Session III.1 - Numerical Linear Algebra
Poster
Achieving scalability with a Hermitian preconditioner for a class of non-Hermitian matrices
Nicole Spillane
CNRS, Ecole Polytechnique, France - This email address is being protected from spambots. You need JavaScript enabled to view it.
This work considers the convergence of GMRES for non-singular problems. Preconditioning and weighted norms within GMRES are considered.
The main focus is on Hermitian preconditioning (even for non-Hermitian problems). It is proposed to choose a Hermitian preconditioner H and to apply GMRES in the inner product induced by H. If moreover, the problem matrix A is positive definite, then a new convergence bound is proved that depends only on how well H preconditions the Hermitian part of A, and on how non-Hermitian A is.
In particular, if a scalable preconditioner is known for the Hermitian part of A, then the proposed method is also scalable. This result is illustrated numerically.
Reference: Nicole Spillane. Hermitian Preconditioning for a class of Non-Hermitian Linear Systems. 2023. https://hal.science/hal-04028590v1/document