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

[shenren_xu at nwpu.edu.cn]

Dear Prof. Roman,

Thank you very much for the swift reply. I agree with your first paragraph. And your second paragraph answers my planned follow-up question. It's good to know that CGS with reorthogonalization would render a similar result compared with MGS with double orthogonalization. I never thought about that as I assuemd that MGS is always the default option in my limited experience.

I really appreciate your help on this.

Cheers,
Shenren


> -----原始邮件-----
> 发件人: "Jose E. Roman" <jroman at dsic.upv.es>
> 发送时间:2025-06-17 00:38:26 (星期二)
> 收件人: "shenren_xu at nwpu.edu.cn" <shenren_xu at nwpu.edu.cn>
> 抄送: "petsc-users at mcs.anl.gov" <petsc-users at mcs.anl.gov>, 赵家资 <zhaojiazi at mail.nwpu.edu.cn>
> 主题: Re: [petsc-users] double orthogonalization for modified gram schmidt in KSPGMRES
> 
> 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
> > 手机/微信：18762660364
> > 电子邮箱：shenren_xu at nwpu.edu.cn
> > 个人主页：https://urldefense.us/v3/__https://teacher.nwpu.edu.cn/xushenren.html__;!!G_uCfscf7eWS!YoYB4_XPhklQn6ykT0zNJh7eMh1sPOYNDSXEb4BjRYNBgoQdlRlqKreuZ2JEGhr4NAMsqsscF_EHoOIMAPn5phleh1vBxQ$  Shenren Xu, PhD
> > Associate Professor
> > School of Power and Energy
> > Northwestern Polytechnical University
> > Xi'an 710129 , China P.R.
> > Tel: +86-18762660364
> > Web: https://urldefense.us/v3/__https://teacher.nwpu.edu.cn/xushenren.html__;!!G_uCfscf7eWS!YoYB4_XPhklQn6ykT0zNJh7eMh1sPOYNDSXEb4BjRYNBgoQdlRlqKreuZ2JEGhr4NAMsqsscF_EHoOIMAPn5phleh1vBxQ$ 
> > Email: shenren_xu at nwpu.edu.cn 
> > 
>


------------------------------
徐慎忍
西北工业大学动力与能源学院 副教授/博导
手机/微信：18762660364
电子邮箱：shenren_xu at nwpu.edu.cn
个人主页：https://urldefense.us/v3/__https://teacher.nwpu.edu.cn/xushenren.html__;!!G_uCfscf7eWS!YoYB4_XPhklQn6ykT0zNJh7eMh1sPOYNDSXEb4BjRYNBgoQdlRlqKreuZ2JEGhr4NAMsqsscF_EHoOIMAPn5phleh1vBxQ$ 
Shenren Xu, PhD
Associate Professor
School of Power and Energy
Northwestern Polytechnical University
Xi'an 710129 , China P.R.
Tel: +86-18762660364
Web: https://urldefense.us/v3/__https://teacher.nwpu.edu.cn/xushenren.html__;!!G_uCfscf7eWS!YoYB4_XPhklQn6ykT0zNJh7eMh1sPOYNDSXEb4BjRYNBgoQdlRlqKreuZ2JEGhr4NAMsqsscF_EHoOIMAPn5phleh1vBxQ$ 
Email: shenren_xu at nwpu.edu.cn