Thank you Matt.<br>I guess I got the impression that M is generally the same as A from reading the manual.(PETSc Users Manual, Reversion 3.3, page 71, under 4.1 Using KSP)<br><br>"<span style="background-color:rgb(255,255,153)"><span style="color:rgb(153,0,0)">Typically the preconditioning matrix (i.e., the matrix from which the preconditioner is to be constructed), Pmat, is the same as the matrix that defines the linear system, Amat</span></span>; however, occasionally these matrices differ (for instance, when a preconditioning matrix is obtained from a lower order method than that employed to form the linear system matrix)."<br>
<br>Best,<br><br>Ling<br><br><div class="gmail_quote">On Mon, Jan 28, 2013 at 12:29 PM, Matthew Knepley <span dir="ltr"><<a href="mailto:knepley@gmail.com" target="_blank">knepley@gmail.com</a>></span> wrote:<br><blockquote class="gmail_quote" style="margin:0 0 0 .8ex;border-left:1px #ccc solid;padding-left:1ex">
<div dir="ltr"><div class="im">On Mon, Jan 28, 2013 at 2:27 PM, Ling Zou <span dir="ltr"><<a href="mailto:lingzou80@gmail.com" target="_blank">lingzou80@gmail.com</a>></span> wrote:<br></div><div class="gmail_extra">
<div class="gmail_quote"><div class="im">
<blockquote class="gmail_quote" style="margin:0 0 0 .8ex;border-left:1px #ccc solid;padding-left:1ex"><br><br><div class="gmail_quote">On Mon, Jan 28, 2013 at 12:01 PM, Matthew Knepley <span dir="ltr"><<a href="mailto:knepley@gmail.com" target="_blank">knepley@gmail.com</a>></span> wrote:<br>
<blockquote class="gmail_quote" style="margin:0 0 0 .8ex;border-left:1px #ccc solid;padding-left:1ex">
<div dir="ltr"><div>On Mon, Jan 28, 2013 at 1:39 PM, Ling Zou <span dir="ltr"><<a href="mailto:lingzou80@gmail.com" target="_blank">lingzou80@gmail.com</a>></span> wrote:<br></div><div class="gmail_extra">
<div class="gmail_quote"><div>
<blockquote class="gmail_quote" style="margin:0 0 0 .8ex;border-left:1px #ccc solid;padding-left:1ex">Hi, All<br><br>I am trying to understand how the preconditioner works when using KSP.<br><br>For example, when using KSP to solve the linear system problem,<br>
<br>Ax = b<br><br>with the default left preconditioning. We actually solve,<br>
<br>M^(-1) * A x = M^(-1) * b<br><br>where, M is the preconditioning matrix and in many cases, we just use A as the preconditioning matrix.<br><br><br>Question:<br>1), Is the understanding above correct?<br></blockquote>
<div><br></div></div><div>This is too simplistic. If you really mean M^{-1}, then no, you (almost) never use A as M. If you mean an</div><div>approximate inverse to M, then yes.</div><div><div> </div><blockquote class="gmail_quote" style="margin:0 0 0 .8ex;border-left:1px #ccc solid;padding-left:1ex">
2), If the understanding above is correct, is it correct to state the different methods provided in PETSc (such as PCLU, PCILU, etc) are to calculate the inverse matrix M^(-1) from M?<br></blockquote><div><br></div></div>
<div>
An approximate inverse.</div><div><div> </div><blockquote class="gmail_quote" style="margin:0 0 0 .8ex;border-left:1px #ccc solid;padding-left:1ex">3), How to understand this sentence in the manual (PETSc Users Manual, Reversion 3.3, page 78, under 4.4 Preconditioners)<br>
"The direct preconditioner, PCLU, is, in fact, a direct solver for the linear system that uses LU factorization. PCLU is included as a preconditioner so that PETSc has a consistent interface among direct and iterative linear solvers."<br>
Does this indicate when using PCLU, we solve Ax = b directly using LU factorization, or, we solve M^(-1) from M using LU factorization?<br></blockquote><div><br></div></div><div>Same thing, if M = A,</div><div><br>
</div><div> M^{-1} A x = A^{-1} A x = x = A^{-1} b</div><div><br></div><div>which is Gaussian elimination for the original problem.</div></div></div></div></blockquote><div><br>Ahh.. that's true! <br>In case M is not A (as you pointed out earlier), does PCLU provide the approximated inverse matrix of M^{-1} using LU factorization on M?<br>
</div></div></blockquote><div><br></div></div><div>Yes.</div><div><br></div><div> Matt</div><div class="im"><div> </div><blockquote class="gmail_quote" style="margin:0 0 0 .8ex;border-left:1px #ccc solid;padding-left:1ex">
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<div></div><blockquote class="gmail_quote" style="margin:0 0 0 .8ex;border-left:1px #ccc solid;padding-left:1ex"><div dir="ltr"><div class="gmail_extra"><div class="gmail_quote"><div><br></div><div> Matt</div><div>
<div> </div><blockquote class="gmail_quote" style="margin:0 0 0 .8ex;border-left:1px #ccc solid;padding-left:1ex">
As a beginner to the PETSc, all questions are probably too simple. I'd appreciate it if someone could answer my questions.<br>
<br>Best,<br><br>Ling<br>
</blockquote></div></div><span><font color="#888888"><br><br clear="all"><span><font color="#888888"><div><br></div>-- <br>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
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</blockquote></div><br>
</blockquote></div></div><div class="im"><br><br clear="all"><div><br></div>-- <br>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
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