[petsc-users] What is the right way to implement a (block) Diagonal ILU as PC?

[Smith, Barry F.]

> On Feb 4, 2020, at 12:41 PM, Hao DONG <dong-hao at outlook.com> wrote:
> 
> Dear all, 
> 
> 
> I have a few questions about the implementation of diagonal ILU PC in PETSc. I want to solve a very simple system with KSP (in parallel), the nature of the system (finite difference time-harmonic Maxwell) is probably not important to the question itself. Long story short, I just need to solve a set of equations of Ax = b with a block diagonal system matrix, like (not sure if the mono font works): 
> 
>    |X    |  
> A =|  Y  |  
>    |    Z| 
> 
> Note that A is not really block-diagonal, it’s just a multi-diagonal matrix determined by the FD mesh, where most elements are close to diagonal. So instead of a full ILU decomposition, a D-ILU is a good approximation as a preconditioner, and the number of blocks should not matter too much: 
> 
>     |Lx      |         |Ux      |
> L = |   Ly   | and U = |   Uy   |
>     |      Lz|         |      Uz|
> 
> Where [Lx, Ux] = ILU0(X), etc. Indeed, the D-ILU preconditioner (with 3N blocks) is quite efficient with Krylov subspace methods like BiCGstab or QMR in my sequential Fortran/Matlab code. 
> 
> So like most users, I am looking for a parallel implement with this problem in PETSc. After looking through the manual, it seems that the most straightforward way to do it is through PCBJACOBI. Not sure I understand it right, I just setup a 3-block PCJACOBI and give each of the block a KSP with PCILU. Is this supposed to be equivalent to my D-ILU preconditioner? My little block of fortran code would look like: 
> ...
>       call PCBJacobiSetTotalBlocks(pc_local,Ntotal,                   &
>      &     isubs,ierr)
>       call PCBJacobiSetLocalBlocks(pc_local, Nsub,                    &
>      &    isubs(istart:iend),ierr)
>       ! set up the block jacobi structure
>       call KSPSetup(ksp_local,ierr)
>       ! allocate sub ksps
>       allocate(ksp_sub(Nsub))
>       call PCBJacobiGetSubKSP(pc_local,Nsub,istart,                   &
>      &     ksp_sub,ierr)
>       do i=1,Nsub
>           call KSPGetPC(ksp_sub(i),pc_sub,ierr)
>           !ILU preconditioner
>           call PCSetType(pc_sub,ptype,ierr)
>           call PCFactorSetLevels(pc_sub,1,ierr) ! use ILU(1) here
>           call KSPSetType(ksp_sub(i),KSPPREONLY,ierr)]
>       end do
>       call KSPSetTolerances(ksp_local,KSSiter%tol,PETSC_DEFAULT_REAL, &
>      &     PETSC_DEFAULT_REAL,KSSiter%maxit,ierr)
> … 

     This code looks essentially right. You should call with -ksp_view to see exactly what PETSc is using for a solver. 

> 
> I understand that the parallel performance may not be comparable, so I first set up a one-process test (with MPIAij, but all the rows are local since there is only one process). The system is solved without any problem (identical results within error). But the performance is actually a lot worse (code built without debugging flags in performance tests) than my own home-brew implementation in Fortran (I wrote my own ILU0 in CSR sparse matrix format), which is hard to believe. I suspect the difference is from the PC as the PETSc version took much more BiCGstab iterations (60-ish vs 100-ish) to converge to the same relative tol. 

   PETSc uses GMRES by default with a restart of 30 and left preconditioning. It could be different exact choices in the solver (which is why -ksp_view is so useful) can explain the differences in the runs between your code and PETSc's
> 
> This is further confirmed when I change the setup of D-ILU (using 6 or 9 blocks instead of 3). While my Fortran/Matlab codes see minimal performance difference (<5%) when I play with the D-ILU setup, increasing the number of D-ILU blocks from 3 to 9 caused the ksp setup with PCBJACOBI to suffer a performance decrease of more than 50% in sequential test.

   This is odd, the more blocks the smaller each block so the quicker the ILU set up should be. You can run various cases with -log_view and send us the output to see what is happening at each part of the computation in time.
 
> So my implementation IS somewhat different in PETSc. Do I miss something in the PCBJACOBI setup? Or do I have some fundamental misunderstanding of how PCBJACOBI works in PETSc? 

   Probably not.
> 
> If this is not the right way to implement a block diagonal ILU as (parallel) PC, please kindly point me to the right direction. I searched through the mail list to find some answers, only to find a couple of similar questions... An example would be nice.

   You approach seems fundamentally right but I cannot be sure of possible bugs.
> 
> On the other hand, does PETSc support a simple way to use explicit L/U matrix as a preconditioner? I can import the  D-ILU matrix (I already converted my A matrix into Mat) constructed in my Fortran code to make a better comparison. Or do I have to construct my own PC using PCshell? If so, is there a good tutorial/example to learn about how to use PCSHELL (in a more advanced sense, like how to setup pc side and transpose)? 

   Not sure what you mean by explicit L/U matrix as a preconditioner. As Hong said, yes you can use a parallel LU from MUMPS or SuperLU_DIST or Pastix as the solver. You do not need any shell code. You simply need to set the PCType to lu 

   You can also set all this options from the command line and don't need to write the code specifically. So call KSPSetFromOptions() and then for example

    -pc_type bjacobi  -pc_bjacobi_local_blocks 3 -pc_sub_type ilu (this last one is applied to each block so you could use -pc_type lu and it would use lu on each block.) 

   -ksp_type_none  -pc_type lu -pc_factor_mat_solver_type mumps  (do parallel LU with mumps)

By not hardwiring in the code and just using options you can test out different cases much quicker

Use -ksp_view to make sure that is using the solver the way you expect.

Barry



   Barry

> 
> Thanks in advance, 
> 
> Hao