[petsc-users] Convergence rate for spatially varying Helmholtz system

[Ramakrishnan Thirumalaisamy]

>
> Several things look off here:
>
> 1) Your true residual norm is 2.4e9, but r_0/b is 4.4e-5. That seems to
> indicate that ||b|| is 1e14. Is this true?
>
Yes. ||b|| is 1e14.  We have verified that.

>
> 2) Your preconditioned residual is 11 orders of magnitude less than the
> true residual. This usually indicates that the system is near singular.
>
The system is diagonal, as shown in Temperature_fill.pdf. So we don't think
it is singular unless we are missing something very obvious.   Diagonal
elements range from 8.8e6 to 5.62896e+10, while off-diagonal terms are 0,
as shown in the spy plot.

>
> 3) The disparity above does not seem possible if C only has elements ~
> 1e4. The preconditioner consistently has norm around 1e-11.
>
The value of C in the Helmholtz system is computed as : *C =
rho*specific_heat/dt* in which dt = 5e-5, specific_heat ~10^3 and rho
ranges from 0.4 to 2700. Hence, C ranges from 8.8e6 to 5.62896e10.
Max_diagonal(C)/Min_diagonal(C) ~ 10^4.

>
> 4) Using numbers that large can be a problem. You lose precision, so that
> you really only have 3-4 correct digits each time, as you see above. It
> appears to be
>      doing iterative refinement with a very low precision solve.
>
Indeed the numbers are large because C =  rho*specific_heat/dt.

On Wed, Sep 29, 2021 at 3:58 PM Matthew Knepley <knepley at gmail.com> wrote:

> On Wed, Sep 29, 2021 at 6:03 PM Ramakrishnan Thirumalaisamy <
> rthirumalaisam1857 at sdsu.edu> wrote:
>
>> Hi all,
>>
>> I am trying to solve the Helmholtz equation for temperature T:
>>
>> (C I  + Div D grad) T = f
>>
>> in IBAMR, in which C is the spatially varying diagonal entries, and D is
>> the spatially varying diffusion coefficient.   I use a matrix-free solver
>> with matrix-based PETSc preconditioner. For the matrix-free solver, I use
>> gmres solver and for the matrix based preconditioner, I use Richardson ksp
>> + Jacobi as a preconditioner. As the simulation progresses, the iterations
>> start to increase. To understand the cause, I set D to be zero, which
>> results in a diagonal system:
>>
>> C T = f.
>>
>> This should result in convergence within a single iteration, but I get
>> convergence in 3 iterations.
>>
>> Residual norms for temperature_ solve.
>>
>>   0 KSP preconditioned resid norm 4.590811647875e-02 true resid norm
>> 2.406067589273e+09 ||r(i)||/||b|| 4.455533946945e-05
>>
>>   1 KSP preconditioned resid norm 2.347767895880e-06 true resid norm
>> 1.210763896685e+05 ||r(i)||/||b|| 2.242081505717e-09
>>
>>   2 KSP preconditioned resid norm 1.245406571896e-10 true resid norm
>> 6.328828824310e+00 ||r(i)||/||b|| 1.171966730978e-13
>>
>> Linear temperature_ solve converged due to CONVERGED_RTOL iterations 2
>>
>
> Several things look off here:
>
> 1) Your true residual norm is 2.4e9, but r_0/b is 4.4e-5. That seems to
> indicate that ||b|| is 1e14. Is this true?
>

> 2) Your preconditioned residual is 11 orders of magnitude less than the
> true residual. This usually indicates that the system is near singular.
>
> 3) The disparity above does not seem possible if C only has elements ~
> 1e4. The preconditioner consistently has norm around 1e-11.
>
> 4) Using numbers that large can be a problem. You lose precision, so that
> you really only have 3-4 correct digits each time, as you see above. It
> appears to be
>      doing iterative refinement with a very low precision solve.
>
>   Thanks,
>
>      Matt
>
>
>> To verify that I am indeed solving a diagonal system I printed the PETSc
>> matrix from the preconditioner and viewed it in Matlab. It indeed shows it
>> to be a diagonal system. Attached is the plot of the spy command on the
>> printed matrix. The matrix in binary form is also attached.
>>
>> My understanding is that because the C coefficient is varying in 4 orders
>> of magnitude, i.e., Max(C)/Min(C) ~ 10^4, the matrix is poorly scaled. When
>> I rescale my matrix by 1/C then the system converges in 1 iteration as
>> expected. Is my understanding correct, and that scaling 1/C should be done
>> even for a diagonal system?
>>
>> When D is non-zero, then scaling by 1/C seems to be very inconvenient as
>> D is stored as side-centered data for the matrix free solver.
>>
>> In the case that I do not scale my equations by 1/C, is there some solver
>> setting that improves the convergence rate? (With D as non-zero, I have
>> also tried gmres as the ksp solver in the matrix-based preconditioner to
>> get better performance, but it didn't matter much.)
>>
>>
>> Thanks,
>> Ramakrishnan Thirumalaisamy
>> San Diego State University.
>>
>
>
> --
> 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
>
> https://www.cse.buffalo.edu/~knepley/
> <http://www.cse.buffalo.edu/~knepley/>
>
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