[petsc-users] KSPSolve convergence and norms

[Torquil Macdonald Sørensen]

Hi!

I'm using PETSc 3.4.3, with only default runtime settings, to solve Ax=b,
for complex numbers (I have used --with-scalar-type=complex).

After reading sections 4.3.1, 4.3.2, 4.3.3 in the manual, I decided to
investigate the convergence in my program, since I'm having some problems
when the system is larger. So, I'm running it with the options

-ksp_monitor_true_residual -ksp_converged_reason -ksp_view

The output shows that the relative tolerance required for convergence is
1e-5, but none of the reported norms are that small. Despite this,
convergence is reported (CONVERGED_RTOL).

I though that CONVERGED_RTOL should indicate that ||r||_2/||b||_2 < 1e-5,
where r is the preconditioned residual?

Here is the output of my program (which includes some code to compute the
l_2 norm of b, just before KSPSolve is run):

*************************************************************
Norm of b: 0.179829
  0 KSP preconditioned resid norm 2.508789303280e+04 true resid norm
1.798290175843e-01 ||r(i)||/||b|| 1.000000000000e+00
  1 KSP preconditioned resid norm 1.856991132478e+00 true resid norm
3.186462510386e-01 ||r(i)||/||b|| 1.771940120227e+00
  2 KSP preconditioned resid norm 1.704678606994e-01 true resid norm
4.004071321891e-02 ||r(i)||/||b|| 2.226599119363e-01
Linear solve converged due to CONVERGED_RTOL iterations 2
KSP Object: 1 MPI processes
  type: gmres
    GMRES: restart=30, using Classical (unmodified) Gram-Schmidt
Orthogonalization with no iterative refinement
    GMRES: happy breakdown tolerance 1e-30
  maximum iterations=10000, initial guess is zero
  tolerances:  relative=1e-05, absolute=1e-50, divergence=10000
  left preconditioning
  using PRECONDITIONED norm type for convergence test
PC Object: 1 MPI processes
  type: ilu
    ILU: out-of-place factorization
    0 levels of fill
    tolerance for zero pivot 2.22045e-14
    using diagonal shift on blocks to prevent zero pivot [INBLOCKS]
    matrix ordering: natural
    factor fill ratio given 1, needed 1
      Factored matrix follows:
        Matrix Object:         1 MPI processes
          type: seqaij
          rows=18, cols=18
          package used to perform factorization: petsc
          total: nonzeros=144, allocated nonzeros=144
          total number of mallocs used during MatSetValues calls =0
            using I-node routines: found 9 nodes, limit used is 5
  linear system matrix = precond matrix:
  Matrix Object:   1 MPI processes
    type: seqaij
    rows=18, cols=18
    total: nonzeros=144, allocated nonzeros=360
    total number of mallocs used during MatSetValues calls =18
      using I-node routines: found 9 nodes, limit used is 5
************************************************************************************************

Here is the code I used to get the l_2 norm of b:

PetscReal val;
VecNorm(b, NORM_2, &val);
PetscPrintf(PETSC_COMM_WORLD, "Norm of b: %G\n", val);

Best regards
Torquil Sørensen
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