<div class="gmail_quote">On Tue, May 15, 2012 at 4:41 PM, Avery Bingham <span dir="ltr"><<a href="mailto:avery.bingham@gmail.com" target="_blank">avery.bingham@gmail.com</a>></span> wrote:<br><blockquote class="gmail_quote" style="margin:0 0 0 .8ex;border-left:1px #ccc solid;padding-left:1ex">
By scaling, I mean to say that a problem is poorly scaled if a change in x in a certain direction produces a much larger variation in f(x) than a change in another direction. For a simple example is the scalar function:<div>
f(x) = 10^10*x1^2+x2^2</div></blockquote><div><br></div><div>This is ill-conditioned, it just happens to be diagonal.</div><div> </div><blockquote class="gmail_quote" style="margin:0 0 0 .8ex;border-left:1px #ccc solid;padding-left:1ex">
<div></div><div>A more practical example would be a physical coupled problem that has Pressure in ~10^6 Pascals and temperature in degrees ~100 C. </div></blockquote><div><br></div><div>So if you have an accurate linear solve, this kind of ill-conditioning should be taken care of (provided you are not up against finite precision). If you can't do an accurate linear solve, or if you define the matrix by finite differencing, then it's very important to fix the scaling in problem formulation, otherwise you will constantly struggle with artificially inaccurate finite differences and tricky convergence tolerances (the residual is a poor indicator of convergence, etc).</div>
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</div><div><br><div class="gmail_quote"><div class="im">On Tue, May 15, 2012 at 3:45 PM, Jed Brown <span dir="ltr"><<a href="mailto:jedbrown@mcs.anl.gov" target="_blank">jedbrown@mcs.anl.gov</a>></span> wrote:<br></div>
<div><div class="h5"><blockquote class="gmail_quote" style="margin:0 0 0 .8ex;border-left:1px #ccc solid;padding-left:1ex">
<div class="gmail_quote"><div>On Tue, May 15, 2012 at 3:32 PM, Avery Bingham <span dir="ltr"><<a href="mailto:avery.bingham@gmail.com" target="_blank">avery.bingham@gmail.com</a>></span> wrote:<br><blockquote class="gmail_quote" style="margin:0 0 0 .8ex;border-left:1px #ccc solid;padding-left:1ex">
<pre><font face="arial, helvetica, sans-serif">I have also seen this behavior, and I think this might be related to the scaling of the variables in the nonlinear system. I am using PETSc through an application of MOOSE which allows for scaling of the variables. This scaling reduces the chance that the default cubic backtracking line search fails, but it is not reliable on all problems. Would it be possible to get a scaling-independent Line Search Method implemented within PETSc? </font></pre>
</blockquote><div><br></div></div><div>I'm not sure what you mean by scaling, but there is -snes_linesearch_type cp (critical point) that might be useful.</div></div>
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