<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;}
/* 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-4128250693350498318p1, li.m-4128250693350498318p1, div.m-4128250693350498318p1
{mso-style-name:m_-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-4128250693350498318gmailsignatureprefix
{mso-style-name:m_-4128250693350498318gmailsignatureprefix;}
span.gmailsignatureprefix
{mso-style-name:gmail_signature_prefix;}
.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">Hello,<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">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><o:p></o:p></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]<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>
<p class="MsoNormal"><span style="font-size:11.0pt;mso-fareast-language:EN-US">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"><span style="font-size:11.0pt;mso-fareast-language:EN-US">Since I have one monolithic matrix K, I was not sure how to split the fields.<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">Best regards,<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>
<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"><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">Chockalingam, Karthikeyan (STFC,DL,HC) <karthikeyan.chockalingam@stfc.ac.uk><br>
<b>Date: </b>Wednesday, 19 April 2023 at 17:52<br>
<b>To: </b>Matthew Knepley <knepley@gmail.com><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>
<p class="MsoNormal" style="background:white"><span style="font-size:11.0pt;color:black"> I have declared the mapping
</span><o:p></o:p></p>
<p class="MsoNormal" style="background:white"><span style="font-size:11.0pt;color:black"> </span><o:p></o:p></p>
<p class="MsoNormal" style="background:white"><span style="font-size:11.0pt;color:black"> ISLocalToGlobalMapping mapping;</span><o:p></o:p></p>
<p class="MsoNormal" style="background:white"><span style="font-size:11.0pt;color:black"> ISLocalToGlobalMappingCreate(PETSC_COMM_WORLD, 1, n, nindices, PETSC_COPY_VALUES, &mapping);</span><o:p></o:p></p>
<p class="MsoNormal" style="background:white"><span style="font-size:11.0pt;color:black"> </span><o:p></o:p></p>
<p class="MsoNormal" style="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><o:p></o:p></p>
<p class="MsoNormal" style="background:white"><span style="font-size:11.0pt;color:black"> </span><o:p></o:p></p>
<p class="MsoNormal" style="background:white"><span style="font-size:11.0pt;color:black">Thank you.</span><o:p></o:p></p>
<p class="MsoNormal" style="background:white"><span style="font-size:11.0pt;color:black"> </span><o:p></o:p></p>
<p class="MsoNormal" style="background:white"><span style="font-size:11.0pt;color:black">Kind regards,</span><o:p></o:p></p>
<p class="MsoNormal" style="background:white"><span style="font-size:11.0pt;color:black">Karthik.</span><o:p></o:p></p>
<p class="MsoNormal"><span style="font-size:11.0pt;mso-fareast-language:EN-US"> </span><o:p></o:p></p>
<p class="MsoNormal"><span style="font-size:11.0pt;mso-fareast-language:EN-US"> </span><o:p></o:p></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>Tuesday, 18 April 2023 at 16:21<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</span><o:p></o:p></p>
</div>
<div>
<div>
<p class="MsoNormal"><span style="font-size:11.0pt">On Tue, Apr 18, 2023 at 11:16 AM Karthikeyan Chockalingam - STFC UKRI <</span><a href="mailto:karthikeyan.chockalingam@stfc.ac.uk"><span style="font-size:11.0pt">karthikeyan.chockalingam@stfc.ac.uk</span></a><span style="font-size:11.0pt">>
wrote:</span><o:p></o:p></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-4128250693350498318p1">Thank you for your response. I spend some time understanding how<o:p></o:p></p>
<p class="m-4128250693350498318p1">MatSetValuesLocal and ISLocalToGlobalMappingCreate work.<o:p></o:p></p>
</div>
</div>
</div>
</blockquote>
<div>
<p class="MsoNormal"><span style="font-size:11.0pt"> </span><o:p></o:p></p>
</div>
<div>
<p class="MsoNormal"><span style="font-size:11.0pt">You can look at SNES ex28 where we do this with DMCOMPOSITE.</span><o:p></o:p></p>
</div>
<div>
<p class="MsoNormal"><span style="font-size:11.0pt"> </span><o:p></o:p></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?</span><o:p></o:p></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><o:p></o:p></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><o:p></o:p></p>
</div>
</div>
</div>
</blockquote>
<div>
<p class="MsoNormal"><span style="font-size:11.0pt"> </span><o:p></o:p></p>
</div>
<div>
<p class="MsoNormal"><span style="font-size:11.0pt">I assume MPIAIJ since IS is only used for Neumann-Neumann decompositions.</span><o:p></o:p></p>
</div>
<div>
<p class="MsoNormal"><span style="font-size:11.0pt"> </span><o:p></o:p></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.</span><o:p></o:p></p>
</div>
</div>
</div>
</blockquote>
<div>
<p class="MsoNormal"><span style="font-size:11.0pt"> </span><o:p></o:p></p>
</div>
<div>
<p class="MsoNormal"><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.</span><o:p></o:p></p>
</div>
<div>
<p class="MsoNormal"><span style="font-size:11.0pt"> </span><o:p></o:p></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?</span><o:p></o:p></p>
</div>
</div>
</div>
</blockquote>
<div>
<p class="MsoNormal"><span style="font-size:11.0pt"> </span><o:p></o:p></p>
</div>
<div>
<p class="MsoNormal"><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.</span><o:p></o:p></p>
</div>
<div>
<p class="MsoNormal"><span style="font-size:11.0pt"> </span><o:p></o:p></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</span><o:p></o:p></p>
</div>
</div>
</div>
</blockquote>
<div>
<p class="MsoNormal"><span style="font-size:11.0pt"> </span><o:p></o:p></p>
</div>
<div>
<p class="MsoNormal"><span style="font-size:11.0pt">You wouldn't in this case.</span><o:p></o:p></p>
</div>
<div>
<p class="MsoNormal"><span style="font-size:11.0pt"> </span><o:p></o:p></p>
</div>
<div>
<p class="MsoNormal"><span style="font-size:11.0pt"> Thanks,</span><o:p></o:p></p>
</div>
<div>
<p class="MsoNormal"><span style="font-size:11.0pt"> </span><o:p></o:p></p>
</div>
<div>
<p class="MsoNormal"><span style="font-size:11.0pt"> Matt</span><o:p></o:p></p>
</div>
<div>
<p class="MsoNormal"><span style="font-size:11.0pt"> </span><o:p></o:p></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,</span><o:p></o:p></p>
<p class="MsoNormal" style="mso-margin-top-alt:auto;mso-margin-bottom-alt:auto"><span style="font-size:11.0pt">Karthik.</span><o:p></o:p></p>
<p class="MsoNormal" style="mso-margin-top-alt:auto;mso-margin-bottom-alt:auto"><span style="font-size:11.0pt"> </span><o:p></o:p></p>
<p class="MsoNormal" style="mso-margin-top-alt:auto;mso-margin-bottom-alt:auto"><span style="font-size:11.0pt"> </span><o:p></o:p></p>
<p class="MsoNormal" style="mso-margin-top-alt:auto;mso-margin-bottom-alt:auto"><span style="font-size:11.0pt"> </span><o:p></o:p></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><a href="mailto:knepley@gmail.com" target="_blank"><span style="font-size:12.0pt">knepley@gmail.com</span></a><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><a href="mailto:karthikeyan.chockalingam@stfc.ac.uk" target="_blank"><span style="font-size:12.0pt">karthikeyan.chockalingam@stfc.ac.uk</span></a><span style="font-size:12.0pt;color:black">><br>
<b>Cc: </b></span><a href="mailto:petsc-users@mcs.anl.gov" target="_blank"><span style="font-size:12.0pt">petsc-users@mcs.anl.gov</span></a><span style="font-size:12.0pt;color:black"> <</span><a href="mailto:petsc-users@mcs.anl.gov" target="_blank"><span style="font-size:12.0pt">petsc-users@mcs.anl.gov</span></a><span style="font-size:12.0pt;color:black">><br>
<b>Subject: </b>Re: [petsc-users] Setting up a matrix for Lagrange multiplier</span><o:p></o:p></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 <</span><a href="mailto:petsc-users@mcs.anl.gov" target="_blank"><span style="font-size:11.0pt">petsc-users@mcs.anl.gov</span></a><span style="font-size:11.0pt">>
wrote:</span><o:p></o:p></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><o:p></o:p></p>
<p class="MsoNormal" style="mso-margin-top-alt:auto;mso-margin-bottom-alt:auto"><span style="font-size:11.0pt;color:black"> </span><o:p></o:p></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><o:p></o:p></p>
<p class="MsoNormal" style="mso-margin-top-alt:auto;mso-margin-bottom-alt:auto"><span style="font-size:11.0pt;color:black"> </span><o:p></o:p></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><o:p></o:p></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><o:p></o:p></p>
<p class="MsoNormal" style="mso-margin-top-alt:auto;mso-margin-bottom-alt:auto"><span style="font-size:11.0pt"> </span><o:p></o:p></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.</span><o:p></o:p></p>
<p class="MsoNormal" style="mso-margin-top-alt:auto;mso-margin-bottom-alt:auto"><span style="font-size:11.0pt"> </span><o:p></o:p></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?</span><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"> </span><o:p></o:p></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:</span><o:p></o:p></p>
</div>
<div>
<p class="MsoNormal" style="mso-margin-top-alt:auto;mso-margin-bottom-alt:auto"><span style="font-size:11.0pt"> </span><o:p></o:p></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</span><o:p></o:p></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.</span><o:p></o:p></p>
</div>
<div>
<p class="MsoNormal" style="mso-margin-top-alt:auto;mso-margin-bottom-alt:auto"><span style="font-size:11.0pt"> </span><o:p></o:p></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.</span><o:p></o:p></p>
</div>
<div>
<p class="MsoNormal" style="mso-margin-top-alt:auto;mso-margin-bottom-alt:auto"><span style="font-size:11.0pt"> </span><o:p></o:p></p>
</div>
<div>
<p class="MsoNormal" style="mso-margin-top-alt:auto;mso-margin-bottom-alt:auto"><span style="font-size:11.0pt"> Thanks,</span><o:p></o:p></p>
</div>
<div>
<p class="MsoNormal" style="mso-margin-top-alt:auto;mso-margin-bottom-alt:auto"><span style="font-size:11.0pt"> </span><o:p></o:p></p>
</div>
<div>
<p class="MsoNormal" style="mso-margin-top-alt:auto;mso-margin-bottom-alt:auto"><span style="font-size:11.0pt"> Matt</span><o:p></o:p></p>
</div>
<div>
<p class="MsoNormal" style="mso-margin-top-alt:auto;mso-margin-bottom-alt:auto"><span style="font-size:11.0pt"> </span><o:p></o:p></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.</span><o:p></o:p></p>
<p class="MsoNormal" style="mso-margin-top-alt:auto;mso-margin-bottom-alt:auto"><span style="font-size:11.0pt"> </span><o:p></o:p></p>
<p class="MsoNormal" style="mso-margin-top-alt:auto;mso-margin-bottom-alt:auto"><span style="font-size:11.0pt">Kind regards,</span><o:p></o:p></p>
<p class="MsoNormal" style="mso-margin-top-alt:auto;mso-margin-bottom-alt:auto"><span style="font-size:11.0pt">Karthik.</span><o:p></o:p></p>
<p class="MsoNormal" style="mso-margin-top-alt:auto;mso-margin-bottom-alt:auto"><span style="font-size:11.0pt"> </span><o:p></o:p></p>
<p class="MsoNormal" style="mso-margin-top-alt:auto;mso-margin-bottom-alt:auto"><span style="font-size:11.0pt"> </span><o:p></o:p></p>
<p class="MsoNormal" style="mso-margin-top-alt:auto;mso-margin-bottom-alt:auto"><span style="font-size:11.0pt"> </span><o:p></o:p></p>
<p class="MsoNormal" style="mso-margin-top-alt:auto;mso-margin-bottom-alt:auto"><span style="font-size:11.0pt"> </span><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"><br clear="all">
</span><o:p></o:p></p>
<div>
<p class="MsoNormal" style="mso-margin-top-alt:auto;mso-margin-bottom-alt:auto"><span style="font-size:11.0pt"> </span><o:p></o:p></p>
</div>
<p class="MsoNormal" style="mso-margin-top-alt:auto;mso-margin-bottom-alt:auto"><span class="m-4128250693350498318gmailsignatureprefix"><span style="font-size:11.0pt">--
</span></span><o:p></o:p></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</span><o:p></o:p></p>
</div>
<div>
<p class="MsoNormal" style="mso-margin-top-alt:auto;mso-margin-bottom-alt:auto"><span style="font-size:11.0pt"> </span><o:p></o:p></p>
</div>
<div>
<p class="MsoNormal" style="mso-margin-top-alt:auto;mso-margin-bottom-alt:auto"><a href="http://www.cse.buffalo.edu/~knepley/" target="_blank"><span style="font-size:11.0pt">https://www.cse.buffalo.edu/~knepley/</span></a><o:p></o:p></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">
</span><o:p></o:p></p>
<div>
<p class="MsoNormal"><span style="font-size:11.0pt"> </span><o:p></o:p></p>
</div>
<p class="MsoNormal"><span class="gmailsignatureprefix"><span style="font-size:11.0pt">--
</span></span><o:p></o:p></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</span><o:p></o:p></p>
</div>
<div>
<p class="MsoNormal"><span style="font-size:11.0pt"> </span><o:p></o:p></p>
</div>
<div>
<p class="MsoNormal"><a href="http://www.cse.buffalo.edu/~knepley/" target="_blank"><span style="font-size:11.0pt">https://www.cse.buffalo.edu/~knepley/</span></a><o:p></o:p></p>
</div>
</div>
</div>
</div>
</div>
</div>
</div>
</div>
</div>
</body>
</html>