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" <jro...@dsic.upv.es> > 发送时间:2025-06-17 00:38:26 (星期二) > 收件人: "shenren...@nwpu.edu.cn" <shenren...@nwpu.edu.cn> > 抄送: "petsc-users@mcs.anl.gov" <petsc-users@mcs.anl.gov>, 赵家资 > <zhaoji...@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...@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...@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...@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...@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...@nwpu.edu.cn
