So Picard iteration is generally only associated with quasi-linear problems? Or always only associated with quasi-linear problems? Or used all over the place for any kind of problem?
On Dec 3, 2012, at 1:51 PM, Jed Brown <jedbrown at mcs.anl.gov> wrote: > On Mon, Dec 3, 2012 at 11:34 AM, Barry Smith <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. > > 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.)
