Krylov subspace card is omitted, then the default dimension is m = 30.

Examples

The following is a sample input card:

Size of Krylov subspace = 128

Technical Discussion

If the size of the subspace is at least as large as the maximum number of iterations permitted by the solver then the gmres iteration will not include any restarts. Depending on the problem, restarts may be beneficial, and then again they may not. Particularly poorly conditioned linear systems may never converge below a certain tolerance if gmres is allowed to restart (i.e. they “level off”). However, some linear systems will admit a converged solution more rapidly with restarts than without. Consequently, the user may wish to experiment with different values of this parameter. See the Orthogonalization card for related information.

gmres’ internal iterations create a Krylov subspace up to dimension m (less in some circumstances, such as convergence). The time and space required by the internal iterations increases nonlinearly with m (but see the Orthogonalization card) - a doubling of m will result in more than a doubling of space and time requirements. So simply choosing a very large dimension is generally not recommended.