<div class="gmail_quote">On Thu, Aug 9, 2012 at 11:28 AM, Jinquan Zhong <span dir="ltr"><<a href="mailto:jzhong@scsolutions.com" target="_blank">jzhong@scsolutions.com</a>></span> wrote:<br><blockquote class="gmail_quote" style="margin:0 0 0 .8ex;border-left:1px #ccc solid;padding-left:1ex">
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<p>Hope this is not too technical. <span style="font-family:Wingdings">J</span><u></u><u></u></p>
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<p class="MsoNormal"><span style="font-size:11.0pt;font-family:"Calibri","sans-serif";color:#1f497d">**Jed, A’=[A^-1 U B], not the transpose of A.</span><u></u><u></u></p>
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<p class="MsoNormal">I understand that it's not the transpose. I still don't have a clue what the notation [A^{-1} U B] means. I would think it means some block decomposed thing, but I don't think you mean just adding columns or taking a product of matrices.
Using some standard mathematical notation would help.<u></u><u></u></p>
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<p class="MsoNormal"><span style="font-size:11.0pt;font-family:"Calibri","sans-serif";color:#1f497d">>>
</span><span style="font-size:11.0pt;font-family:"Calibri","sans-serif";color:#1f497d">A’=[A^-1 U B] means matrix A’ consists all elements from A^-1, the inverted A, and all elements from B. One simple equation is
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<p class="MsoNormal" style="text-indent:11.55pt"><span style="font-size:11.0pt;font-family:"Calibri","sans-serif";color:#1f497d"> K=[Kst] =A’ = ASSEMBLE_ij (A^-1_ij) + ASSEMBLE_i’j’(B_i’j’), here (s,t), (i,j) and (i’,j’) = different index sets<u></u><u></u></span></p>
<p class="MsoNormal" style="text-indent:11.55pt"><span style="font-size:11.0pt;font-family:"Calibri","sans-serif";color:#1f497d"><u></u> <u></u></span></p>
<p class="MsoNormal" style="text-indent:11.55pt"><span style="font-size:11.0pt;font-family:"Calibri","sans-serif";color:#1f497d">i.e., entry K_st= A^-1_st+ B_st,
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<p class="MsoNormal" style="margin-left:.5in;text-indent:.5in"><span style="font-size:11.0pt;font-family:"Calibri","sans-serif";color:#1f497d"> = 0 when A^-1_st=0 and B_st=0</span><span style="font-size:11.0pt;font-family:"Calibri","sans-serif";color:#1f497d"><u></u><u></u></span></p>
<p class="MsoNormal" style="margin-left:.5in;text-indent:.5in"><span style="font-size:11.0pt;font-family:"Calibri","sans-serif";color:#1f497d">= A^-1_st when B_st =0</span><span style="font-size:11.0pt;font-family:"Calibri","sans-serif";color:#1f497d"><u></u><u></u></span></p>
<p class="MsoNormal" style="margin-left:.5in;text-indent:.5in"><span style="font-size:11.0pt;font-family:"Calibri","sans-serif";color:#1f497d">= B_st when A^-1_st =0</span></p></div></div></div></div>
</blockquote><div><br></div><div>Oh, so U meant "union" (in some funny sense). We would often write things like</div><div><br></div><div>K = \sum_i R_i^T A_i^{-1} R_i + B</div><div><br></div><div>where R_i is a restriction to "part" i. The great thing about mathematical notation is that we can communicate precisely what we mean. In this case, we would have been able to ask how the restriction operators R_i are connected (e.g. how much they overlap).</div>
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<p class="MsoNormal"><span style="font-size:11.0pt;font-family:"Calibri","sans-serif";color:#1f497d"> We need to solve for A’*x=b. A is dense matrix.</span><u></u><u></u></p>
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<p class="MsoNormal">Where does A come from? Why is it dense?<u></u><u></u></p>
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</div><p style="margin-left:0in"><span style="font-size:11.0pt;font-family:"Calibri","sans-serif";color:#1f497d">>> A comes from wave propagation. It is dense since it denotes a function of Green function.</span></p>
</div></div></div></div></blockquote><div>What is the kernel? Is it a pure integral operator (a discretized first kind Fredholm integral equation), identity plus a compact integral operator (a second kind Fredholm integral operator), or something else? This matters a great deal.</div>
<div> </div><blockquote class="gmail_quote" style="margin:0 0 0 .8ex;border-left:1px #ccc solid;padding-left:1ex"><div lang="EN-US" link="blue" vlink="purple"><div><div><div><div><p class="MsoNormal"></p></div><div>
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<p class="MsoNormal">Is A^{-1} an operator on some subdomain? Are you trying to implement a substructuring algorithm? What is B physically?<u></u><u></u></p>
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</div><p class="MsoNormal"><span style="font-size:11.0pt;font-family:"Calibri","sans-serif";color:#1f497d">>> NO.
</span>A^{-1} denotes the inverted A. B is a sparse matrix of much larger order.</p></div></div></div></div></blockquote><div><br></div><div>What is B physically? In what way is it coupled to A?</div><div><br></div><blockquote class="gmail_quote" style="margin:0 0 0 .8ex;border-left:1px #ccc solid;padding-left:1ex">
<div lang="EN-US" link="blue" vlink="purple"><div><div><div><div><p class="MsoNormal">What does QA stand for? Can you explain why B needs the entries of A^{-1}?<u></u><u></u></p>
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</div><p class="MsoNormal"><span style="font-size:11.0pt;font-family:"Calibri","sans-serif";color:#1f497d">>> QA means quality assurance. It is a procedure to ensure product quality. In Eq.
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<p class="MsoNormal" style="text-indent:.5in"><span style="font-size:11.0pt;font-family:"Calibri","sans-serif";color:#1f497d">K=A’=ASSEMBLE_ij (A^-1_ij)+ ASSEMBLE_i’j’(B_i’j’)<u></u><u></u></span></p>
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</span><span style="font-size:11.0pt;font-family:"Calibri","sans-serif";color:#1f497d"><u></u><u></u></span></p>
<p class="MsoNormal" style="text-indent:.5in"><span style="font-size:11.0pt;font-family:"Calibri","sans-serif";color:#1f497d">Entry B_i’j’ and
</span><span style="font-size:11.0pt;font-family:"Calibri","sans-serif";color:#1f497d">A^-1_ij may or may not locate at the same row and col. That why we need explicitly each entry in
</span><span style="font-size:11.0pt;font-family:"Calibri","sans-serif";color:#1f497d">B_i’j’ and
</span><span style="font-size:11.0pt;font-family:"Calibri","sans-serif";color:#1f497d">A^-1_ij to assemble K. The big picture is that K is the final sparse matrix we need to solve K*x=A’*x=b. However, K indexed by (s,t) needs to be constructed in terms of
dense matrix A and sparse matrix B using index sets (i,j) and (i’,j’). </span><span style="font-size:11.0pt;font-family:"Calibri","sans-serif";color:#1f497d"><u></u><u></u></span></p>
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<div>It is very likely that you can (and should) solve K x = b without computing explicit entries for K. You would apply its action by some other procedure, likely not involving explicit construction of the A_i^{-1} matrices either.</div>