The idea goes back to the alternate-block-factorization (ABF) method https://link.springer.com/article/10.1007/BF01932753

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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. https://link.springer.com/article/10.1007/s00791-016-0273-3 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 > transformations before calling the solvers. I have to think about this. > > 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 > > > > > > > > > > > > > > > > > >