[petsc-users] Fast LU solver for block diagonal matrix

[Luc Berger-Vergiat]

Hi Barry,
I would like to compute S explicitly, I have a few good option to 
precondition S but they are based on S not an approximation of S (i.e. I 
don't want to compute my preconditioner using Sp different from S).

Also Jee is obtained using MatGetSubmatrix(J,isrow_e,iscol_e) and J is 
an AIJ matrix so I assume that Jee is AIJ too.
I can convert Jee to BAIJ to take advantage of the block structure but 
from your conversation with Chung-Kan it might require to reassemble?
In that case what about the following:

    MatCreateBAIJ
    <http://www.mcs.anl.gov/petsc/petsc-current/docs/manualpages/Mat/MatCreateBAIJ.html#MatCreateBAIJ>(comm,
    5, Jee_inv)
    for(i=0,i<nblocks,i++)
    {
    MatGetValues
    <http://www.mcs.anl.gov/petsc/petsc-current/docs/manualpages/Mat/MatGetValues.html#MatGetValues>(Jee,5,[5*i+0,5*i+1,5*i+2,5*i+3,5*i+4],5,[5*i+0,5*i+1,5*i+2,5*i+3,5*i+4],
    block_values)
         some_package_inverts( block_values )
    MatSetValuesBlocked
    <http://www.mcs.anl.gov/petsc/petsc-current/docs/manualpages/Mat/MatSetValuesBlocked.html#MatSetValuesBlocked>(Jee_inv,5,[5*i+0,5*i+1,5*i+2,5*i+3,5*i+4],5,[5*i+0,5*i+1,5*i+2,5*i+3,5*i+4],
    block_values)
    }
    MatAssemblyBegin
    <http://www.mcs.anl.gov/petsc/petsc-current/docs/manualpages/Mat/MatAssemblyBegin.html#MatAssemblyBegin>()
    MatAssemblyEnd
    <http://www.mcs.anl.gov/petsc/petsc-current/docs/manualpages/Mat/MatAssemblyBegin.html#MatAssemblyBegin>()

With this I could then just go on and do matrix multiplications to 
finish my Schur complement calculations.
Would this be decently efficient?

Best,
Luc

On 03/18/2015 03:22 PM, Barry Smith wrote:
>    Do you want to explicitly compute (as a matrix) S = Joo - joe * inv(Jee) jeo   or do you want to just have an efficient computation of
> S y  for any y vector?
>
>    Here is some possibly useful information. If you create Jee as a BAIJ matrix of block size 5 and use MatILUFactor() it will efficiently factor this matrix (each 5 by 5 block factorization is done with custom code) then you can use MatSolve() efficiently with the result (note that internally when factoring a BAIJ matrix PETSc actually stores the inverse of the diagonal blocks so in your case the MatSolve() actually ends up doing little matrix-vector products (and there are no triangular solves).
>
>    To use this with the MatCreateSchurComplement() object you can do
>
>     MatCreateSchurComplement(...,&S)
>     MatSchurComplementGetKSP(S,&ksp)
>     KSPSetType(ksp,KSPPREONLY);
>
>     now MatMult(S,y,z) will be efficient.
>
>     Of course you still have the question, how do you plan to solve S? This depends on its structure and if you have a good way of preconditioning it.
>
>     If you want to explicitly form S you can use MatMatSolve( fact,jeo) but this requires making jeo dense which seems to defeat the purpose.
>
>    Barry
>
>
>
>> On Mar 18, 2015, at 1:41 PM, Luc Berger-Vergiat <lb2653 at columbia.edu> wrote:
>>
>> Hi all,
>> I am solving multi-physics problem that leads to a jacobian of the form:
>>
>> [ Jee  Jeo ]
>> [ Joe  Joo ]
>>
>> where Jee is 5by5 block diagonal. This feature makes it a very good candidate for a Schur complement.
>> Indeed, Jee could be inverted in parallel with no inter-nodes communication.
>> My only issue is the fact that the Schur complement is not accessible directly with PETSC, only an approximation is available, for direct solvers (usually S~Joo or S~Joo-Joe* diag(Jee)^-1 *Jeo).
>>
>> Any advice on how I could efficiently compute Jee^-1 for the given structure?
>> I am currently thinking about hard coding the formula for the inverse of a 5by5 and sweeping through Jee (with some threading) and storing the inverse in-place. Instead of hard coding the formula for a 5by5 I could also do a MatLUFactorSym on a 5by5 matrix but it would not give me an inverse, only a factorization...
>>
>> Thanks in advance for your suggestions!
>>
>> -- 
>> Best,
>> Luc
>>
>>

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