Sparse Matrix Inversion using PETSc

[Barry Smith]

  Tim,

   Ok, this is a bit more reasonable :-)

   The proceedure I discussed previously is the same. Just
use for the right hand side matrix a dense vector with
M columns whose entries are the first M columns of the identity,
then solve with it.

   Barry


On Thu, 16 Aug 2007, Dr. Timothy Stitt wrote:

> Barry,
> 
> If it helps I was speaking to some of the project members and they mention
> that they actually only need the first M columns of the inverse. The equations
> can also be rewritten so that they only require the last M columns also or
> first/last M rows. I believe the retarded Green's Function forms an integral.
> For the integral in the real plane (which is repeated for thousands of energy
> points and hence requires thousands of inversions) they need only the first
> few columns as described above.
> 
> I would be grateful if you could suggest a possible approach based on this
> extra information. I much appreciate your comments already.
> 
> Thanks,
> 
> Tim.
> 
> Barry Smith wrote:
> >   Tim,
> > 
> >    Do you know what the G_r is then used for?
> > 
> >    Thanks
> > 
> >      Barry
> > 
> > 
> > On Wed, 15 Aug 2007, Dr. Timothy Stitt wrote:
> > 
> >   
> > > Barry,
> > > 
> > > The group I am working with are calculating what they call retarded
> > > Green's
> > > Functions of the form:
> > > 
> > > G_r=(E-H)^(-1)
> > > 
> > > where (E-H) is a matrix. Apparently they say there is no way to avoid this
> > > calculation.
> > > 
> > > Tim.
> > > 
> > > Barry Smith wrote:
> > >     
> > > >   Tim,
> > > > 
> > > >     A dense matrix with 100,000 rows and columns requires 80 gigabytes
> > > > to
> > > > store
> > > > the result. 1,000,000 rows and columns requires 8,000 gigabytes.    With
> > > > this range of sizes I wouldn't even consider iterative solvers and
> > > > would only use direct solvers. I would divide MPI_COMM_WORLD into N
> > > > subcommunicators of size n each and have each subcommunicator work on a
> > > > collection of columns of the inverse. For matrix sizes of 5,000 to say
> > > > 20,000?
> > > > I'd make n be 1 and just use PETSc's native LU solver and use
> > > > MatMatSolve()
> > > > and not use KSP at all. For larger matrices you may be able to use an n
> > > > of 2
> > > > to possibly as large as 8?
> > > > 
> > > >    Now my numerical analysis training :-) requires me to state the
> > > > following.
> > > > It is completely insane to compute the EXPLICIT inverse of large sparse
> > > > matrices
> > > > since they are dense. Please tell me what the inverses are used for and
> > > > perhaps
> > > > we can come up with an approach the does not require computing them.
> > > > 
> > > >    Barry
> > > > 
> > > > 
> > > > 
> > > > On Wed, 15 Aug 2007, Dr. Timothy Stitt wrote:
> > > > 
> > > >         
> > > > > Firstly, many thanks to everyone who has replied with information. It
> > > > > has
> > > > > been
> > > > > very useful indeed. Much appreciated.
> > > > > 
> > > > > Barry, in this case the sparse matrices would be of ~ order 5000x5000.
> > > > > They
> > > > > could grow in size but this is the sample matrices I am working with
> > > > > right
> > > > > now. We would love a scalable approach so we can deal with more
> > > > > interesting
> > > > > problems and hence larger sparse matrices. Hope that helps.
> > > > > 
> > > > > Many thanks again.
> > > > > 
> > > > > Tim.
> > > > > 
> > > > > Barry Smith wrote:
> > > > >             
> > > > > >   Tim,
> > > > > > 
> > > > > >     How large are you matrices?
> > > > > > 
> > > > > >    Barry
> > > > > > 
> > > > > > 
> > > > > > On Wed, 15 Aug 2007, Dr. Timothy Stitt wrote:
> > > > > > 
> > > > > >                   
> > > > > > > Hi all,
> > > > > > > 
> > > > > > > I am currently investigating the best way to perform the inversion
> > > > > > > of
> > > > > > > a
> > > > > > > large
> > > > > > > sparse matrix and came upon the idea of using PETSc as a framework
> > > > > > > for
> > > > > > > testing
> > > > > > > various strategies from direct to iterative methods on my sample
> > > > > > > matrices.
> > > > > > > In
> > > > > > > this setup for an NxN sparse matrix A I would have N rhs's
> > > > > > > representing
> > > > > > > the
> > > > > > > Identity matrix and then solve for X. I wanted to experiment with
> > > > > > > both
> > > > > > > parallel and serial strategies ranging from LU Decomposition using
> > > > > > > SuperLU,
> > > > > > > MUMPS etc. to iterative methods using GMRES etc. Am I right in
> > > > > > > thinking
> > > > > > > that
> > > > > > > all this can be done in PETSc by setting up a core framework and
> > > > > > > then
> > > > > > > varying
> > > > > > > the solver methods etc?
> > > > > > > 
> > > > > > > I have looked over the sample KSP Solver codes although they only
> > > > > > > seem
> > > > > > > to
> > > > > > > suggest single vectors for x and b. Can this be changed to accept
> > > > > > > multiple
> > > > > > > vectors? Can anyone suggest a sample code that maybe demonstrates
> > > > > > > the
> > > > > > > sort
> > > > > > > of
> > > > > > > thing I want to achieve...if it is in fact possible.
> > > > > > > 
> > > > > > > Thanks in advance for any assistance given,
> > > > > > > 
> > > > > > > Regards,
> > > > > > > 
> > > > > > > Tim.
> > > > > > > 
> > > > > > > 
> > > > > > >                         
> > > > > >                   
> > > > >             
> > > >         
> > >     
> > 
> >   
> 
> 
>