On Tue, Aug 21, 2012 at 6:22 PM, Jack Poulson <jack.poulson at gmail.com>wrote:
> On Tue, Aug 21, 2012 at 3:37 PM, Matthew Knepley <knepley at gmail.com>wrote: > >> On Tue, Aug 21, 2012 at 11:47 AM, Jack Poulson <jack.poulson at >> gmail.com>wrote: >> >>> On Tue, Aug 21, 2012 at 11:42 AM, Alexander Grayver < >>> agrayver at gfz-potsdam.de> wrote: >>> >>>> On 21.08.2012 18:32, Matthew Knepley wrote: >>>> >>>> MUMPS takes only several minutes and 6 GB of memory to factorize it. >>>>> This factorization gives residual on the order of 10e-12 and solution >>>>> is indeed correct. >>>>> >>>>> Nevertheless, you're right, there is numerical null-space in this >>>>> matrix since it comes >>>>> from the discretization of equation that contains curl curl operator, >>>>> but practically this >>>>> case is not really the worst one. >>>>> >>>> >>>> This makes no sense whatsoever. How can you LU factor a matrix that >>>> has a null space? >>>> >>>> >>>> Matt, >>>> >>>> I'm not sure that I correctly used term numerical null-space in my post. >>>> The equation is >>>> >>>> curl curl E + kE = -J, >>>> >>>> where k is a function of frequency and conductivity, whenever one of >>>> them becomes small this term gets vanishingly small thus we have problems >>>> since curl curl operator has nontrivial null-space by definition. So let's >>>> say solving this equation for low frequencies and for models containing air >>>> is difficult. >>>> >>>> What kind of magic is inside MUMPS I don't know, but it is able to >>>> handle such cases (e.g. SuperLU and PaStiX fail). >>>> >>>> Also, if it matters, I'm talking about LDLt factorization in MUMPS. >>>> >>>> -- >>>> Regards, >>>> Alexander >>>> >>>> You can find Vasseur's talk on this exact subject here: >>> http://graal.ens-lyon.fr/MUMPS/doc/ud_2010/Vasseur_talk.pdf >> >> >> I was wrong, this is not nonsense. However, for curl curl the null space >> grows with matrix dimension, and >> as far as I can tell from the slides, the null space determination is not >> scalable (Jack correct me if I am wrong). >> Also, they gave no timings, so I suspect null space determination is slow. >> >> I don't think any other LU we have will do this, so if you have null >> spaces you are stuck with MUMPS. >> >> Matt >> >> > > It's not something that I've studied in detail, but I believe that it > isn't that the behavior will not be that different from "difficult" > nonsingular cases (i.e., where a large number of pivots do not satisfy the > thresholding condition and must be delayed to the parent front). If the > null space is large, then I would expect this to impact performance > significantly. I would expect it to make load balancing much more > difficult. In practice, this might lead to nonscalability, as it is > sophisticated algorithm. > > Jack > Please ignore the atrocity that was my attempt at a first sentence in the previous email: the point is that the delayed pivot mechanism is also used within the standard threshold pivoted LU factorization. Jack -------------- next part -------------- An HTML attachment was scrubbed... URL: <http://lists.mcs.anl.gov/pipermail/petsc-users/attachments/20120821/23436482/attachment-0001.html>
