<html aria-label="message body"><head><meta http-equiv="content-type" content="text/html; charset=utf-8"></head><body style="overflow-wrap: break-word; -webkit-nbsp-mode: space; line-break: after-white-space;"><div><br></div> Yes the more the values are near the diagonal likely the better scaling. Also the number of nonzeros per row, the higher that number the better the scaling.<div><br></div><div> Barry</div><div><br id="lineBreakAtBeginningOfMessage"><div><br><blockquote type="cite"><div>On Feb 13, 2026, at 8:43 AM, Matthew Knepley <knepley@gmail.com> wrote:</div><br class="Apple-interchange-newline"><div><meta charset="UTF-8"><div dir="ltr" style="caret-color: rgb(0, 0, 0); font-family: Helvetica; font-size: 12px; font-style: normal; font-variant-caps: normal; font-weight: 400; letter-spacing: normal; orphans: 2; text-align: start; text-indent: 0px; text-transform: none; white-space: normal; widows: 2; word-spacing: 0px; -webkit-text-stroke-width: 0px; text-decoration-line: none; text-decoration-thickness: auto; text-decoration-style: solid;"><div dir="ltr">On Fri, Feb 13, 2026 at 2:14 AM SCOTTO Alexandre <<a href="mailto:alexandre.scotto@irt-saintexupery.com">alexandre.scotto@irt-saintexupery.com</a>> wrote:</div><div class="gmail_quote gmail_quote_container"><blockquote class="gmail_quote" style="margin: 0px 0px 0px 0.8ex; border-left-width: 1px; border-left-style: solid; border-left-color: rgb(204, 204, 204); padding-left: 1ex;"><div class="msg-4283471384865917870"><div lang="EN-US"><div class="m_-4283471384865917870WordSection1"><p class="MsoNormal"><span lang="FR" style="font-size: 11pt; font-family: Calibri, sans-serif; color: rgb(31, 73, 125);">Dear Matthew, Barry,<u></u><u></u></span></p><p class="MsoNormal"><span lang="FR" style="font-size: 11pt; font-family: Calibri, sans-serif; color: rgb(31, 73, 125);"><u></u> <u></u></span></p><p class="MsoNormal"><span style="font-size: 11pt; font-family: Calibri, sans-serif; color: rgb(31, 73, 125);">Thank you for your answers. The question of the problem size was part of my concern regarding the relevance of the quick test setup, I am going to increase the size in the suggested way to see the difference.<u></u><u></u></span></p><p class="MsoNormal"><span style="font-size: 11pt; font-family: Calibri, sans-serif; color: rgb(31, 73, 125);"><u></u> <u></u></span></p><p class="MsoNormal"><span style="font-size: 11pt; font-family: Calibri, sans-serif; color: rgb(31, 73, 125);">Regarding the sparsity pattern, I assume that the more “diagonal” the matrix is the better the speedup, is this is a correct rule of thumb?</span></p></div></div></div></blockquote><div><br></div><div>What I was referring to was the density. The pattern has implications for the cache efficiency. Here is a good paper explaining what is going on:</div><div><br></div><div> <a href="https://urldefense.us/v3/__https://d1wqtxts1xzle7.cloudfront.net/40652293/Toward_Realistic_Performance_Bounds_for_20151205-5192-8jxqcg-libre.pdf?1449311168=&response-content-disposition=inline*3B*filename*3DToward_realistic_performance_bounds_for.pdf&Expires=1770993790&Signature=ITtMQ-YNb5x*ZZnYof32wXbghpN9y5Bf50*ioozZi6O7GXATT4e4wApHuDX0qsrED1Pv--bv*rXFkMz9BpeGHP491X-qcDdKbRNxp7tg2zhKMwTeGpzzUCDV6UGjWcof39UCWzBSgNDhC35BVObFeDelhewIvn0dNI9O-Msr3wOjO51yDYzh1KJO-oTZ6mIDIYDL8S8ioLhnL0z6ec-3dQOmdDJfV6Vty3gkMJAjAhkhUNst2JEqIuRuygYGizCuVhYksH3p-51et7FWtu043MTmBO6lRCbKodbWMGBXvKe8Kox03NDQ2fs5-ClAWTwjd6VTiGpPq6PxP0a9UPvWZQ__&Key-Pair-Id=APKAJLOHF5GGSLRBV4ZA__;JSslfn5-!!G_uCfscf7eWS!axZVqLk0h37e_aRGUtvMl_doja_7Vw4wdRhxvWhzyvFMozVXizBarj_RQS-_qxWkNTGjz50ihtujnoqWm896D_g$">https://d1wqtxts1xzle7.cloudfront.net/40652293/Toward_Realistic_Performance_Bounds_for_20151205-5192-8jxqcg-libre.pdf?1449311168=&response-content-disposition=inline%3B+filename%3DToward_realistic_performance_bounds_for.pdf&Expires=1770993790&Signature=ITtMQ-YNb5x~ZZnYof32wXbghpN9y5Bf50~ioozZi6O7GXATT4e4wApHuDX0qsrED1Pv--bv~rXFkMz9BpeGHP491X-qcDdKbRNxp7tg2zhKMwTeGpzzUCDV6UGjWcof39UCWzBSgNDhC35BVObFeDelhewIvn0dNI9O-Msr3wOjO51yDYzh1KJO-oTZ6mIDIYDL8S8ioLhnL0z6ec-3dQOmdDJfV6Vty3gkMJAjAhkhUNst2JEqIuRuygYGizCuVhYksH3p-51et7FWtu043MTmBO6lRCbKodbWMGBXvKe8Kox03NDQ2fs5-ClAWTwjd6VTiGpPq6PxP0a9UPvWZQ__&Key-Pair-Id=APKAJLOHF5GGSLRBV4ZA</a></div><div><br></div><div> Thanks,</div><div><br></div><div> Matt</div><div> </div><blockquote class="gmail_quote" style="margin: 0px 0px 0px 0.8ex; border-left-width: 1px; border-left-style: solid; border-left-color: rgb(204, 204, 204); padding-left: 1ex;"><div class="msg-4283471384865917870"><div lang="EN-US"><div class="m_-4283471384865917870WordSection1"><p class="MsoNormal"><span style="font-size: 11pt; font-family: Calibri, sans-serif; color: rgb(31, 73, 125);">Best regards,<u></u><u></u></span></p><p class="MsoNormal"><span style="font-size: 11pt; font-family: Calibri, sans-serif; color: rgb(31, 73, 125);">Alexandre.<u></u><u></u></span></p><p class="MsoNormal"><span style="font-size: 11pt; font-family: Calibri, sans-serif; color: rgb(31, 73, 125);"><u></u> <u></u></span></p><div><div style="border-width: 1pt medium medium; border-style: solid none none; border-color: rgb(225, 225, 225) currentcolor currentcolor; padding: 3pt 0cm 0cm;"><p class="MsoNormal"><b><span style="font-size: 11pt; font-family: Calibri, sans-serif;">De :</span></b><span style="font-size: 11pt; font-family: Calibri, sans-serif;"><span class="Apple-converted-space"> </span>Barry Smith <<a href="mailto:bsmith@petsc.dev" target="_blank">bsmith@petsc.dev</a>><br><b>Envoyé :</b><span class="Apple-converted-space"> </span>jeudi 12 février 2026 17:10<br><b>À :</b><span class="Apple-converted-space"> </span>Matthew Knepley <<a href="mailto:knepley@gmail.com" target="_blank">knepley@gmail.com</a>><br><b>Cc :</b><span class="Apple-converted-space"> </span>SCOTTO A</span><span lang="FR" style="font-size: 11pt; font-family: Calibri, sans-serif;">lexandre <<a href="mailto:alexandre.scotto@irt-saintexupery.com" target="_blank">alexandre.scotto@irt-saintexupery.com</a>>;<span class="Apple-converted-space"> </span><a href="mailto:petsc-users@mcs.anl.gov" target="_blank">petsc-users@mcs.anl.gov</a><br><b>Objet :</b><span class="Apple-converted-space"> </span>Re: [petsc-users] Scalability<u></u><u></u></span></p></div></div><p class="MsoNormal"><u></u> <u></u></p><div><p class="MsoNormal"><u></u> <u></u></p></div><p class="MsoNormal"> The problem size is also very small. Typically one cannot get speedup when the number of variables per MPI rank is below on the order of 10,000. In your 64 process case you only have 390 variables. I would be stunned with any kind of speedup for such sizes. Run a problem at least 10 times bigger, better yet 20 times.<u></u><u></u></p><div><p class="MsoNormal"><u></u> <u></u></p><div><p class="MsoNormal"><br><br><u></u><u></u></p><blockquote style="margin-top: 5pt; margin-bottom: 5pt;"><div><p class="MsoNormal">On Feb 12, 2026, at 9:00 AM, Matthew Knepley <<a href="mailto:knepley@gmail.com" target="_blank">knepley@gmail.com</a>> wrote:<u></u><u></u></p></div><p class="MsoNormal"><u></u> <u></u></p><div><div><div><p class="MsoNormal">On Thu, Feb 12, 2026 at 6:48 AM SCOTTO Alexandre via petsc-users <<a href="mailto:petsc-users@mcs.anl.gov" target="_blank">petsc-users@mcs.anl.gov</a>> wrote:<u></u><u></u></p></div><div><blockquote style="border-width: medium medium medium 1pt; border-style: none none none solid; border-color: currentcolor currentcolor currentcolor rgb(204, 204, 204); padding: 0cm 0cm 0cm 6pt; margin-left: 4.8pt; margin-right: 0cm;"><div><div><div><p class="MsoNormal">Dear PETSc community,<u></u><u></u></p><p class="MsoNormal"> <u></u><u></u></p><p class="MsoNormal">I have conducted a quick strong scalability-like test on direct and adjoint matrix-vector product with a 25,000 x 25,000 sparse matrix, distributed over 2, 4, …, 32 and 64 processes and the results I obtained were not so great.<u></u><u></u></p><p class="MsoNormal">I am not very confident in my setup, so a as a matter of reference, is there any available results on weak and strong scalability of PETSc.Mat mult() and multTranspose() operations?<u></u><u></u></p></div></div></div></blockquote><div><p class="MsoNormal"><u></u> <u></u></p></div><div><p class="MsoNormal">1. This behavior depends on available bandwidth, not on cores. Do you know the bandwidth for your configurations?<u></u><u></u></p></div><div><p class="MsoNormal"><u></u> <u></u></p></div><div><p class="MsoNormal">2. Strong scaling depends heavily on matrix sparsity. If inevitably declines, but slower with more work to do.<u></u><u></u></p></div><div><p class="MsoNormal"><u></u> <u></u></p></div><div><p class="MsoNormal">3. We published a paper on performance recently: <a href="https://urldefense.us/v3/__https:/www.sciencedirect.com/science/article/abs/pii/S016781912100079X__;!!G_uCfscf7eWS!Zr5jUpk1srGDF2h9mXmw_GIn1OFZ2g3APzC0JHZREcxRzzy-2Oz2yyBWtzSI6F21kV4W_ubmc7A0NIIoVXnb$" target="_blank">https://www.sciencedirect.com/science/article/abs/pii/S016781912100079X</a><u></u><u></u></p></div><div><p class="MsoNormal"><u></u> <u></u></p></div><div><p class="MsoNormal"> <span class="Apple-converted-space"> </span>Thanks,<u></u><u></u></p></div><div><p class="MsoNormal"><u></u> <u></u></p></div><div><p class="MsoNormal"> <span class="Apple-converted-space"> </span>Matt <u></u><u></u></p></div><blockquote style="border-width: medium medium medium 1pt; border-style: none none none solid; border-color: currentcolor currentcolor currentcolor rgb(204, 204, 204); padding: 0cm 0cm 0cm 6pt; margin-left: 4.8pt; margin-right: 0cm;"><div><div><div><div><p class="MsoNormal"> <u></u><u></u></p></div><p class="MsoNormal">Best regards,<u></u><u></u></p><p class="MsoNormal">Alexandre.<u></u><u></u></p></div></div></div></blockquote></div><div><p class="MsoNormal"><br clear="all"><u></u><u></u></p></div><div><p class="MsoNormal"><u></u> <u></u></p></div><p class="MsoNormal"><span class="m_-4283471384865917870gmailsignatureprefix">--<span class="Apple-converted-space"> </span></span><u></u><u></u></p><div><div><div><div><div><div><div><p class="MsoNormal">What most experimenters take for granted before they begin their experiments is infinitely more interesting than any results to which their experiments lead.<br>-- Norbert Wiener<u></u><u></u></p></div><div><p class="MsoNormal"><u></u> <u></u></p></div><div><p class="MsoNormal"><a href="https://urldefense.us/v3/__http:/www.cse.buffalo.edu/*knepley/__;fg!!G_uCfscf7eWS!Zr5jUpk1srGDF2h9mXmw_GIn1OFZ2g3APzC0JHZREcxRzzy-2Oz2yyBWtzSI6F21kV4W_ubmc7A0NEclfLI_$" target="_blank">https://www.cse.buffalo.edu/~knepley/</a><u></u><u></u></p></div></div></div></div></div></div></div></div></div></blockquote></div><p class="MsoNormal"><u></u> <u></u></p></div></div></div></div></blockquote></div><div><br clear="all"></div><div><br></div><span class="gmail_signature_prefix">--<span class="Apple-converted-space"> </span></span><br><div dir="ltr" class="gmail_signature"><div dir="ltr"><div><div dir="ltr"><div><div dir="ltr"><div>What most experimenters take for granted before they begin their experiments is infinitely more interesting than any results to which their experiments lead.<br>-- Norbert Wiener</div><div><br></div><div><a href="https://urldefense.us/v3/__http://www.cse.buffalo.edu/*knepley/__;fg!!G_uCfscf7eWS!axZVqLk0h37e_aRGUtvMl_doja_7Vw4wdRhxvWhzyvFMozVXizBarj_RQS-_qxWkNTGjz50ihtujnoqWfV1bHoU$" target="_blank">https://www.cse.buffalo.edu/~knepley/</a></div></div></div></div></div></div></div></div></div></blockquote></div><br></div></body></html>