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<p><span style="font-size:11.0pt;font-family:Symbol;color:#1F497D">·</span><span style="font-size:7.0pt;color:#1F497D">
</span><span style="font-size:11.0pt;font-family:"Calibri","sans-serif";color:#1F497D">A’x=b, where A’=[A^-1 U B]=
</span><o:p></o:p></p>
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<p class="MsoNormal" style="mso-margin-top-alt:auto;mso-margin-bottom-alt:auto">What does the notation above mean? <o:p></o:p></p>
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<blockquote style="border:none;border-left:solid #CCCCCC 1.0pt;padding:0in 0in 0in 6.0pt;margin-left:4.8pt;margin-top:5.0pt;margin-right:0in;margin-bottom:5.0pt">
<p><span style="font-family:Wingdings">ü</span><span style="font-size:7.0pt"> </span>
<span style="font-size:11.0pt;font-family:"Calibri","sans-serif";color:#1F497D">That means A’ consists A^-1 and B. A’ is a sparse matrix at the order of 2 million. B = sparse matrix at the order of 2 million to be mixed and mingled with A^-1.</span><o:p></o:p></p>
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<p class="MsoNormal"><o:p> </o:p></p>
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<p class="MsoNormal">"mingled" is not a very useful term. How about using an actual equation so we can have some clue what you mean.<o:p></o:p></p>
<p class="MsoListParagraph" style="text-indent:-.25in;mso-list:l0 level1 lfo1"><![if !supportLists]><span style="font-size:11.0pt;font-family:"Courier New";color:#1F497D"><span style="mso-list:Ignore">o<span style="font:7.0pt "Times New Roman"">
</span></span></span><![endif]><span style="font-size:11.0pt;font-family:"Calibri","sans-serif";color:#1F497D">I agree. It is hard to have an accurate equation to for the structure of A’. But if you are familiar with FEM, you may know global stiffness matrix
K=SIGMA_ij( K_ij ), the assembling process of elemental stiffness. Here A’=K. <o:p>
</o:p></span></p>
<p class="MsoListParagraph" style="text-indent:-.25in;mso-list:l0 level1 lfo1"><![if !supportLists]><span style="font-size:11.0pt;font-family:"Courier New";color:#1F497D"><span style="mso-list:Ignore">o<span style="font:7.0pt "Times New Roman"">
</span></span></span><![endif]><span style="font-size:11.0pt;font-family:"Calibri","sans-serif";color:#1F497D">We partition the FEM domain into two physical pieces due to different operations they are defined for. After completing each operation, we need to
operate on A^-1_ij and B_i’j’ such that we can have K=A’=SIGMA_ij (A^-1_ij)+SIGMA_i’j’(B_i’j’).<o:p></o:p></span></p>
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<p class="MsoNormal"> <span style="color:#1F497D"><o:p></o:p></span></p>
<p class="MsoNormal"><span style="font-size:11.0pt;font-family:"Calibri","sans-serif";color:#1F497D"><o:p> </o:p></span></p>
<p class="MsoNormal" style="mso-margin-top-alt:auto;mso-margin-bottom-alt:auto">If you have an "inverse" floating around, what sort of operator is it the inverse of.<o:p></o:p></p>
<p><span style="font-size:11.0pt;font-family:Wingdings;color:#1F497D">ü</span><span style="font-size:7.0pt;color:#1F497D">
</span><span style="font-size:11.0pt;font-family:"Calibri","sans-serif";color:#1F497D">A is a double complex matrix we have generated from a practical application using FEM. </span><o:p></o:p></p>
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<p class="MsoNormal">Then there is no way A^{-1} should be stored as a dense matrix.
<o:p></o:p></p>
<p class="MsoListParagraph" style="text-indent:-.25in;mso-list:l0 level1 lfo1"><![if !supportLists]><span style="font-size:11.0pt;font-family:"Courier New";color:#1F497D"><span style="mso-list:Ignore">o<span style="font:7.0pt "Times New Roman"">
</span></span></span><![endif]><span style="font-size:11.0pt;font-family:"Calibri","sans-serif";color:#1F497D">It is stored as a dense matrix in scalapack. It stays in the same way in cores as scalapack finishes its inversion. We could define A^-1 as dense
matrix in PETSc. <o:p></o:p></span></p>
<p class="MsoListParagraph" style="text-indent:-.25in;mso-list:l0 level1 lfo1"><![if !supportLists]><span style="font-size:11.0pt;font-family:"Courier New";color:#1F497D"><span style="mso-list:Ignore">o<span style="font:7.0pt "Times New Roman"">
</span></span></span><![endif]><span style="font-size:11.0pt;font-family:"Calibri","sans-serif";color:#1F497D">Another constraint we have is we have to use direct solver for QA purposes. This brings up the needs to convert the dense matrix into MATAIJ or MATMPIAIJ.
However, in PETSc, converting dense matrix-formatted A^-1 into MATAIJ/ MATMPIAIJ also takes lots of time.
<o:p></o:p></span></p>
<p class="MsoNormal"><span style="font-size:11.0pt;font-family:"Calibri","sans-serif";color:#1F497D"><o:p> </o:p></span></p>
<p class="MsoNormal"><b><span style="font-family:"Calibri","sans-serif";color:#1F497D">Here is what I am looking at after all these discussion:<o:p></o:p></span></b></p>
<p class="MsoNormal"><b><span style="font-family:"Calibri","sans-serif";color:#1F497D">~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~<o:p></o:p></span></b></p>
<p class="MsoListParagraph" style="margin-left:.75in;text-indent:-.25in;mso-list:l1 level1 lfo3">
<![if !supportLists]><span style="font-size:11.0pt;font-family:Symbol;color:#1F497D"><span style="mso-list:Ignore">·<span style="font:7.0pt "Times New Roman"">
</span></span></span><![endif]>Some function similar to <a href="http://www.mcs.anl.gov/petsc/petsc-current/docs/manualpages/Mat/MatCreateMPIAIJWithArrays.html">
MatCreateMPIAIJWithArrays</a>(..,A^-1) with the additional capability to specify the location of upper-left corner of distributed A^-1 in A’, the global matrix.<span style="font-size:11.0pt;font-family:"Calibri","sans-serif";color:#1F497D"><o:p></o:p></span></p>
<p class="MsoListParagraph" style="margin-left:.75in;text-indent:-.25in;mso-list:l1 level1 lfo3">
<![if !supportLists]><span style="font-size:11.0pt;font-family:Symbol;color:#1F497D"><span style="mso-list:Ignore">·<span style="font:7.0pt "Times New Roman"">
</span></span></span><![endif]>The reason is that in CSR format defaulted in PETSc, the entire row has to be input in one rank. In this application, we have partitioned this row into different pieces due to the operation on ScaLAPACK. Due to large amount of
data, there is no practical way that we can gather the entire portioned rows of A^-1 into on rank.<span style="font-size:11.0pt;font-family:"Calibri","sans-serif";color:#1F497D"><o:p></o:p></span></p>
<p class="MsoNormal"><span style="font-size:11.0pt;font-family:"Calibri","sans-serif";color:#1F497D"><o:p> </o:p></span></p>
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