Preconditioner

Preconditioner = {char_string}

Description / Usage

Iterative techniques for solving a linear matrix system (see above) often benefit from preconditioning to aid convergence. This optional card provides for the selection of a preconditioner from those available through Aztec. For direct factorization Solution Algorithm specifications, the Preconditioner specification is immaterial since none is performed; in such cases, this card should be omitted.

Valid options for {char_string} are listed below.

none

No preconditioning is performed. This is the default specification if no preconditioner has been specified.

Jacobi

A k-step Jacobi preconditioner is used (block Jacobi for VBR matrices). The number of Jacobi steps, k, is set using the Matrix polynomial order card.

Neumann

A Neumann series polynomial preconditioner is used, where the order of the polynomial, k, is set using the Matrix polynomial order card.

ls

A least-squares polynomial preconditioner is used, where the order of the polynomial, k, is set using the Matrix polynomial order card.

sym_GS

A k-step symmetric Gauss-Seidel preconditioner is used for non-overlapping domain decomposition (additive Schwarz). In parallel, each processor performs one step of symmetric Gauss-Seidel on its local matrix, followed by communication to update boundary values from adjacent processors before performing the next local symmetric Gauss-Seidel step. The number of steps, k, is set using the Matrix polynomial order card.

lu

Approximately solve the processor’s local matrix via direct factorization using Sparse 1.3 in conjunction with a userspecified Matrix drop tolerance.

dom_decomp

A domain-decomposition-based preconditioner (additive Schwarz). Each processor augments its local matrix according to the Matrix factorization overlap card and then approximately solves the resulting linear system using the solver specified by the Matrix subdomain solver card. This is the most often used Preconditioner card.

Examples

Following is a sample card:

Preconditioner = dom_decomp

Technical Discussion

Note that prior to Aztec 2.x, certain subdomain solvers were specified simply as arguments to the Preconditioner card. While this historical usage is permitted via limited backward compatibility in order to ease the transition from Aztec 1 usage, the preferred usage is to specify ILU (and similar) preconditioners as a subdomain solver using the more powerful and flexible options that are available using Aztec 2.x together with this option for the preconditioner. Since subdomain solvers such as ILU and ILUT are powerful and frequently used, this preconditioner option will predominate when iterative solvers are being used, even in serial execution.

The most popular setting is dom_decomp, with a subdomain solver specified in the Matrix Subdomain Solver card. For further details, consult Mike Heroux’s recipe for applying preconditioners and what to dial the knobs to (in Schunk, et. al., 2002).

References

SAND2001-3512J: Iterative Solvers and Preconditioners for Fully-coupled Finite Element Formulations of Incompressible Fluid Mechanics and Related Transport Problems, P. R. Schunk, M. A. Heroux, R. R. Rao, T. A. Baer, S. R. Subia and A. C. Sun, March 2002.