<html><head><meta http-equiv="Content-Type" content="text/html; charset=utf-8"></head><body style="word-wrap: break-word; -webkit-nbsp-mode: space; line-break: after-white-space;" class=""><div class=""><br class=""></div> Aiden,<div class=""><br class=""></div><div class=""> Do you have true degrees of freedom on both the edges and the vertices? If only have edge variables and no vertex variables but only use "work" values on vertices than maybe DMNETWORK might behave as you need. But my trivial example defintely shows a problem with only edge vertex variables. The DMPLEX object believes there are more than two vertices when there are only two because it cannot determine that some vertices are actually the same.</div><div class=""><br class=""></div><div class=""> Barry<br class=""><div><br class=""><blockquote type="cite" class=""><div class="">On Dec 2, 2021, at 5:31 PM, Zhang, Hong via petsc-dev <<a href="mailto:petsc-dev@mcs.anl.gov" class="">petsc-dev@mcs.anl.gov</a>> wrote:</div><br class="Apple-interchange-newline"><div class=""><meta charset="UTF-8" class=""><div style="font-style: normal; font-variant-caps: normal; font-weight: normal; letter-spacing: normal; text-align: start; text-indent: 0px; text-transform: none; white-space: normal; word-spacing: 0px; -webkit-text-stroke-width: 0px; text-decoration: none; font-family: Calibri, Arial, Helvetica, sans-serif; font-size: 12pt;" class="">All,</div><div style="font-style: normal; font-variant-caps: normal; font-weight: normal; letter-spacing: normal; text-align: start; text-indent: 0px; text-transform: none; white-space: normal; word-spacing: 0px; -webkit-text-stroke-width: 0px; text-decoration: none; font-family: Calibri, Arial, Helvetica, sans-serif; font-size: 12pt;" class="">I'm forwarding Aidan's comments on this issue (see below). Aidan has done extensive study on DMPlex and has been working on hyperbolic flow applications. </div><div style="font-style: normal; font-variant-caps: normal; font-weight: normal; letter-spacing: normal; text-align: start; text-indent: 0px; text-transform: none; white-space: normal; word-spacing: 0px; -webkit-text-stroke-width: 0px; text-decoration: none; font-family: Calibri, Arial, Helvetica, sans-serif; font-size: 12pt;" class=""><br class=""></div><div style="font-style: normal; font-variant-caps: normal; font-weight: normal; letter-spacing: normal; text-align: start; text-indent: 0px; text-transform: none; white-space: normal; word-spacing: 0px; -webkit-text-stroke-width: 0px; text-decoration: none; font-family: Calibri, Arial, Helvetica, sans-serif; font-size: 12pt;" class="">Satish:</div><div style="font-style: normal; font-variant-caps: normal; font-weight: normal; letter-spacing: normal; text-align: start; text-indent: 0px; text-transform: none; white-space: normal; word-spacing: 0px; -webkit-text-stroke-width: 0px; text-decoration: none; font-family: Calibri, Arial, Helvetica, sans-serif; font-size: 12pt;" class="">Please add Aidan to petsc-dev mailing list.</div><div style="font-style: normal; font-variant-caps: normal; font-weight: normal; letter-spacing: normal; text-align: start; text-indent: 0px; text-transform: none; white-space: normal; word-spacing: 0px; -webkit-text-stroke-width: 0px; text-decoration: none; font-family: Calibri, Arial, Helvetica, sans-serif; font-size: 12pt;" class=""><br class=""></div><div style="font-style: normal; font-variant-caps: normal; font-weight: normal; letter-spacing: normal; text-align: start; text-indent: 0px; text-transform: none; white-space: normal; word-spacing: 0px; -webkit-text-stroke-width: 0px; text-decoration: none; font-family: Calibri, Arial, Helvetica, sans-serif; font-size: 12pt;" class="">Hong</div><div style="caret-color: rgb(0, 0, 0); font-family: Helvetica; font-size: 18px; font-style: normal; font-variant-caps: normal; font-weight: normal; letter-spacing: normal; text-align: start; text-indent: 0px; text-transform: none; white-space: normal; word-spacing: 0px; -webkit-text-stroke-width: 0px; text-decoration: none;" class=""><div id="appendonsend" class=""></div><hr tabindex="-1" style="display: inline-block; width: 1110.328125px;" class=""><div id="divRplyFwdMsg" dir="ltr" class=""><font face="Calibri, sans-serif" style="font-size: 11pt;" class=""><b class="">From:</b><span class="Apple-converted-space"> </span>Aidan Hamilton <<a href="mailto:aidan@udel.edu" class="">aidan@udel.edu</a>><br class=""><b class="">Sent:</b><span class="Apple-converted-space"> </span>Thursday, December 2, 2021 2:34 PM<br class=""><b class="">To:</b><span class="Apple-converted-space"> </span>Zhang, Hong <<a href="mailto:hzhang@mcs.anl.gov" class="">hzhang@mcs.anl.gov</a>><br class=""><b class="">Cc:</b><span class="Apple-converted-space"> </span>Sundermann, Rylee Edwin - SDSU Student <<a href="mailto:rylee.sundermann@jacks.sdstate.edu" class="">rylee.sundermann@jacks.sdstate.edu</a>><br class=""><b class="">Subject:</b><span class="Apple-converted-space"> </span>Re: [petsc-dev] DMPLEX cannot support two different edges for the same two vertices, hence DMPLEX cannot?</font><div class=""> </div></div><div lang="EN-US" style="word-wrap: break-word;" class=""><div class="x_WordSection1"><div style="margin: 0in; font-size: 11pt; font-family: Calibri, sans-serif;" class="">Hong,<span class="Apple-converted-space"> </span></div><p class="x_MsoNormal" style="margin: 0in; font-size: 11pt; font-family: Calibri, sans-serif;"> </p><div style="margin: 0in; font-size: 11pt; font-family: Calibri, sans-serif;" class="">I am not on petsc-dev, I guess I should be. Also you can push, you’re always free to push changes.</div><p class="x_MsoNormal" style="margin: 0in; font-size: 11pt; font-family: Calibri, sans-serif;"> </p><div style="margin: 0in; font-size: 11pt; font-family: Calibri, sans-serif;" class="">And it doesn’t matter for anything on my end. Each edge is distinct topological entity, so I have the directed graph</div><p class="x_MsoNormal" style="margin: 0in; font-size: 11pt; font-family: Calibri, sans-serif;"> </p><div style="margin: 0in; font-size: 11pt; font-family: Calibri, sans-serif;" class="">V0 ---- E0 -<span style="font-family: Wingdings;" class=""></span><span class="Apple-converted-space"> </span>V1 ----- E1 ----<span style="font-family: Wingdings;" class=""></span><span class="Apple-converted-space"> </span>V0</div><p class="x_MsoNormal" style="margin: 0in; font-size: 11pt; font-family: Calibri, sans-serif;"> </p><div style="margin: 0in; font-size: 11pt; font-family: Calibri, sans-serif;" class="">I don’t need any special identification. And it works in parallel with globaltolocal seemingly working . Now this is not a standard graph, it follows the more general notion of directed graph, see Diestel page 25, that allows for multi-edges.</div><p class="x_MsoNormal" style="margin: 0in; font-size: 11pt; font-family: Calibri, sans-serif;"> </p><div style="margin: 0in; font-size: 11pt; font-family: Calibri, sans-serif;" class="">Now for my case I don’t need to distinguish between the two edges at all beyond each one having their own unique edge id. In all of our use cases for traffic I don’t think we need to care (and honestly I’m not sure if Shri actually needs to care either).</div><p class="x_MsoNormal" style="margin: 0in; font-size: 11pt; font-family: Calibri, sans-serif;"> </p><div style="margin: 0in; font-size: 11pt; font-family: Calibri, sans-serif;" class="">Now we do have the topological distinction of direction for all of our graphs, which we can use to assign direction to traffic flow. And that will always work. And this notion of direction is the only distinction we need to care about, and we have it.</div><p class="x_MsoNormal" style="margin: 0in; font-size: 11pt; font-family: Calibri, sans-serif;"> </p><div style="margin: 0in; font-size: 11pt; font-family: Calibri, sans-serif;" class="">It’s worth noting that a CW-Complex does support this notion of multiedges, no reason for it not to, so DMPlex should be supporting its topological operations with this case in mind.</div><div style="margin: 0in; font-size: 11pt; font-family: Calibri, sans-serif;" class="">If DMplex can’t support this then dmplex isn’t doing what it claims it does.<span class="Apple-converted-space"> </span></div><p class="x_MsoNormal" style="margin: 0in; font-size: 11pt; font-family: Calibri, sans-serif;"> </p><div style="margin: 0in; font-size: 11pt; font-family: Calibri, sans-serif;" class="">As long as you as the user know that you are dealing with a multi-edge graph, I don’t see why there would be any issues. Any topological query of a multi-edge graph should behave as you would expect a multi-edge graph to behave, though it may behave differently than single edge graphs, as long as you are aware of the difference when you code your residuals and other functions on dmnetwork it should be fine.</div><p class="x_MsoNormal" style="margin: 0in; font-size: 11pt; font-family: Calibri, sans-serif;"> </p><div style="margin: 0in; font-size: 11pt; font-family: Calibri, sans-serif;" class="">Best,<span class="Apple-converted-space"> </span></div><p class="x_MsoNormal" style="margin: 0in; font-size: 11pt; font-family: Calibri, sans-serif;"> </p><div style="margin: 0in; font-size: 11pt; font-family: Calibri, sans-serif;" class="">Aidan<span class="Apple-converted-space"> </span></div><p class="x_MsoNormal" style="margin: 0in; font-size: 11pt; font-family: Calibri, sans-serif;"> </p><p class="x_MsoNormal" style="margin: 0in; font-size: 11pt; font-family: Calibri, sans-serif;"> </p><div style="border-style: solid none none; border-top-width: 1pt; border-top-color: rgb(181, 196, 223); padding: 3pt 0in 0in;" class=""><p class="x_MsoNormal" style="margin: 0in 0in 12pt; font-size: 11pt; font-family: Calibri, sans-serif;"><b class=""><span style="font-size: 12pt;" class="">From:<span class="Apple-converted-space"> </span></span></b><span style="font-size: 12pt;" class="">Zhang, Hong <<a href="mailto:hzhang@mcs.anl.gov" class="">hzhang@mcs.anl.gov</a>><br class=""><b class="">Date:<span class="Apple-converted-space"> </span></b>Thursday, December 2, 2021 at 1:56 PM<br class=""><b class="">To:<span class="Apple-converted-space"> </span></b>Aidan Hamilton <<a href="mailto:aidan@udel.edu" class="">aidan@udel.edu</a>><br class=""><b class="">Cc:<span class="Apple-converted-space"> </span></b>Sundermann, Rylee Edwin - SDSU Student <<a href="mailto:rylee.sundermann@jacks.sdstate.edu" class="">rylee.sundermann@jacks.sdstate.edu</a>><br class=""><b class="">Subject:<span class="Apple-converted-space"> </span></b>Fw: [petsc-dev] DMPLEX cannot support two different edges for the same two vertices, hence DMPLEX cannot?</span></p></div><div class=""><div style="margin: 0in; font-size: 11pt; font-family: Calibri, sans-serif;" class=""><span style="font-size: 12pt;" class="">Aidan,</span></div></div><div class=""><div style="margin: 0in; font-size: 11pt; font-family: Calibri, sans-serif;" class=""><span style="font-size: 12pt;" class="">Are you on petsc-dev email list? </span></div></div><div class=""><div style="margin: 0in; font-size: 11pt; font-family: Calibri, sans-serif;" class=""><span style="font-size: 12pt;" class="">How do you distinguish two parallel edges connecting the same junctions in your traffic code? Do you use </span></div></div><div class=""><div style="margin: 0in; font-size: 11pt; font-family: Calibri, sans-serif;" class=""><span style="font-size: 12pt;" class="">offset_vto, offset_vfrom in EdgeFE?</span></div></div><div class=""><p class="x_MsoNormal" style="margin: 0in; font-size: 11pt; font-family: Calibri, sans-serif;"><span style="font-size: 12pt;" class=""> </span></p></div><div class=""><div style="margin: 0in; font-size: 11pt; font-family: Calibri, sans-serif;" class=""><span style="font-size: 12pt;" class="">I removed white spaces in dgnet.c and dgnet.h. Can I push the changes?</span></div></div><div class=""><div style="margin: 0in; font-size: 11pt; font-family: Calibri, sans-serif;" class=""><span style="font-size: 12pt;" class="">Hong</span></div></div><div class=""><div class="x_MsoNormal" align="center" style="margin: 0in; font-size: 11pt; font-family: Calibri, sans-serif; text-align: center;"><hr size="0" width="86%" align="center" class=""></div><div id="x_divRplyFwdMsg" class=""><div style="margin: 0in; font-size: 11pt; font-family: Calibri, sans-serif;" class=""><b class=""><span style="" class="">From:</span></b><span style="" class=""><span class="Apple-converted-space"> </span>petsc-dev <<a href="mailto:petsc-dev-bounces@mcs.anl.gov" class="">petsc-dev-bounces@mcs.anl.gov</a>> on behalf of Barry Smith <<a href="mailto:bsmith@petsc.dev" class="">bsmith@petsc.dev</a>><br class=""><b class="">Sent:</b><span class="Apple-converted-space"> </span>Wednesday, December 1, 2021 7:41 PM<br class=""><b class="">To:</b><span class="Apple-converted-space"> </span>Abhyankar, Shrirang G <<a href="mailto:shrirang.abhyankar@pnnl.gov" class="">shrirang.abhyankar@pnnl.gov</a>><br class=""><b class="">Cc:</b><span class="Apple-converted-space"> </span>petsc-dev <<a href="mailto:petsc-dev@mcs.anl.gov" class="">petsc-dev@mcs.anl.gov</a>>; Betrie, Getnet <<a href="mailto:gbetrie@anl.gov" class="">gbetrie@anl.gov</a>><br class=""><b class="">Subject:</b><span class="Apple-converted-space"> </span>Re: [petsc-dev] DMPLEX cannot support two different edges for the same two vertices, hence DMPLEX cannot?</span></div><div class=""><p class="x_MsoNormal" style="margin: 0in; font-size: 11pt; font-family: Calibri, sans-serif;"> </p></div></div><div class=""><div class=""><p class="x_MsoNormal" style="margin: 0in; font-size: 11pt; font-family: Calibri, sans-serif;"> </p></div><div style="margin: 0in; font-size: 11pt; font-family: Calibri, sans-serif;" class=""> Shri,<span class="Apple-converted-space"> </span></div><div class=""><p class="x_MsoNormal" style="margin: 0in; font-size: 11pt; font-family: Calibri, sans-serif;"> </p></div><div class=""><div style="margin: 0in; font-size: 11pt; font-family: Calibri, sans-serif;" class=""> If you provided your graph in parallel I suspect that the GlobalToLocal and LocalToGlobal would not work as you expect and put everything in the appropriate place.</div></div><div class=""><p class="x_MsoNormal" style="margin: 0in; font-size: 11pt; font-family: Calibri, sans-serif;"> </p></div><div class=""><div style="margin: 0in; font-size: 11pt; font-family: Calibri, sans-serif;" class=""> I'll set up a simple code that takes exactly Get's configuration and send it out so everyone can run it and see exactly what happens.</div></div><div class=""><p class="x_MsoNormal" style="margin: 0in; font-size: 11pt; font-family: Calibri, sans-serif;"> </p></div><div class=""><div style="margin: 0in; font-size: 11pt; font-family: Calibri, sans-serif;" class=""> Barry</div></div><div class=""><p class="x_MsoNormal" style="margin: 0in; font-size: 11pt; font-family: Calibri, sans-serif;"> </p><div class=""><div style="margin: 0in; font-size: 11pt; font-family: Calibri, sans-serif;" class=""><br class=""><br class=""></div><blockquote style="margin-top: 5pt; margin-bottom: 5pt;" class=""><div class=""><div style="margin: 0in; font-size: 11pt; font-family: Calibri, sans-serif;" class="">On Dec 1, 2021, at 8:00 PM, Abhyankar, Shrirang G via petsc-dev <<a href="mailto:petsc-dev@mcs.anl.gov" class="">petsc-dev@mcs.anl.gov</a>> wrote:</div></div><p class="x_MsoNormal" style="margin: 0in; font-size: 11pt; font-family: Calibri, sans-serif;"> </p><div class=""><div class=""><div class=""><p class="x_MsoNormal" style="margin: 0in; font-size: 11pt; font-family: Calibri, sans-serif;"><span style="font-size: 12pt; font-family: "Times New Roman", serif;" class=""> </span></p></div><div class=""><p class="x_MsoNormal" style="margin: 0in; font-size: 11pt; font-family: Calibri, sans-serif;"><span style="font-size: 12pt; font-family: "Times New Roman", serif;" class=""> </span></p></div><div style="border-style: solid none none; border-top-width: 1pt; border-top-color: rgb(181, 196, 223); padding: 3pt 0in 0in;" class=""><div style="margin-left: 0.5in;" class=""><div style="margin: 0in; font-size: 11pt; font-family: Calibri, sans-serif;" class=""><b class=""><span style="font-size: 12pt;" class="">From:<span class="x_xapple-converted-space"> </span></span></b><span style="font-size: 12pt;" class="">Matthew Knepley <<a href="mailto:knepley@gmail.com" class="">knepley@gmail.com</a>><br class=""><b class="">Date:<span class="x_xapple-converted-space"> </span></b>Wednesday, December 1, 2021 at 5:57 PM<br class=""><b class="">To:<span class="x_xapple-converted-space"> </span></b>"Brown, Jedediah A (VISIT)" <<a href="mailto:jed@jedbrown.org" class="">jed@jedbrown.org</a>><br class=""><b class="">Cc:<span class="x_xapple-converted-space"> </span></b>"Abhyankar, Shrirang G" <<a href="mailto:shrirang.abhyankar@pnnl.gov" class="">shrirang.abhyankar@pnnl.gov</a>>, PETSc Development <<a href="mailto:petsc-dev@mcs.anl.gov" class="">petsc-dev@mcs.anl.gov</a>>, Getnet Betrie <<a href="mailto:gbetrie@anl.gov" class="">gbetrie@anl.gov</a>><br class=""><b class="">Subject:<span class="x_xapple-converted-space"> </span></b>Re: [petsc-dev] DMPLEX cannot support two different edges for the same two vertices, hence DMPLEX cannot?</span></div></div></div><div class=""><div style="margin-left: 0.5in;" class=""><p class="x_MsoNormal" style="margin: 0in; font-size: 11pt; font-family: Calibri, sans-serif;"> </p></div></div><div style="border-style: none none none solid; border-left-width: 6pt; border-left-color: rgb(215, 118, 0); padding: 0in; font-size: 1.15rem;" class=""><div style="margin-left: 0.5in;" class=""><div style="margin: 0in; font-size: 11pt; font-family: Calibri, sans-serif; text-align: center; background-color: rgb(247, 227, 204);" class=""><span style="font-family: Arial, sans-serif;" class="">Check twice before you click! This email originated from outside PNNL.</span></div></div></div><div style="margin-left: 0.5in;" class=""><p class="x_MsoNormal" style="margin: 0in; font-size: 11pt; font-family: Calibri, sans-serif;"> </p></div><div class=""><div class=""><div class=""><div style="margin-left: 0.5in;" class=""><div style="margin: 0in; font-size: 11pt; font-family: Calibri, sans-serif;" class="">On Wed, Dec 1, 2021 at 6:55 PM Jed Brown <<a href="mailto:jed@jedbrown.org" class="">jed@jedbrown.org</a>> wrote:</div></div></div><div class=""><blockquote style="border-style: none none none solid; border-left-width: 1pt; border-left-color: rgb(204, 204, 204); padding: 0in 0in 0in 6pt; margin: 5pt 0in 5pt 4.8pt;" class=""><div style="margin-left: 0.5in;" class=""><div style="margin: 0in; font-size: 11pt; font-family: Calibri, sans-serif;" class="">Matthew Knepley <<a href="mailto:knepley@gmail.com" target="_blank" class="">knepley@gmail.com</a>> writes:<br class=""><br class="">> On Wed, Dec 1, 2021 at 5:17 PM Abhyankar, Shrirang G <<br class="">><span class="x_xapple-converted-space"> </span><a href="mailto:shrirang.abhyankar@pnnl.gov" target="_blank" class="">shrirang.abhyankar@pnnl.gov</a>> wrote:<br class="">><br class="">>> “You can certainly have many fields on a given edge, but I don't know<br class="">>> what it would mean to have two edges since no topological query could tell<br class="">>> the difference.”<br class="">>><br class="">>><br class="">>><br class="">>> The two edges in a power grid represent two parallel power lines that are<br class="">>> connected between two locations (vertices). There are line ids (stored in<br class="">>> the component data) to distinguish the two lines.<br class="">>><br class="">><br class="">> Yes, so you can tell the difference in the function space (since difference<br class="">> current passes down each one), but _topologically_ you cannot. If you put<br class="">> duplicate cells in, then<br class="">> some topological queries will give unexpected results, like the join of the<br class="">> two vertices.<br class=""><br class="">This could be modeled with some ghost vertices. So instead of<br class=""><br class=""> a ------ b<br class=""> \_____/<br class=""><br class="">you would set up<br class=""><br class=""> a ---o---- b<br class=""> \___o___/<br class=""><br class="">Those ghost vertices don't have to "do" anything, but they make the edges topologically distinct.<br class=""><br class="">Shri, what problems might this cause?</div></div></blockquote></div><div style="margin-left: 0.5in;" class=""><p class="x_MsoNormal" style="margin: 0in; font-size: 11pt; font-family: Calibri, sans-serif;"> </p></div><div class=""><div style="margin: 0in; font-size: 11pt; font-family: Calibri, sans-serif;" class=""><span style="font-size: 12pt; font-family: "Times New Roman", serif;" class="">I don’t understand the figure you’ve drawn above. Sorry.</span></div></div><div class=""><p class="x_MsoNormal" style="margin: 0in; font-size: 11pt; font-family: Calibri, sans-serif;"><span style="font-size: 12pt; font-family: "Times New Roman", serif;" class=""> </span></p></div><div class=""><div style="margin: 0in; font-size: 11pt; font-family: Calibri, sans-serif;" class=""><span style="font-size: 12pt; font-family: "Times New Roman", serif;" class="">As a user, would I need to add anything to the way I am setting up the network/plex or any additional equations in the residual evaluation?</span></div></div><div class=""><p class="x_MsoNormal" style="margin: 0in; font-size: 11pt; font-family: Calibri, sans-serif;"><span style="font-size: 12pt; font-family: "Times New Roman", serif;" class=""> </span></p></div><div class=""><div style="margin: 0in; font-size: 11pt; font-family: Calibri, sans-serif;" class=""><span style="font-size: 12pt; font-family: "Times New Roman", serif;" class="">I do not have any issue right now for the power grid problem since I don’t require DMNetwork or DMPLEX to do the topological distinction between parallel edges. There are unique edge identifiers in my dataset through which I can make this distinction.</span></div></div><div class=""><p class="x_MsoNormal" style="margin: 0in; font-size: 11pt; font-family: Calibri, sans-serif;"><span style="font-size: 12pt; font-family: "Times New Roman", serif;" class=""> </span></p></div><div class=""><p class="x_MsoNormal" style="margin: 0in; font-size: 11pt; font-family: Calibri, sans-serif;"><span style="font-size: 12pt; font-family: "Times New Roman", serif;" class=""> </span></p></div><div class=""><div style="margin-left: 0.5in;" class=""><div style="margin: 0in; font-size: 11pt; font-family: Calibri, sans-serif;" class="">Yes, this would work, but it looks like the multiple cells are not causing them problems right now with the questions they are asking the mesh.</div></div></div><div class=""><div style="margin-left: 0.5in;" class=""><p class="x_MsoNormal" style="margin: 0in; font-size: 11pt; font-family: Calibri, sans-serif;"> </p></div></div><div class=""><div style="margin-left: 0.5in;" class=""><div style="margin: 0in; font-size: 11pt; font-family: Calibri, sans-serif;" class=""> Matt</div></div></div><div class=""><div style="margin-left: 0.5in;" class=""><p class="x_MsoNormal" style="margin: 0in; font-size: 11pt; font-family: Calibri, sans-serif;"> </p></div></div><div style="margin-left: 0.5in;" class=""><div style="margin: 0in; font-size: 11pt; font-family: Calibri, sans-serif;" class="">--<span class="x_xapple-converted-space"> </span></div></div><div class=""><div class=""><div class=""><div class=""><div class=""><div class=""><div class=""><div style="margin-left: 0.5in;" class=""><div style="margin: 0in; font-size: 11pt; font-family: Calibri, sans-serif;" class="">What most experimenters take for granted before they begin their experiments is infinitely more interesting than any results to which their experiments lead.<br class="">-- Norbert Wiener</div></div></div><div class=""><div style="margin-left: 0.5in;" class=""><p class="x_MsoNormal" style="margin: 0in; font-size: 11pt; font-family: Calibri, sans-serif;"> </p></div></div><div class=""><div style="margin-left: 0.5in;" class=""><div style="margin: 0in; font-size: 11pt; font-family: Calibri, sans-serif;" class=""><a href="https://www.google.com/url?q=https://gcc02.safelinks.protection.outlook.com/?url%3Dhttp%253A%252F%252Fwww.cse.buffalo.edu%252F~knepley%252F%26data%3D04%257C01%257Cshrirang.abhyankar%2540pnnl.gov%257Cbcbb40fc51f6428ac17708d9b5264361%257Cd6faa5f90ae240338c0130048a38deeb%257C0%257C0%257C637739998211230005%257CUnknown%257CTWFpbGZsb3d8eyJWIjoiMC4wLjAwMDAiLCJQIjoiV2luMzIiLCJBTiI6Ik1haWwiLCJXVCI6Mn0%253D%257C3000%26sdata%3DSg7eEbjzkSqVd6Sjg8IpTN3iXxMAvih0UNV0fkolO8w%253D%26reserved%3D0&source=gmail-imap&ust=1639079810000000&usg=AOvVaw2q1URmUrcN7rBglSVGeqIH" target="_blank" class="">https://www.cse.buffalo.edu/~knepley/</a></div></div></div></div></div></div></div></div></div></div></div></div></div></blockquote></div><p class="x_MsoNormal" style="margin: 0in; font-size: 11pt; font-family: Calibri, sans-serif;"> </p></div></div></div></div></div></div><span id="cid:EC07DA99CD79D8409CDF329BC1CADDD9@namprd09.prod.outlook.com"><DiestelGT.pdf></span></div></blockquote></div><br class=""></div></body></html>