TY - JOUR
T1 - Projection Improved SPAI Preconditioner for FGMRES
AU - Geng , Mingguang
AU - Sun , Shuli
JO - Numerical Mathematics: Theory, Methods and Applications
VL - 4
SP - 1035
EP - 1052
PY - 2023
DA - 2023/11
SN - 16
DO - http://doi.org/10.4208/nmtma.OA-2022-0202
UR - https://global-sci.org/intro/article_detail/nmtma/22122.html
KW - Generalized minimal residual, preconditioner, sparse approximate inverse, oblique projection, orthogonal projection.
AB - <p style="text-align: justify;">Krylov subspace methods are widely used for solving sparse linear algebraic equations, but they rely heavily on preconditioners, and it is difficult to find
an effective preconditioner that is efficient and stable for all problems. In this paper, a novel projection strategy including the orthogonal and the oblique projection
is proposed to improve the preconditioner, which can enhance the efficiency and
stability of iteration. The proposed strategy can be considered as a minimization
process, where the orthogonal projection minimizes the energy norm of error and
the oblique projection minimizes the 2-norm of the residual, meanwhile they can be
regarded as approaches to correct the approximation by solving low-rank inverse of
the matrices. The strategy is a wide-ranging approach and provides a way to transform the constant preconditioner into a variable one. The paper discusses in detail
the projection strategy for sparse approximate inverse (SPAI) preconditioner applied
to flexible GMRES and conducts the numerical test for problems from different applications. The results show that the proposed projection strategy can significantly
improve the solving process, especially for some non-converging and slowly convergence systems.</p>