On 12/3/12 8:51 PM, Jed Brown wrote: > On Mon, Dec 3, 2012 at 11:34 AM, Barry Smith <bsmith at mcs.anl.gov > <mailto:bsmith at mcs.anl.gov>> wrote: > > 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. > > 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.
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. 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). I know, these are just words, it's a bit difficult to generalize it for all possible cases. Anton > 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). > > 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.) -------------- next part -------------- An HTML attachment was scrubbed... URL: <http://lists.mcs.anl.gov/pipermail/petsc-dev/attachments/20121205/2caa3145/attachment.html>
