[petsc-users] double orthogonalization for modified gram schmidt in KSPGMRES

[Jose E. Roman]

It is well known that MGS will not guarantee a fully orthogonal basis. However, the Krylov basis is usually good enough when you solve linear systems (GMRES). A different story is when you want to approximate eigenvalues, in which case the quality of the orthogonal basis is more critical.

On the other hand, both MGS with reorthogonalization and CGS with reorthogonalization will give you a similar level of orthogonality. In that scenario, CGS is preferred because its performance is much better (both sequentially and in parallel).

An implementation of MGS with reorthogonalization is available in SLEPc (for eigenvalues).

Jose


> El 16 jun 2025, a las 11:25, shenren_xu at nwpu.edu.cn escribió:
> 
> Dear PETSC moderator, 
> 
> I found there are classical and modified gram-schmidt options for KSPGMRES solver. For classical GS, one could define 
> additional orthogonalization sweeps, while for modified GS there is no such option. I got the impression that PETSC 
> GMRES implementation assumes that the MGS is so robust that double orthogonalization is unnecessary. However, our 
> recent experience indicated that even for MGS, double orth. is nenecessary. Otherwise, GMRES would produce an increase 
> residual convergence history.  
> I'm emailing for clarification on this: was it because I did not use the option correctly with some misunderstanding 
> about the user guide, or is this indeed the current situation for the KSPGMRES solver implemention? 
> 
> As a side note, it was written in Yousef Saad's book 'Iterative methods for sparse linear systems, second editon'  
> (page 162) that 
> "However, there are cases where cancellations are so severe in the orthogonalization steps that even the Modified Gram-Schmidt option is inadequate." It seems that Prof Saad was well aware of this, which backs 
> our finding. 
> 
> Thanks and look forward to further discussion on this. 
> 
> Best regards, 
> Shenren 
> 
> 徐慎忍
> 西北工业大学动力与能源学院 副教授/博导
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