On Mar 2, 2013, at 12:45 PM, Barry Smith <bsmith at mcs.anl.gov> wrote:
> > On Mar 2, 2013, at 11:16 AM, "Mark F. Adams" <mark.adams at columbia.edu> > wrote: > >>> >>> An alternative description for the method in the paper is "two-level >>> unsmoothed aggregation applied to the ASM-preconditioned operator". >>> >> >> Isn't this paper just doing two level unsmoothed aggregation with a V(0,k) >> cycle, where k (and the smoother) is K in the paper? >> >> It looks to me like you are moving the multigrid coarse grid correction >> (projection) from the PC into the operator, in which case this method is >> identical to two level MG with the V(0,k) cycle. > > Well, its not "identical" is it? My thinking is that with Q = P (RAP)^-1 R, and smoother S, then a two level V(0,1) PC (M) looks like: S (I - Q). (I think I'm missing an A in here, before the Q maybe) The preconditioned system being solved looks like: MAx = Mb with MG PC we have: (S (I-Q)) A x = ? Move the brackets around S ((I-Q)A) x = .. S is K in the paper and (I-Q)A is the new operator. The preconditioned system does not have a null space and thats all that matters. Anyway, I'm sure its not identical, I'm just not seeing it in this paper and don't have the patients work through it ? and maybe I'm having a bad arithmetic day but on page 935, the definition of P (first equation on page) and Pw (last equation on page) don't look consistent. > In the deflation approach it introduces a null space into the operator, in > the multigrid approach it does not. So identical only in analogy? So what I > am interested in EXACTLY how are they related, relationship between > eigenvalues or convergence rate. > > Barry > >> >> I'm sure I'm missing something. Jie's writeup has an orthogonality >> condition on the restriction operator, which I don't see in the Vuik paper. > >
