<html xmlns:o="urn:schemas-microsoft-com:office:office" xmlns:w="urn:schemas-microsoft-com:office:word" xmlns:m="http://schemas.microsoft.com/office/2004/12/omml" xmlns="http://www.w3.org/TR/REC-html40">
<head>
<meta http-equiv="Content-Type" content="text/html; charset=utf-8">
<meta name="Generator" content="Microsoft Word 15 (filtered medium)">
<style><!--
/* Font Definitions */
@font-face
{font-family:"Cambria Math";
panose-1:2 4 5 3 5 4 6 3 2 4;}
@font-face
{font-family:Calibri;
panose-1:2 15 5 2 2 2 4 3 2 4;}
@font-face
{font-family:Menlo;
panose-1:2 11 6 9 3 8 4 2 2 4;}
/* Style Definitions */
p.MsoNormal, li.MsoNormal, div.MsoNormal
{margin:0cm;
font-size:10.0pt;
font-family:"Calibri",sans-serif;}
a:link, span.MsoHyperlink
{mso-style-priority:99;
color:blue;
text-decoration:underline;}
p.m-968874379040273580m-4128250693350498318p1, li.m-968874379040273580m-4128250693350498318p1, div.m-968874379040273580m-4128250693350498318p1
{mso-style-name:m_-968874379040273580m-4128250693350498318p1;
mso-margin-top-alt:auto;
margin-right:0cm;
mso-margin-bottom-alt:auto;
margin-left:0cm;
font-size:11.0pt;
font-family:"Calibri",sans-serif;}
span.m-968874379040273580m-4128250693350498318gmailsignatureprefix
{mso-style-name:m_-968874379040273580m-4128250693350498318gmailsignatureprefix;}
span.m-968874379040273580gmailsignatureprefix
{mso-style-name:m_-968874379040273580gmailsignatureprefix;}
span.gmailsignatureprefix
{mso-style-name:gmail_signature_prefix;}
span.EmailStyle22
{mso-style-type:personal-reply;
font-family:"Calibri",sans-serif;
color:windowtext;}
p.p1, li.p1, div.p1
{mso-style-name:p1;
margin:0cm;
background:white;
font-size:8.5pt;
font-family:Menlo;
color:black;}
p.p2, li.p2, div.p2
{mso-style-name:p2;
margin:0cm;
background:white;
font-size:8.5pt;
font-family:Menlo;
color:black;}
p.p3, li.p3, div.p3
{mso-style-name:p3;
margin:0cm;
background:white;
font-size:8.5pt;
font-family:Menlo;
color:#5D6C79;}
p.p4, li.p4, div.p4
{mso-style-name:p4;
margin:0cm;
background:white;
font-size:8.5pt;
font-family:Menlo;
color:#C41A16;}
span.s1
{mso-style-name:s1;
color:#3900A0;}
span.s2
{mso-style-name:s2;
color:#643820;}
span.s3
{mso-style-name:s3;
color:#6C36A9;}
span.s4
{mso-style-name:s4;
color:#C41A16;}
span.s5
{mso-style-name:s5;
color:#9B2393;}
span.s6
{mso-style-name:s6;
color:#5D6C79;}
span.apple-converted-space
{mso-style-name:apple-converted-space;}
span.s7
{mso-style-name:s7;}
.MsoChpDefault
{mso-style-type:export-only;
font-size:10.0pt;
mso-ligatures:none;}
@page WordSection1
{size:612.0pt 792.0pt;
margin:72.0pt 72.0pt 72.0pt 72.0pt;}
div.WordSection1
{page:WordSection1;}
--></style>
</head>
<body lang="EN-GB" link="blue" vlink="purple" style="word-wrap:break-word">
<div class="WordSection1">
<p class="MsoNormal"><span style="font-size:11.0pt;mso-fareast-language:EN-US">Great to know there is an example now.<o:p></o:p></span></p>
<p class="MsoNormal"><span style="font-size:11.0pt;mso-fareast-language:EN-US"><o:p> </o:p></span></p>
<p class="MsoNormal"><span style="font-size:11.0pt;mso-fareast-language:EN-US">Yes, -pc_fieldsplit_defect_saddle_point worked.<o:p></o:p></span></p>
<p class="MsoNormal"><span style="font-size:11.0pt;mso-fareast-language:EN-US"><o:p> </o:p></span></p>
<p class="MsoNormal"><span style="font-size:11.0pt;mso-fareast-language:EN-US">But I would like to understand what I did wrong (and why it didn’t work).
<o:p></o:p></span></p>
<p class="MsoNormal"><span style="font-size:11.0pt;mso-fareast-language:EN-US">After reading many posted posts, I needed to create PCFieldSplitSetIS to split the fields.<o:p></o:p></span></p>
<p class="MsoNormal"><span style="font-size:11.0pt;mso-fareast-language:EN-US">I set the first N indices to the field phi and the next C indices (starting from N) to the field lambda.
<o:p></o:p></span></p>
<p class="MsoNormal"><span style="font-size:11.0pt;mso-fareast-language:EN-US">Please tell me what I did wrong and how I can fix the lines which are now commented?<o:p></o:p></span></p>
<p class="MsoNormal"><span style="font-size:11.0pt;mso-fareast-language:EN-US"><o:p> </o:p></span></p>
<p class="MsoNormal" style="background:white"><span style="font-size:8.5pt;font-family:Menlo;color:#3900A0"> PetscErrorCode ierr;<o:p></o:p></span></p>
<p class="MsoNormal" style="background:white"><span style="font-size:8.5pt;font-family:Menlo;color:black">
</span><span style="font-size:8.5pt;font-family:Menlo;color:#3900A0">KSP</span><span style="font-size:8.5pt;font-family:Menlo;color:black"> ksp;<o:p></o:p></span></p>
<p class="MsoNormal" style="background:white"><span style="font-size:8.5pt;font-family:Menlo;color:black">
</span><span style="font-size:8.5pt;font-family:Menlo;color:#3900A0">PC</span><span style="font-size:8.5pt;font-family:Menlo;color:black"> pc;<o:p></o:p></span></p>
<p class="MsoNormal" style="background:white"><span style="font-size:8.5pt;font-family:Menlo;color:black"><o:p> </o:p></span></p>
<p class="MsoNormal" style="background:white"><span style="font-size:8.5pt;font-family:Menlo;color:#6C36A9"> KSPCreate(PETSC_COMM_WORLD, &ksp);<o:p></o:p></span></p>
<p class="MsoNormal" style="background:white"><span style="font-size:8.5pt;font-family:Menlo;color:black">
</span><span style="font-size:8.5pt;font-family:Menlo;color:#6C36A9">KSPSetType</span><span style="font-size:8.5pt;font-family:Menlo;color:black">(ksp,
</span><span style="font-size:8.5pt;font-family:Menlo;color:#643820">KSPGMRES</span><span style="font-size:8.5pt;font-family:Menlo;color:black">);<o:p></o:p></span></p>
<p class="MsoNormal" style="background:white"><span style="font-size:8.5pt;font-family:Menlo;color:black">
</span><span style="font-size:8.5pt;font-family:Menlo;color:#6C36A9">KSPSetOperators</span><span style="font-size:8.5pt;font-family:Menlo;color:black">(ksp,
</span><span style="font-size:8.5pt;font-family:Menlo;color:#326D74">A</span><span style="font-size:8.5pt;font-family:Menlo;color:black">[level],
</span><span style="font-size:8.5pt;font-family:Menlo;color:#326D74">A</span><span style="font-size:8.5pt;font-family:Menlo;color:black">[level]);<o:p></o:p></span></p>
<p class="MsoNormal" style="background:white"><span style="font-size:8.5pt;font-family:Menlo;color:black"> ierr =
</span><span style="font-size:8.5pt;font-family:Menlo;color:#6C36A9">KSPGetPC</span><span style="font-size:8.5pt;font-family:Menlo;color:black">(ksp,&pc);</span><span style="font-size:8.5pt;font-family:Menlo;color:#643820">CHKERRQ</span><span style="font-size:8.5pt;font-family:Menlo;color:black">(ierr);<o:p></o:p></span></p>
<p class="MsoNormal" style="background:white"><span style="font-size:8.5pt;font-family:Menlo;color:black"> ierr =
</span><span style="font-size:8.5pt;font-family:Menlo;color:#6C36A9">PCSetType</span><span style="font-size:8.5pt;font-family:Menlo;color:black">(pc,</span><span style="font-size:8.5pt;font-family:Menlo;color:#643820">PCFIELDSPLIT</span><span style="font-size:8.5pt;font-family:Menlo;color:black">);</span><span style="font-size:8.5pt;font-family:Menlo;color:#643820">CHKERRQ</span><span style="font-size:8.5pt;font-family:Menlo;color:black">(ierr);<o:p></o:p></span></p>
<p class="p1"><o:p> </o:p></p>
<p class="p1"> ……..<o:p></o:p></p>
<p class="p1"><o:p> </o:p></p>
<p class="MsoNormal" style="background:white"><b><span style="font-size:8.5pt;font-family:Menlo;color:#9B2393"> for</span></b><span style="font-size:8.5pt;font-family:Menlo;color:black"> (</span><b><span style="font-size:8.5pt;font-family:Menlo;color:#9B2393">int</span></b><span style="font-size:8.5pt;font-family:Menlo;color:black">
i = </span><span style="font-size:8.5pt;font-family:Menlo;color:#1C00CF">0</span><span style="font-size:8.5pt;font-family:Menlo;color:black">; i < N; i++)<o:p></o:p></span></p>
<p class="MsoNormal" style="background:white"><span style="font-size:8.5pt;font-family:Menlo;color:black"> vec_zero_field_index[i] = i;<o:p></o:p></span></p>
<p class="MsoNormal" style="background:white"><span style="font-size:8.5pt;font-family:Menlo;color:black"> <o:p></o:p></span></p>
<p class="MsoNormal" style="background:white"><span style="font-size:8.5pt;font-family:Menlo;color:black">
</span><span style="font-size:8.5pt;font-family:Menlo;color:#3900A0">PetscInt</span><span style="font-size:8.5pt;font-family:Menlo;color:black"> * vec_first_field_index =
</span><b><span style="font-size:8.5pt;font-family:Menlo;color:#9B2393">NULL</span></b><span style="font-size:8.5pt;font-family:Menlo;color:black">;<o:p></o:p></span></p>
<p class="MsoNormal" style="background:white"><span style="font-size:8.5pt;font-family:Menlo;color:black">
</span><span style="font-size:8.5pt;font-family:Menlo;color:#643820">PetscMalloc</span><span style="font-size:8.5pt;font-family:Menlo;color:black">(n *
</span><b><span style="font-size:8.5pt;font-family:Menlo;color:#9B2393">sizeof</span></b><span style="font-size:8.5pt;font-family:Menlo;color:black">(</span><span style="font-size:8.5pt;font-family:Menlo;color:#3900A0">PetscInt</span><span style="font-size:8.5pt;font-family:Menlo;color:black">),
&vec_first_field_index);<o:p></o:p></span></p>
<p class="MsoNormal" style="background:white"><span style="font-size:8.5pt;font-family:Menlo;color:black"> <o:p></o:p></span></p>
<p class="MsoNormal" style="background:white"><span style="font-size:8.5pt;font-family:Menlo;color:black">
</span><b><span style="font-size:8.5pt;font-family:Menlo;color:#9B2393">for</span></b><span style="font-size:8.5pt;font-family:Menlo;color:black"> (</span><b><span style="font-size:8.5pt;font-family:Menlo;color:#9B2393">int</span></b><span style="font-size:8.5pt;font-family:Menlo;color:black">
i = </span><span style="font-size:8.5pt;font-family:Menlo;color:#1C00CF">0</span><span style="font-size:8.5pt;font-family:Menlo;color:black">; i < C; i++)<o:p></o:p></span></p>
<p class="MsoNormal" style="background:white"><span style="font-size:8.5pt;font-family:Menlo;color:black"> vec_first_field_index[i] = N + i;<o:p></o:p></span></p>
<p class="p1"><o:p> </o:p></p>
<p class="p1"><span class="s1"> IS</span> zero_field_isx, first_field_isx;<o:p></o:p></p>
<p class="p1"><span class="apple-converted-space"> </span><span class="s2">CHKERRQ</span>(<span class="s3">ISCreateGeneral</span>(<span class="s3">PETSC_COMM_WORLD</span>, N, vec_zero_field_index,
<span class="s3">PETSC_COPY_VALUES</span>, &zero_field_isx));<o:p></o:p></p>
<p class="p1"><span class="apple-converted-space"> </span><span class="s2">CHKERRQ</span>(<span class="s3">ISCreateGeneral</span>(<span class="s3">PETSC_COMM_WORLD</span>, C, vec_first_field_index,
<span class="s3">PETSC_COPY_VALUES</span>, &first_field_isx));<o:p></o:p></p>
<p class="p2"><span class="apple-converted-space"> </span><o:p></o:p></p>
<p class="p1"><span class="apple-converted-space"> </span>ierr = <span class="s3">
PCFieldSplitSetIS</span>(pc,<span class="s4">"0"</span>,zero_field_isx);<span class="apple-converted-space">
</span><span class="s2">CHKERRQ</span>(ierr);<o:p></o:p></p>
<p class="p1"><span class="apple-converted-space"> </span>ierr = <span class="s3">
PCFieldSplitSetIS</span>(pc,<span class="s4">"1"</span>,first_field_isx);<span class="apple-converted-space">
</span><span class="s2">CHKERRQ</span>(ierr);<o:p></o:p></p>
<p class="p2"><span class="apple-converted-space"> </span><o:p></o:p></p>
<p class="p1"><span class="apple-converted-space"> </span>ierr = <span class="s3">
PetscOptionsSetValue</span>(<span class="s5"><b>NULL</b></span>,<span class="s4">"-ksp_type"</span>,
<span class="s4">"fgmres"</span>); <span class="s2">CHKERRQ</span>(ierr);<o:p></o:p></p>
<p class="p1"><span class="apple-converted-space"> </span>ierr = <span class="s3">
PetscOptionsSetValue</span>(<span class="s5"><b>NULL</b></span>,<span class="s4">"-pc_type"</span>,
<span class="s4">"fieldsplit"</span>); <span class="s2">CHKERRQ</span>(ierr);<o:p></o:p></p>
<p class="p1"><o:p> </o:p></p>
<p class="p1"><span class="apple-converted-space"> </span><span class="s6">/*</span><o:p></o:p></p>
<p class="p3"><span class="apple-converted-space"> </span>ierr = PetscOptionsSetValue(NULL,"-fieldsplit_0_ksp_type", "gmres"); CHKERRQ(ierr);<o:p></o:p></p>
<p class="p3"><span class="apple-converted-space"> </span>ierr = PetscOptionsSetValue(NULL,"-fieldsplit_0_pc_type", "jacobi"); CHKERRQ(ierr);<o:p></o:p></p>
<p class="p3"><span class="apple-converted-space"> </span>ierr = PetscOptionsSetValue(NULL,"-fieldsplit_1_ksp_type", "preonly"); CHKERRQ(ierr);<o:p></o:p></p>
<p class="p3"><span class="apple-converted-space"> </span>ierr = PetscOptionsSetValue(NULL,"-fieldsplit_1_pc_type", "lu"); CHKERRQ(ierr);*/<o:p></o:p></p>
<p class="p3"><o:p> </o:p></p>
<p class="p4"><span class="apple-converted-space"> </span><span class="s7">ierr =
</span><span class="s3">PetscOptionsSetValue</span><span class="s7">(</span><span class="s5"><b>NULL</b></span><span class="s7">,
</span>"-pc_fieldsplit_detect_saddle_point"<span class="s7">, </span><span class="s5"><b>NULL</b></span><span class="s7">);</span><span class="s2">CHKERRQ</span><span class="s7">(ierr);</span><o:p></o:p></p>
<p class="p1"><span class="apple-converted-space"> </span>ierr = <span class="s3">
PetscOptionsSetValue</span>(<span class="s5"><b>NULL</b></span>, <span class="s4">
"-options_left"</span>, <span class="s5"><b>NULL</b></span>);<span class="s2">CHKERRQ</span>(ierr);<o:p></o:p></p>
<p class="p1"><span class="apple-converted-space"> </span>ierr = <span class="s3">
KSPSetFromOptions</span>(ksp); <span class="s2">CHKERRQ</span>(ierr);<o:p></o:p></p>
<p class="MsoNormal" style="background:white"><span style="font-size:8.5pt;font-family:Menlo;color:#6C36A9"> KSPSolve</span><span style="font-size:8.5pt;font-family:Menlo;color:black">(ksp,
</span><span style="font-size:8.5pt;font-family:Menlo;color:#326D74">b</span><span style="font-size:8.5pt;font-family:Menlo;color:black">,
</span><span style="font-size:8.5pt;font-family:Menlo;color:#326D74">x</span><span style="font-size:8.5pt;font-family:Menlo;color:black">);<o:p></o:p></span></p>
<p class="p1"><o:p> </o:p></p>
<p class="MsoNormal"><span style="font-size:11.0pt;mso-fareast-language:EN-US">Best,<o:p></o:p></span></p>
<p class="MsoNormal"><span style="font-size:11.0pt;mso-fareast-language:EN-US">Karthik.<o:p></o:p></span></p>
<div style="border:none;border-top:solid #B5C4DF 1.0pt;padding:3.0pt 0cm 0cm 0cm">
<p class="MsoNormal" style="margin-bottom:12.0pt"><b><span style="font-size:12.0pt;color:black">From:
</span></b><span style="font-size:12.0pt;color:black">Matthew Knepley <knepley@gmail.com><br>
<b>Date: </b>Monday, 24 April 2023 at 17:57<br>
<b>To: </b>Chockalingam, Karthikeyan (STFC,DL,HC) <karthikeyan.chockalingam@stfc.ac.uk><br>
<b>Cc: </b>petsc-users@mcs.anl.gov <petsc-users@mcs.anl.gov><br>
<b>Subject: </b>Re: [petsc-users] Setting up a matrix for Lagrange multiplier<o:p></o:p></span></p>
</div>
<div>
<div>
<p class="MsoNormal"><span style="font-size:11.0pt">On Mon, Apr 24, 2023 at 10:22 AM Karthikeyan Chockalingam - STFC UKRI <<a href="mailto:karthikeyan.chockalingam@stfc.ac.uk">karthikeyan.chockalingam@stfc.ac.uk</a>> wrote:<o:p></o:p></span></p>
</div>
<div>
<blockquote style="border:none;border-left:solid #CCCCCC 1.0pt;padding:0cm 0cm 0cm 6.0pt;margin-left:4.8pt;margin-right:0cm">
<div>
<div>
<div>
<p class="MsoNormal" style="mso-margin-top-alt:auto;mso-margin-bottom-alt:auto"><span style="font-size:11.0pt">Hello,<o:p></o:p></span></p>
<p class="MsoNormal" style="mso-margin-top-alt:auto;mso-margin-bottom-alt:auto"><span style="font-size:11.0pt"> <o:p></o:p></span></p>
<p class="MsoNormal" style="mso-margin-top-alt:auto;mso-margin-bottom-alt:auto"><span style="font-size:11.0pt">I was able to construct the below K matrix (using submatrices P and P^T), which is of type MATAIJ
<o:p></o:p></span></p>
<p class="MsoNormal" style="mso-margin-top-alt:auto;mso-margin-bottom-alt:auto"><span style="font-size:11.0pt;color:black">K = [A P^T</span><span style="font-size:11.0pt"><o:p></o:p></span></p>
<p class="MsoNormal" style="mso-margin-top-alt:auto;mso-margin-bottom-alt:auto"><span style="font-size:11.0pt;color:black"> P 0]</span><span style="font-size:11.0pt"><o:p></o:p></span></p>
<p class="MsoNormal" style="mso-margin-top-alt:auto;mso-margin-bottom-alt:auto"><span style="font-size:11.0pt">and solved them using a direct solver.<o:p></o:p></span></p>
</div>
</div>
</div>
</blockquote>
<div>
<p class="MsoNormal"><span style="font-size:11.0pt"><o:p> </o:p></span></p>
</div>
<div>
<p class="MsoNormal"><span style="font-size:11.0pt">I modified your example to create either AIJ or Nest matrices, but use the same assembly code:<o:p></o:p></span></p>
</div>
<div>
<p class="MsoNormal"><span style="font-size:11.0pt"><o:p> </o:p></span></p>
</div>
<div>
<p class="MsoNormal"><span style="font-size:11.0pt"> <a href="https://gitlab.com/petsc/petsc/-/merge_requests/6368">https://gitlab.com/petsc/petsc/-/merge_requests/6368</a><o:p></o:p></span></p>
</div>
<div>
<p class="MsoNormal"><span style="font-size:11.0pt"> <o:p></o:p></span></p>
</div>
<blockquote style="border:none;border-left:solid #CCCCCC 1.0pt;padding:0cm 0cm 0cm 6.0pt;margin-left:4.8pt;margin-right:0cm">
<div>
<div>
<div>
<p class="MsoNormal" style="mso-margin-top-alt:auto;mso-margin-bottom-alt:auto"><span style="font-size:11.0pt">However, I was reading online that this is a saddle point problem and I should be employing PCFIELDSPLIT.
<o:p></o:p></span></p>
<p class="MsoNormal" style="mso-margin-top-alt:auto;mso-margin-bottom-alt:auto"><span style="font-size:11.0pt">Since I have one monolithic matrix K, I was not sure how to split the fields.<o:p></o:p></span></p>
</div>
</div>
</div>
</blockquote>
<div>
<p class="MsoNormal"><span style="font-size:11.0pt"><o:p> </o:p></span></p>
</div>
<div>
<p class="MsoNormal"><span style="font-size:11.0pt">With this particular matrix, you can use<o:p></o:p></span></p>
</div>
<div>
<p class="MsoNormal"><span style="font-size:11.0pt"><o:p> </o:p></span></p>
</div>
<div>
<p class="MsoNormal"><span style="font-size:11.0pt"> -pc_fieldsplit_detect_saddle_point<o:p></o:p></span></p>
</div>
<div>
<p class="MsoNormal"><span style="font-size:11.0pt"><o:p> </o:p></span></p>
</div>
<div>
<p class="MsoNormal"><span style="font-size:11.0pt">and it will split it automatically.<o:p></o:p></span></p>
</div>
<div>
<p class="MsoNormal"><span style="font-size:11.0pt"><o:p> </o:p></span></p>
</div>
<div>
<p class="MsoNormal"><span style="font-size:11.0pt"> Thanks,<o:p></o:p></span></p>
</div>
<div>
<p class="MsoNormal"><span style="font-size:11.0pt"><o:p> </o:p></span></p>
</div>
<div>
<p class="MsoNormal"><span style="font-size:11.0pt"> Matt<o:p></o:p></span></p>
</div>
<div>
<p class="MsoNormal"><span style="font-size:11.0pt"> <o:p></o:p></span></p>
</div>
<blockquote style="border:none;border-left:solid #CCCCCC 1.0pt;padding:0cm 0cm 0cm 6.0pt;margin-left:4.8pt;margin-right:0cm">
<div>
<div>
<div>
<p class="MsoNormal" style="mso-margin-top-alt:auto;mso-margin-bottom-alt:auto"><span style="font-size:11.0pt">Best regards,<o:p></o:p></span></p>
<p class="MsoNormal" style="mso-margin-top-alt:auto;mso-margin-bottom-alt:auto"><span style="font-size:11.0pt">Karthik.<o:p></o:p></span></p>
<p class="MsoNormal" style="mso-margin-top-alt:auto;mso-margin-bottom-alt:auto"><span style="font-size:11.0pt"> <o:p></o:p></span></p>
<p class="MsoNormal" style="mso-margin-top-alt:auto;mso-margin-bottom-alt:auto"><span style="font-size:11.0pt"> <o:p></o:p></span></p>
<div style="border:none;border-top:solid #B5C4DF 1.0pt;padding:3.0pt 0cm 0cm 0cm">
<p class="MsoNormal" style="mso-margin-top-alt:auto;margin-bottom:12.0pt"><b><span style="font-size:12.0pt;color:black">From:
</span></b><span style="font-size:12.0pt;color:black">Chockalingam, Karthikeyan (STFC,DL,HC) <<a href="mailto:karthikeyan.chockalingam@stfc.ac.uk" target="_blank">karthikeyan.chockalingam@stfc.ac.uk</a>><br>
<b>Date: </b>Wednesday, 19 April 2023 at 17:52<br>
<b>To: </b>Matthew Knepley <<a href="mailto:knepley@gmail.com" target="_blank">knepley@gmail.com</a>><br>
<b>Cc: </b><a href="mailto:petsc-users@mcs.anl.gov" target="_blank">petsc-users@mcs.anl.gov</a> <<a href="mailto:petsc-users@mcs.anl.gov" target="_blank">petsc-users@mcs.anl.gov</a>><br>
<b>Subject: </b>Re: [petsc-users] Setting up a matrix for Lagrange multiplier</span><span style="font-size:11.0pt"><o:p></o:p></span></p>
</div>
<p class="MsoNormal" style="mso-margin-top-alt:auto;mso-margin-bottom-alt:auto;background:white">
<span style="font-size:11.0pt;color:black"> I have declared the mapping </span><span style="font-size:11.0pt"><o:p></o:p></span></p>
<p class="MsoNormal" style="mso-margin-top-alt:auto;mso-margin-bottom-alt:auto;background:white">
<span style="font-size:11.0pt;color:black"> </span><span style="font-size:11.0pt"><o:p></o:p></span></p>
<p class="MsoNormal" style="mso-margin-top-alt:auto;mso-margin-bottom-alt:auto;background:white">
<span style="font-size:11.0pt;color:black"> ISLocalToGlobalMapping mapping;</span><span style="font-size:11.0pt"><o:p></o:p></span></p>
<p class="MsoNormal" style="mso-margin-top-alt:auto;mso-margin-bottom-alt:auto;background:white">
<span style="font-size:11.0pt;color:black"> ISLocalToGlobalMappingCreate(PETSC_COMM_WORLD, 1, n, nindices, PETSC_COPY_VALUES, &mapping);</span><span style="font-size:11.0pt"><o:p></o:p></span></p>
<p class="MsoNormal" style="mso-margin-top-alt:auto;mso-margin-bottom-alt:auto;background:white">
<span style="font-size:11.0pt;color:black"> </span><span style="font-size:11.0pt"><o:p></o:p></span></p>
<p class="MsoNormal" style="mso-margin-top-alt:auto;mso-margin-bottom-alt:auto;background:white">
<span style="font-size:11.0pt;color:black">But when I use MatSetValuesLocal(), how do I know the above mapping is employed because it is not one of the parameters passed to the function?</span><span style="font-size:11.0pt"><o:p></o:p></span></p>
<p class="MsoNormal" style="mso-margin-top-alt:auto;mso-margin-bottom-alt:auto;background:white">
<span style="font-size:11.0pt;color:black"> </span><span style="font-size:11.0pt"><o:p></o:p></span></p>
<p class="MsoNormal" style="mso-margin-top-alt:auto;mso-margin-bottom-alt:auto;background:white">
<span style="font-size:11.0pt;color:black">Thank you.</span><span style="font-size:11.0pt"><o:p></o:p></span></p>
<p class="MsoNormal" style="mso-margin-top-alt:auto;mso-margin-bottom-alt:auto;background:white">
<span style="font-size:11.0pt;color:black"> </span><span style="font-size:11.0pt"><o:p></o:p></span></p>
<p class="MsoNormal" style="mso-margin-top-alt:auto;mso-margin-bottom-alt:auto;background:white">
<span style="font-size:11.0pt;color:black">Kind regards,</span><span style="font-size:11.0pt"><o:p></o:p></span></p>
<p class="MsoNormal" style="mso-margin-top-alt:auto;mso-margin-bottom-alt:auto;background:white">
<span style="font-size:11.0pt;color:black">Karthik.</span><span style="font-size:11.0pt"><o:p></o:p></span></p>
<p class="MsoNormal" style="mso-margin-top-alt:auto;mso-margin-bottom-alt:auto"><span style="font-size:11.0pt"> <o:p></o:p></span></p>
<p class="MsoNormal" style="mso-margin-top-alt:auto;mso-margin-bottom-alt:auto"><span style="font-size:11.0pt"> <o:p></o:p></span></p>
<div style="border:none;border-top:solid #B5C4DF 1.0pt;padding:3.0pt 0cm 0cm 0cm">
<p class="MsoNormal" style="mso-margin-top-alt:auto;margin-bottom:12.0pt"><b><span style="font-size:12.0pt;color:black">From:
</span></b><span style="font-size:12.0pt;color:black">Matthew Knepley <<a href="mailto:knepley@gmail.com" target="_blank">knepley@gmail.com</a>><br>
<b>Date: </b>Tuesday, 18 April 2023 at 16:21<br>
<b>To: </b>Chockalingam, Karthikeyan (STFC,DL,HC) <<a href="mailto:karthikeyan.chockalingam@stfc.ac.uk" target="_blank">karthikeyan.chockalingam@stfc.ac.uk</a>><br>
<b>Cc: </b><a href="mailto:petsc-users@mcs.anl.gov" target="_blank">petsc-users@mcs.anl.gov</a> <<a href="mailto:petsc-users@mcs.anl.gov" target="_blank">petsc-users@mcs.anl.gov</a>><br>
<b>Subject: </b>Re: [petsc-users] Setting up a matrix for Lagrange multiplier</span><span style="font-size:11.0pt"><o:p></o:p></span></p>
</div>
<div>
<div>
<p class="MsoNormal" style="mso-margin-top-alt:auto;mso-margin-bottom-alt:auto"><span style="font-size:11.0pt">On Tue, Apr 18, 2023 at 11:16 AM Karthikeyan Chockalingam - STFC UKRI <<a href="mailto:karthikeyan.chockalingam@stfc.ac.uk" target="_blank">karthikeyan.chockalingam@stfc.ac.uk</a>>
wrote:<o:p></o:p></span></p>
</div>
<div>
<blockquote style="border:none;border-left:solid #CCCCCC 1.0pt;padding:0cm 0cm 0cm 6.0pt;margin-left:4.8pt;margin-top:5.0pt;margin-right:0cm;margin-bottom:5.0pt">
<div>
<div>
<div>
<p class="m-968874379040273580m-4128250693350498318p1">Thank you for your response. I spend some time understanding how<o:p></o:p></p>
<p class="m-968874379040273580m-4128250693350498318p1">MatSetValuesLocal and ISLocalToGlobalMappingCreate work.<o:p></o:p></p>
</div>
</div>
</div>
</blockquote>
<div>
<p class="MsoNormal" style="mso-margin-top-alt:auto;mso-margin-bottom-alt:auto"><span style="font-size:11.0pt"> <o:p></o:p></span></p>
</div>
<div>
<p class="MsoNormal" style="mso-margin-top-alt:auto;mso-margin-bottom-alt:auto"><span style="font-size:11.0pt">You can look at SNES ex28 where we do this with DMCOMPOSITE.<o:p></o:p></span></p>
</div>
<div>
<p class="MsoNormal" style="mso-margin-top-alt:auto;mso-margin-bottom-alt:auto"><span style="font-size:11.0pt"> <o:p></o:p></span></p>
</div>
<blockquote style="border:none;border-left:solid #CCCCCC 1.0pt;padding:0cm 0cm 0cm 6.0pt;margin-left:4.8pt;margin-top:5.0pt;margin-right:0cm;margin-bottom:5.0pt">
<div>
<div>
<div>
<p class="MsoNormal" style="mso-margin-top-alt:auto;mso-margin-bottom-alt:auto"><span style="font-size:11.0pt">Q1) Will the matrix K be of type MATMPIAIJ or MATIS?<o:p></o:p></span></p>
<p class="MsoNormal" style="mso-margin-top-alt:auto;mso-margin-bottom-alt:auto"><span style="font-size:11.0pt;color:black">K = [A P^T</span><span style="font-size:11.0pt"><o:p></o:p></span></p>
<p class="MsoNormal" style="mso-margin-top-alt:auto;mso-margin-bottom-alt:auto"><span style="font-size:11.0pt;color:black"> P 0]</span><span style="font-size:11.0pt"><o:p></o:p></span></p>
</div>
</div>
</div>
</blockquote>
<div>
<p class="MsoNormal" style="mso-margin-top-alt:auto;mso-margin-bottom-alt:auto"><span style="font-size:11.0pt"> <o:p></o:p></span></p>
</div>
<div>
<p class="MsoNormal" style="mso-margin-top-alt:auto;mso-margin-bottom-alt:auto"><span style="font-size:11.0pt">I assume MPIAIJ since IS is only used for Neumann-Neumann decompositions.<o:p></o:p></span></p>
</div>
<div>
<p class="MsoNormal" style="mso-margin-top-alt:auto;mso-margin-bottom-alt:auto"><span style="font-size:11.0pt"> <o:p></o:p></span></p>
</div>
<blockquote style="border:none;border-left:solid #CCCCCC 1.0pt;padding:0cm 0cm 0cm 6.0pt;margin-left:4.8pt;margin-top:5.0pt;margin-right:0cm;margin-bottom:5.0pt">
<div>
<div>
<div>
<p class="MsoNormal" style="mso-margin-top-alt:auto;mso-margin-bottom-alt:auto"><span style="font-size:11.0pt">Q2) Can I use both MatSetValues() to MatSetValuesLocal() to populate K? Since I have already used MatSetValues() to construct A.<o:p></o:p></span></p>
</div>
</div>
</div>
</blockquote>
<div>
<p class="MsoNormal" style="mso-margin-top-alt:auto;mso-margin-bottom-alt:auto"><span style="font-size:11.0pt"> <o:p></o:p></span></p>
</div>
<div>
<p class="MsoNormal" style="mso-margin-top-alt:auto;mso-margin-bottom-alt:auto"><span style="font-size:11.0pt">You can, and there would be no changes in serial if K is exactly the upper left block, but in parallel global indices would change.<o:p></o:p></span></p>
</div>
<div>
<p class="MsoNormal" style="mso-margin-top-alt:auto;mso-margin-bottom-alt:auto"><span style="font-size:11.0pt"> <o:p></o:p></span></p>
</div>
<blockquote style="border:none;border-left:solid #CCCCCC 1.0pt;padding:0cm 0cm 0cm 6.0pt;margin-left:4.8pt;margin-top:5.0pt;margin-right:0cm;margin-bottom:5.0pt">
<div>
<div>
<div>
<p class="MsoNormal" style="mso-margin-top-alt:auto;mso-margin-bottom-alt:auto"><span style="font-size:11.0pt">Q3) What are the advantages of using MatSetValuesLocal()? Is it that I can construct P directly using local indies and map the entrees to the global
index in K?<o:p></o:p></span></p>
</div>
</div>
</div>
</blockquote>
<div>
<p class="MsoNormal" style="mso-margin-top-alt:auto;mso-margin-bottom-alt:auto"><span style="font-size:11.0pt"> <o:p></o:p></span></p>
</div>
<div>
<p class="MsoNormal" style="mso-margin-top-alt:auto;mso-margin-bottom-alt:auto"><span style="font-size:11.0pt">You have a monolithic K, so that you can use sparse direct solvers to check things. THis is impossible with separate storage.<o:p></o:p></span></p>
</div>
<div>
<p class="MsoNormal" style="mso-margin-top-alt:auto;mso-margin-bottom-alt:auto"><span style="font-size:11.0pt"> <o:p></o:p></span></p>
</div>
<blockquote style="border:none;border-left:solid #CCCCCC 1.0pt;padding:0cm 0cm 0cm 6.0pt;margin-left:4.8pt;margin-top:5.0pt;margin-right:0cm;margin-bottom:5.0pt">
<div>
<div>
<div>
<p class="MsoNormal" style="mso-margin-top-alt:auto;mso-margin-bottom-alt:auto"><span style="font-size:11.0pt">Q4) I probably don’t have to construct an independent P matrix<o:p></o:p></span></p>
</div>
</div>
</div>
</blockquote>
<div>
<p class="MsoNormal" style="mso-margin-top-alt:auto;mso-margin-bottom-alt:auto"><span style="font-size:11.0pt"> <o:p></o:p></span></p>
</div>
<div>
<p class="MsoNormal" style="mso-margin-top-alt:auto;mso-margin-bottom-alt:auto"><span style="font-size:11.0pt">You wouldn't in this case.<o:p></o:p></span></p>
</div>
<div>
<p class="MsoNormal" style="mso-margin-top-alt:auto;mso-margin-bottom-alt:auto"><span style="font-size:11.0pt"> <o:p></o:p></span></p>
</div>
<div>
<p class="MsoNormal" style="mso-margin-top-alt:auto;mso-margin-bottom-alt:auto"><span style="font-size:11.0pt"> Thanks,<o:p></o:p></span></p>
</div>
<div>
<p class="MsoNormal" style="mso-margin-top-alt:auto;mso-margin-bottom-alt:auto"><span style="font-size:11.0pt"> <o:p></o:p></span></p>
</div>
<div>
<p class="MsoNormal" style="mso-margin-top-alt:auto;mso-margin-bottom-alt:auto"><span style="font-size:11.0pt"> Matt<o:p></o:p></span></p>
</div>
<div>
<p class="MsoNormal" style="mso-margin-top-alt:auto;mso-margin-bottom-alt:auto"><span style="font-size:11.0pt"> <o:p></o:p></span></p>
</div>
<blockquote style="border:none;border-left:solid #CCCCCC 1.0pt;padding:0cm 0cm 0cm 6.0pt;margin-left:4.8pt;margin-top:5.0pt;margin-right:0cm;margin-bottom:5.0pt">
<div>
<div>
<div>
<p class="MsoNormal" style="mso-margin-top-alt:auto;mso-margin-bottom-alt:auto"><span style="font-size:11.0pt">Best regards,<o:p></o:p></span></p>
<p class="MsoNormal" style="mso-margin-top-alt:auto;mso-margin-bottom-alt:auto"><span style="font-size:11.0pt">Karthik.<o:p></o:p></span></p>
<p class="MsoNormal" style="mso-margin-top-alt:auto;mso-margin-bottom-alt:auto"><span style="font-size:11.0pt"> <o:p></o:p></span></p>
<p class="MsoNormal" style="mso-margin-top-alt:auto;mso-margin-bottom-alt:auto"><span style="font-size:11.0pt"> <o:p></o:p></span></p>
<p class="MsoNormal" style="mso-margin-top-alt:auto;mso-margin-bottom-alt:auto"><span style="font-size:11.0pt"> <o:p></o:p></span></p>
<div style="border:none;border-top:solid #B5C4DF 1.0pt;padding:3.0pt 0cm 0cm 0cm">
<p class="MsoNormal" style="mso-margin-top-alt:auto;margin-bottom:12.0pt"><b><span style="font-size:12.0pt;color:black">From:
</span></b><span style="font-size:12.0pt;color:black">Matthew Knepley <</span><span style="font-size:11.0pt"><a href="mailto:knepley@gmail.com" target="_blank"><span style="font-size:12.0pt">knepley@gmail.com</span></a></span><span style="font-size:12.0pt;color:black">><br>
<b>Date: </b>Tuesday, 18 April 2023 at 11:08<br>
<b>To: </b>Chockalingam, Karthikeyan (STFC,DL,HC) <</span><span style="font-size:11.0pt"><a href="mailto:karthikeyan.chockalingam@stfc.ac.uk" target="_blank"><span style="font-size:12.0pt">karthikeyan.chockalingam@stfc.ac.uk</span></a></span><span style="font-size:12.0pt;color:black">><br>
<b>Cc: </b></span><span style="font-size:11.0pt"><a href="mailto:petsc-users@mcs.anl.gov" target="_blank"><span style="font-size:12.0pt">petsc-users@mcs.anl.gov</span></a></span><span style="font-size:12.0pt;color:black"> <</span><span style="font-size:11.0pt"><a href="mailto:petsc-users@mcs.anl.gov" target="_blank"><span style="font-size:12.0pt">petsc-users@mcs.anl.gov</span></a></span><span style="font-size:12.0pt;color:black">><br>
<b>Subject: </b>Re: [petsc-users] Setting up a matrix for Lagrange multiplier</span><span style="font-size:11.0pt"><o:p></o:p></span></p>
</div>
<div>
<div>
<p class="MsoNormal" style="mso-margin-top-alt:auto;mso-margin-bottom-alt:auto"><span style="font-size:11.0pt">On Tue, Apr 18, 2023 at 5:24 AM Karthikeyan Chockalingam - STFC UKRI via petsc-users <<a href="mailto:petsc-users@mcs.anl.gov" target="_blank">petsc-users@mcs.anl.gov</a>>
wrote:<o:p></o:p></span></p>
</div>
<div>
<blockquote style="border:none;border-left:solid #CCCCCC 1.0pt;padding:0cm 0cm 0cm 6.0pt;margin-left:4.8pt;margin-top:5.0pt;margin-right:0cm;margin-bottom:5.0pt">
<div>
<div>
<div>
<p class="MsoNormal" style="mso-margin-top-alt:auto;mso-margin-bottom-alt:auto"><span style="font-size:11.0pt;color:black">Hello,</span><span style="font-size:11.0pt"><o:p></o:p></span></p>
<p class="MsoNormal" style="mso-margin-top-alt:auto;mso-margin-bottom-alt:auto"><span style="font-size:11.0pt;color:black"> </span><span style="font-size:11.0pt"><o:p></o:p></span></p>
<p class="MsoNormal" style="mso-margin-top-alt:auto;mso-margin-bottom-alt:auto"><span style="font-size:11.0pt;color:black">I'm solving a problem using the Lagrange multiplier, the matrix has the form</span><span style="font-size:11.0pt"><o:p></o:p></span></p>
<p class="MsoNormal" style="mso-margin-top-alt:auto;mso-margin-bottom-alt:auto"><span style="font-size:11.0pt;color:black"> </span><span style="font-size:11.0pt"><o:p></o:p></span></p>
<p class="MsoNormal" style="mso-margin-top-alt:auto;mso-margin-bottom-alt:auto"><span style="font-size:11.0pt;color:black">K = [A P^T</span><span style="font-size:11.0pt"><o:p></o:p></span></p>
<p class="MsoNormal" style="mso-margin-top-alt:auto;mso-margin-bottom-alt:auto"><span style="font-size:11.0pt;color:black"> P 0]</span><span style="font-size:11.0pt"><o:p></o:p></span></p>
<p class="MsoNormal" style="mso-margin-top-alt:auto;mso-margin-bottom-alt:auto"><span style="font-size:11.0pt"> <o:p></o:p></span></p>
<p class="MsoNormal" style="mso-margin-top-alt:auto;mso-margin-bottom-alt:auto"><span style="font-size:11.0pt">I am familiar with constructing K using MATMPIAIJ. However, I would like to know if had [A], can I augment it with [P], [P^T] and [0] of type MATMPIAIJ?
Likewise for vectors as well.<o:p></o:p></span></p>
<p class="MsoNormal" style="mso-margin-top-alt:auto;mso-margin-bottom-alt:auto"><span style="font-size:11.0pt"> <o:p></o:p></span></p>
<p class="MsoNormal" style="mso-margin-top-alt:auto;mso-margin-bottom-alt:auto"><span style="font-size:11.0pt">Can you please point me to the right resource, if it is a common operation in PETSc?<o:p></o:p></span></p>
</div>
</div>
</div>
</blockquote>
<div>
<p class="MsoNormal" style="mso-margin-top-alt:auto;mso-margin-bottom-alt:auto"><span style="font-size:11.0pt"> <o:p></o:p></span></p>
</div>
<div>
<p class="MsoNormal" style="mso-margin-top-alt:auto;mso-margin-bottom-alt:auto"><span style="font-size:11.0pt">You can do this at least 2 ways:<o:p></o:p></span></p>
</div>
<div>
<p class="MsoNormal" style="mso-margin-top-alt:auto;mso-margin-bottom-alt:auto"><span style="font-size:11.0pt"> <o:p></o:p></span></p>
</div>
<div>
<p class="MsoNormal" style="mso-margin-top-alt:auto;mso-margin-bottom-alt:auto"><span style="font-size:11.0pt"> 1) Assemble you submatrices directly into the larger matrix by constructing local-to-global maps for the emplacement. so that you do<o:p></o:p></span></p>
</div>
<div>
<p class="MsoNormal" style="mso-margin-top-alt:auto;mso-margin-bottom-alt:auto"><span style="font-size:11.0pt"> not change your assembly code, except to change MatSetValues() to MatSetValuesLocal(). This is usually preferable.<o:p></o:p></span></p>
</div>
<div>
<p class="MsoNormal" style="mso-margin-top-alt:auto;mso-margin-bottom-alt:auto"><span style="font-size:11.0pt"> <o:p></o:p></span></p>
</div>
<div>
<p class="MsoNormal" style="mso-margin-top-alt:auto;mso-margin-bottom-alt:auto"><span style="font-size:11.0pt"> 2) Use MATNEST and VecNEST to put pointers to submatrices and subvectors directly in.<o:p></o:p></span></p>
</div>
<div>
<p class="MsoNormal" style="mso-margin-top-alt:auto;mso-margin-bottom-alt:auto"><span style="font-size:11.0pt"> <o:p></o:p></span></p>
</div>
<div>
<p class="MsoNormal" style="mso-margin-top-alt:auto;mso-margin-bottom-alt:auto"><span style="font-size:11.0pt"> Thanks,<o:p></o:p></span></p>
</div>
<div>
<p class="MsoNormal" style="mso-margin-top-alt:auto;mso-margin-bottom-alt:auto"><span style="font-size:11.0pt"> <o:p></o:p></span></p>
</div>
<div>
<p class="MsoNormal" style="mso-margin-top-alt:auto;mso-margin-bottom-alt:auto"><span style="font-size:11.0pt"> Matt<o:p></o:p></span></p>
</div>
<div>
<p class="MsoNormal" style="mso-margin-top-alt:auto;mso-margin-bottom-alt:auto"><span style="font-size:11.0pt"> <o:p></o:p></span></p>
</div>
<blockquote style="border:none;border-left:solid #CCCCCC 1.0pt;padding:0cm 0cm 0cm 6.0pt;margin-left:4.8pt;margin-top:5.0pt;margin-right:0cm;margin-bottom:5.0pt">
<div>
<div>
<div>
<p class="MsoNormal" style="mso-margin-top-alt:auto;mso-margin-bottom-alt:auto"><span style="font-size:11.0pt">Many thanks.<o:p></o:p></span></p>
<p class="MsoNormal" style="mso-margin-top-alt:auto;mso-margin-bottom-alt:auto"><span style="font-size:11.0pt"> <o:p></o:p></span></p>
<p class="MsoNormal" style="mso-margin-top-alt:auto;mso-margin-bottom-alt:auto"><span style="font-size:11.0pt">Kind regards,<o:p></o:p></span></p>
<p class="MsoNormal" style="mso-margin-top-alt:auto;mso-margin-bottom-alt:auto"><span style="font-size:11.0pt">Karthik.<o:p></o:p></span></p>
<p class="MsoNormal" style="mso-margin-top-alt:auto;mso-margin-bottom-alt:auto"><span style="font-size:11.0pt"> <o:p></o:p></span></p>
<p class="MsoNormal" style="mso-margin-top-alt:auto;mso-margin-bottom-alt:auto"><span style="font-size:11.0pt"> <o:p></o:p></span></p>
<p class="MsoNormal" style="mso-margin-top-alt:auto;mso-margin-bottom-alt:auto"><span style="font-size:11.0pt"> <o:p></o:p></span></p>
<p class="MsoNormal" style="mso-margin-top-alt:auto;mso-margin-bottom-alt:auto"><span style="font-size:11.0pt"> <o:p></o:p></span></p>
</div>
</div>
</div>
</blockquote>
</div>
<p class="MsoNormal" style="mso-margin-top-alt:auto;mso-margin-bottom-alt:auto"><span style="font-size:11.0pt"><br clear="all">
<o:p></o:p></span></p>
<div>
<p class="MsoNormal" style="mso-margin-top-alt:auto;mso-margin-bottom-alt:auto"><span style="font-size:11.0pt"> <o:p></o:p></span></p>
</div>
<p class="MsoNormal" style="mso-margin-top-alt:auto;mso-margin-bottom-alt:auto"><span class="m-968874379040273580m-4128250693350498318gmailsignatureprefix"><span style="font-size:11.0pt">--
</span></span><span style="font-size:11.0pt"><o:p></o:p></span></p>
<div>
<div>
<div>
<div>
<div>
<div>
<div>
<p class="MsoNormal" style="mso-margin-top-alt:auto;mso-margin-bottom-alt:auto"><span style="font-size:11.0pt">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<o:p></o:p></span></p>
</div>
<div>
<p class="MsoNormal" style="mso-margin-top-alt:auto;mso-margin-bottom-alt:auto"><span style="font-size:11.0pt"> <o:p></o:p></span></p>
</div>
<div>
<p class="MsoNormal" style="mso-margin-top-alt:auto;mso-margin-bottom-alt:auto"><span style="font-size:11.0pt"><a href="http://www.cse.buffalo.edu/~knepley/" target="_blank">https://www.cse.buffalo.edu/~knepley/</a><o:p></o:p></span></p>
</div>
</div>
</div>
</div>
</div>
</div>
</div>
</div>
</div>
</div>
</div>
</blockquote>
</div>
<p class="MsoNormal" style="mso-margin-top-alt:auto;mso-margin-bottom-alt:auto"><span style="font-size:11.0pt"><br clear="all">
<o:p></o:p></span></p>
<div>
<p class="MsoNormal" style="mso-margin-top-alt:auto;mso-margin-bottom-alt:auto"><span style="font-size:11.0pt"> <o:p></o:p></span></p>
</div>
<p class="MsoNormal" style="mso-margin-top-alt:auto;mso-margin-bottom-alt:auto"><span class="m-968874379040273580gmailsignatureprefix"><span style="font-size:11.0pt">--
</span></span><span style="font-size:11.0pt"><o:p></o:p></span></p>
<div>
<div>
<div>
<div>
<div>
<div>
<div>
<p class="MsoNormal" style="mso-margin-top-alt:auto;mso-margin-bottom-alt:auto"><span style="font-size:11.0pt">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<o:p></o:p></span></p>
</div>
<div>
<p class="MsoNormal" style="mso-margin-top-alt:auto;mso-margin-bottom-alt:auto"><span style="font-size:11.0pt"> <o:p></o:p></span></p>
</div>
<div>
<p class="MsoNormal" style="mso-margin-top-alt:auto;mso-margin-bottom-alt:auto"><span style="font-size:11.0pt"><a href="http://www.cse.buffalo.edu/~knepley/" target="_blank">https://www.cse.buffalo.edu/~knepley/</a><o:p></o:p></span></p>
</div>
</div>
</div>
</div>
</div>
</div>
</div>
</div>
</div>
</div>
</div>
</blockquote>
</div>
<p class="MsoNormal"><span style="font-size:11.0pt"><br clear="all">
<o:p></o:p></span></p>
<div>
<p class="MsoNormal"><span style="font-size:11.0pt"><o:p> </o:p></span></p>
</div>
<p class="MsoNormal"><span class="gmailsignatureprefix"><span style="font-size:11.0pt">--
</span></span><span style="font-size:11.0pt"><o:p></o:p></span></p>
<div>
<div>
<div>
<div>
<div>
<div>
<div>
<p class="MsoNormal"><span style="font-size:11.0pt">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<o:p></o:p></span></p>
</div>
<div>
<p class="MsoNormal"><span style="font-size:11.0pt"><o:p> </o:p></span></p>
</div>
<div>
<p class="MsoNormal"><span style="font-size:11.0pt"><a href="http://www.cse.buffalo.edu/~knepley/" target="_blank">https://www.cse.buffalo.edu/~knepley/</a><o:p></o:p></span></p>
</div>
</div>
</div>
</div>
</div>
</div>
</div>
</div>
</div>
</body>
</html>