[petsc-users] Mixed matrices MatMatMult

[Luc Berger-Vergiat]

The difference between IS(1), IS(2) and IS(1)',IS(2)' is that they 
operate on two different matrices. The former operate on A whereas the 
operates on S and these matrices have different sizes so it's not 
obvious to me that they IS would be identical? I guess it depends on the 
initial ordering of the blocks of A, if A00 is actually the bottom right 
block then I can see how the two are the same.

But just to get a confirmation here is a simple example: let's assume I 
solve a 2D problem with momentum and energy conservation.
In that case I get three equations per nodes:

C1*d^2 u_x/dx^2 + C2*d^2 u_y/dxdy + a*dT/dt= fx      ---> IS(0)
C3*d^2 u_x/dxdy + C4*d^2 u_y/dy^2 + a*dT/dt = fy     ---> IS(1)
dT/dt + kappa*(d^2 T/dx^2 + d^2 T/dy^2) = fT            ---> IS(2)

This leads to a block matrix with 3x3 block per nodes. If I extract the 
temperature problem, invert it and compute the schur complement I will 
end up with S with the same size as the elasticity part of the problem 
but will the IS(0) and IS(1) be the same for the jacobian and the Schur 
complement?
If that's the case that's awesome that you programmed it to keep track 
of these things!

Best,
Luc

On 03/26/2015 06:05 PM, Matthew Knepley wrote:
> On Thu, Mar 26, 2015 at 4:06 PM, Luc Berger-Vergiat 
> <lb2653 at columbia.edu <mailto:lb2653 at columbia.edu>> wrote:
>
>     Ok,
>     this work is still part of my Schur complement approach using the
>     full schur but with a block diagonal A00^-1.
>     I implemented the computation of A00^-1 by extracting each
>     diagonal block and inverting them individually.
>     This works quite well and does not cost some much, especially
>     since I can still use threads to accelerate this process (I might
>     send a question about this in the future...).
>
>     At the moment the most expensive part of the procedure is
>     inverting S (I'm using LU at the moment to make sure that
>     everything is implemented correctly) and the second most expensive
>     procedure is MatMatMult. I'm doing two of these: A10 * A00^-1 and
>     then a right multiplication by A01.
>     Decreasing that cost would be nice (I attached the output of
>     -log_summary for reference).
>     I think I need to look for the objects that are not Destroyed too.
>
>
> We will look, but this is a notoriously hard thing to speed up. You 
> might try the RAP code instead.
>
>     Finally I now would like to split the Schur complement into two
>     submatrices. I have an IS that tracks location of these
>     sub-matrices in the global system:
>
>            [ A00  A01  A02 ]  --> IS(0)
>     A = [ A10  A11  A12 ]  --> IS(1)
>            [ A20  A21  A22 ]  --> IS(2)
>
>     How can I use IS(1) and IS(2) to track:
>
>     S =   [ A11  A12 ]  _  [ A10] * [A00]^-1 * [ A01 A02 ] = [ S11 
>     S12 ]  --> IS(1)'
>              [ A21  A22 ]     [
>     A20]                                          = [ S21 S22 ]  -->
>     IS(2)'
>
>     or is there a simple way to compute IS(1)' and IS(2)' based on
>     IS(1) and IS(2)?
>
>
> Aren't they identical?
>
>   Thanks,
>
>      Matt
>
>     Thanks!
>
>     Best,
>     Luc
>
>     On 03/26/2015 04:12 PM, Matthew Knepley wrote:
>>     On Thu, Mar 26, 2015 at 3:07 PM, Luc Berger-Vergiat
>>     <lb2653 at columbia.edu <mailto:lb2653 at columbia.edu>> wrote:
>>
>>         Hi all,
>>         I want to multiply two matrices together, one is MATAIJ and
>>         the second is MATBAIJ, is there a way to leverage the
>>         properties of the blocked matrix in the BAIJ format or should
>>         I just assemble the BAIJ matrix as AIJ?
>>
>>
>>     I am afraid you are currently stuck with the latter.
>>
>>       Thanks,
>>
>>         Matt
>>
>>
>>         -- 
>>         Best,
>>         Luc
>>
>>
>>
>>
>>
>>     -- 
>>     What most experimenters take for granted before they begin their
>>     experiments is infinitely more interesting than any results to
>>     which their experiments lead.
>>     -- Norbert Wiener
>
>
>
>
> -- 
> What most experimenters take for granted before they begin their 
> experiments is infinitely more interesting than any results to which 
> their experiments lead.
> -- Norbert Wiener

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