<html>
<head>
<meta content="text/html; charset=UTF-8" http-equiv="Content-Type">
</head>
<body bgcolor="#FFFFFF" text="#000000">
<div class="moz-cite-prefix">On 12/3/12 8:51 PM, Jed Brown wrote:<br>
</div>
<blockquote
cite="mid:CAM9tzSmW+6CTmRoF-uivW_EaiyNNW2-x_H6MBm-bVASYfEn+jQ@mail.gmail.com"
type="cite">
<meta http-equiv="Content-Type" content="text/html; charset=UTF-8">
<div class="gmail_extra">On Mon, Dec 3, 2012 at 11:34 AM, Barry
Smith <span dir="ltr"><<a moz-do-not-send="true"
href="mailto:bsmith@mcs.anl.gov" target="_blank">bsmith@mcs.anl.gov</a>></span>
wrote:<br>
<div class="gmail_quote">
<blockquote class="gmail_quote" style="margin:0 0 0
.8ex;border-left:1px #ccc solid;padding-left:1ex">
What is "a Picard linearization"? As opposed to a
non-Picard linearization? Also if you phrase it as in my
other email isn't Newton "a Picard linearization"? You act
as if the term "a Picard linearization" has a well defined
meaning, but Matt never found it in any book in history.</blockquote>
</div>
<br>
</div>
</blockquote>
Some info on Picard linearization can be found for example in
Chapter 2, Volume 2 of the book by Zienkiewicz & Taylor on
Finite Elements. Couple of equations are given on page 29 (5-th
edition), although they're quite unclear. Nevertheless it can be
helpful. <br>
<br>
The idea is that one approximates total nonlinear solution vector
(not just defect correction) from a linear system with a secant
matrix that itself depends on the latest solution, and a fixed (at
least on a time step) right hand side.<br>
<br>
When no satisfactory approximation for solution is given, doing
couple of Picard steps is a good strategy to start with. Then one
can switch to Newton with/without line search. In general, the
advantage of Picard is stability at the expense of linear vs.
potentially quadratic convergence (for Newton, when exact Jacobian
and blah-blah-blah is known).<br>
<br>
I know, these are just words, it's a bit difficult to generalize it
for all possible cases.<br>
<br>
Anton<br>
<br>
<blockquote
cite="mid:CAM9tzSmW+6CTmRoF-uivW_EaiyNNW2-x_H6MBm-bVASYfEn+jQ@mail.gmail.com"
type="cite">
<div class="gmail_extra">If you have a quasi-linear problem, then
you can write the homogenous part of the operator as A(u) u.
That A(u) is the Picard linearization. Achi calls it the
"principle linearization" in some FAS papers because it's
provably all that is necessary in the smoother (the other terms
in Newton linearization involve lower frequencies, thus are not
needed in the smoother).</div>
<div class="gmail_extra"><br>
</div>
<div class="gmail_extra">Some equations, perhaps most notably the
Euler flux, satisfy the "homogeneity property" that F(u) = F'(u)
u, i.e., A(u) _is_ the Jacobian, in which case Picard would be
equal to Newton. (People don't normally "solve" the flux
equation.)</div>
</blockquote>
<br>
</body>
</html>