petsc-3.13.6 2020-09-29

KSPPIPEGCR

Implements a Pipelined Generalized Conjugate Residual method.

Options Database Keys

	-ksp_pipegcr_mmax <N>	- the max number of Krylov directions to orthogonalize against
	-ksp_pipegcr_unroll_w	- unroll w at the storage cost of a maximum of (mmax+1) extra vectors with the benefit of better pipelining (default: PETSC_TRUE)
	-ksp_pipegcr_nprealloc <N>	- the number of vectors to preallocated for storing Krylov directions. Once exhausted new directions are allocated blockwise (default: 5)
	-ksp_pipegcr_truncation_type <standard,notay>	- which previous search directions to orthogonalize against

Notes

The PIPEGCR Krylov method supports non-symmetric matrices and permits the use of a preconditioner which may vary from one iteration to the next. Users can can define a method to vary the preconditioner between iterates via KSPPIPEGCRSetModifyPC(). Restarts are solves with x0 not equal to zero. When a restart occurs, the initial starting solution is given by the current estimate for x which was obtained by the last restart iterations of the PIPEGCR algorithm. The method implemented requires at most the storage of 4 x mmax + 5 vectors, roughly twice as much as GCR.

Only supports left preconditioning.

The natural "norm" for this method is (u,Au), where u is the preconditioned residual. This norm is available at no additional computational cost, as with standard CG. Choosing preconditioned or unpreconditioned norm types involves a blocking reduction which prevents any benefit from pipelining.

Reference

P. Sanan, S.M. Schnepp, and D.A. May, "Pipelined, Flexible Krylov Subspace Methods," SIAM Journal on Scientific Computing 2016 38:5, C441-C470, DOI: 10.1137/15M1049130

Level

intermediate

Location