# Re: [petsc-users] multiply a mpibaij matrix by its block diagonal inverse

```  In the PETSc git branch barry/feature-baij-blockdiagonal-scale I have done
the "heavy lifting" for what you need. See
https://bitbucket.org/petsc/petsc/branch/barry/feature-baij-blockdiagonal-scale ```
```
It scales the Seq BAIJ matrix by its block diagonal. You will need to write a
routine to also scale the right hand side vector by the block diagonal and then
you can try the preconditioner for sequential code. Write something like
VecBlockDiagonalScale(Vec,const PetscScalar *). You get the block size from the
vector.

Later you or I can add the parallel version (not much more difficult). I
don't have time to work on it now.

Let us know if you have any difficulties.

Barry

> On Feb 14, 2018, at 9:10 AM, Xiangdong <epsco...@gmail.com> wrote:
>
> The idea goes back to the alternate-block-factorization (ABF) method
>
>
> and is widely used in the reservoir simulation, where the unknowns are
> pressure and saturation. Although the coupled equations are parabolic, the
> pressure equations/variables are more elliptic and the saturation equations
> are more hyperbolic. People always decouple the transformed linear equation
> to obtain a better (more elliptical) pressure matrix and then apply the AMG
> preconditioner on the decoupled matrix.
>
>
> Thanks.
>
> Xiangdong
>
> On Wed, Feb 14, 2018 at 9:49 AM, Smith, Barry F. <bsm...@mcs.anl.gov> wrote:
>
>   Hmm, I never had this idea presented to me, I have no way to know if it is
> particularly good or bad. So essentially you transform the matrix "decoupling
> the physics alone the diagonal" and then do PCFIELDSPLIT instead of using
> PCFIELDSPLIT directly on the original equations.
>
>   Maybe in the long run this should be an option to PCFIEDLSPLIT. In general
> we like the solvers to manage any transformations, not require
>
>    Barry
>
>
> > On Feb 14, 2018, at 8:29 AM, Xiangdong <epsco...@gmail.com> wrote:
> >
> > The reason for the operation invdiag(A)*A is to have a decoupled
> > matrix/physics for preconditioning. For example, after the transformation,
> > the diagonal block is identity matrix ( e.g. [1,0,0;0,1,0;0,0,1]  for
> > bs=3). One can extract a submatrix (e.g. corresponding to only first
> > unknown) and apply special preconditioners for the extracted/decoupled
> > matrix. The motivation is that after the transformation, one can get a
> > better decoupled matrix to preserve the properties of the unknowns.
> >
> > Thanks.
> >
> > Xiangdong
> >
> > On Tue, Feb 13, 2018 at 6:27 PM, Smith, Barry F. <bsm...@mcs.anl.gov> wrote:
> >
> >  In general you probably don't want to do this. Most good preconditioners
> > (like AMG) rely on the matrix having the "natural" scaling that arises from
> > the discretization and doing a scaling like you describe destroys that
> > natural scaling. You can use PCPBJACOBI to use point block Jacobi
> > preconditioner on the matrix without needing to do the scaling up front.
> > The ILU preconditioners for BAIJ matrices work directly with the block
> > structure so again pre-scaling the matrix buys you nothing. PETSc doesn't
> > have any particularly efficient routines for computing what you desire, the
> > only way to get something truly efficient is to write the code directly
> > using the BAIJ data structure, doable but probably not worth it.
> >
> >   Barry
> >
> >
> > > On Feb 13, 2018, at 5:21 PM, Xiangdong <epsco...@gmail.com> wrote:
> > >
> > > Hello everyone,
> > >
> > > I have a block sparse matrices A created from the DMDA3d. Before passing
> > > the matrix to ksp solver, I want to apply a transformation to this
> > > matrix: namely A:= invdiag(A)*A. Here invdiag(A) is the inverse of the
> > > block diagonal of A. What is the best way to get the transformed matrix?
> > >
> > > At this moment, I created a new mat IDA=inv(diag(A)) by looping through
> > > each row and call MatMatMult to get B=invdiag(A)*A, then destroy the
> > > temporary matrix B. However, I prefer the in-place transformation if
> > > possible, namely, without the additional matrix B for memory saving
> > > purpose.
> > >
> > > Do you have any suggestion on compute invdiag(A)*A for mpibaij matrix?
> > >
> > > Thanks for your help.
> > >
> > > Best,
> > > Xiangdong
> > >
> > >
> > >
> > >
> >
> >
>
>

```