[petsc-users] Increasing norm with finer mesh

Mon Oct 8 17:13:29 CDT 2018

Sorry I was caught up with midterms for the last few days. I tried the lu
decomposition today and the 2-norm is pretty stable at ~ 10^-15, which is
expected for double precision. Since the discretization error is so small,
it would be reasonable to assume the rest is majority the algebraic error.
Then looking at the result without the -pc_type lu flag(second graph), the
error is asymptoting to a constant several magnitudes larger than the
tolerance set for the solver. (atol=1e-12, rtol=1e-9) Is this the expected
behavior? Shouldn't it decrease with finer grid?
[image: LU.png]
[image: Total.png]

On Tue, Oct 2, 2018 at 6:52 PM Matthew Knepley <knepley at gmail.com> wrote:

> On Tue, Oct 2, 2018 at 5:26 PM Weizhuo Wang <weizhuo2 at illinois.edu> wrote:
>
>> I didn't specify a tolerance, it was using the default tolerance. Doesn't
>> the asymptoting norm implies finer grid won't help to get finer solution?
>>
>
> There are two things going on in your test, discretization error
> controlled by the grid, and algebraic error controlled by the solver. This
> makes it difficult to isolate what is happening. However, it seems clear
> that your plot is looking at algebraic error. You can confirm this by using
>
>   -pc_type lu
>
> for the solve. Then all you have is discretization error.
>
>   Thanks,
>
>      Matt
>
>
>> Mark Adams <mfadams at lbl.gov> ：
>>
>>>
>>>
>>> On Tue, Oct 2, 2018 at 5:04 PM Weizhuo Wang <weizhuo2 at illinois.edu>
>>> wrote:
>>>
>>>> Yes I was using one norm in my Helmholtz code, the example code used 2
>>>> norm. But now I am using 2 norm in both code.
>>>>
>>>>   /*
>>>>      Check the error
>>>>   */
>>>>   ierr = VecAXPY(x,-1.0,u); CHKERRQ(ierr);
>>>>   ierr = VecNorm(x,NORM_1,&norm); CHKERRQ(ierr);
>>>>   ierr = KSPGetIterationNumber(ksp,&its); CHKERRQ(ierr);
>>>>   ierr = PetscPrintf(PETSC_COMM_WORLD,"Norm of error %g iterations
>>>> %D\n",(double)norm/(m*n),its); CHKERRQ(ierr);
>>>>
>>>>  I made a plot to show the increase:
>>>>
>>>
>>>
>>> FYI, this is asymptoting to a constant.  What solver tolerance are
>>> you using?
>>>
>>>
>>>>
>>>> [image: Norm comparison.png]
>>>>
>>>> Mark Adams <mfadams at lbl.gov>：
>>>>
>>>>>
>>>>>
>>>>> On Tue, Oct 2, 2018 at 2:24 PM Weizhuo Wang <weizhuo2 at illinois.edu>
>>>>> wrote:
>>>>>
>>>>>> The example code and makefile are attached below. The whole thing
>>>>>> started as I tried to build a Helmholtz solver, and the mean error
>>>>>> (calculated by: sum( | numerical_sol - analytical_sol | / analytical_sol )
>>>>>> )
>>>>>>
>>>>>
>>>>> This is a one norm. If you use max (instead of sum) then you don't
>>>>> need to scale. You do have to be careful about dividing by (near) zero.
>>>>>
>>>>>
>>>>>> increases as I use finer and finer grids.
>>>>>>
>>>>>
>>>>> What was the rate of increase?
>>>>>
>>>>>
>>>>>> Then I looked at the example 12 (Laplacian solver) which is similar
>>>>>> to what I did to see if I have missed something. The example is using
>>>>>> 2_norm. I have made some minor modifications (3 places) on the code, you
>>>>>> can search 'Modified' in the code to see them.
>>>>>>
>>>>>> If this helps: I configured the PETSc to use real and double
>>>>>> precision. Changed the name of the example code from ex12.c to ex12c.c
>>>>>>
>>>>>> Thanks for all your reply!
>>>>>>
>>>>>> Weizhuo
>>>>>>
>>>>>>
>>>>>> Smith, Barry F. <bsmith at mcs.anl.gov>
>>>>>>
>>>>>>
>>>>>>>    Please send your version of the example that computes the mean
>>>>>>> norm of the grid; I suspect we are talking apples and oranges
>>>>>>>
>>>>>>>    Barry
>>>>>>>
>>>>>>>
>>>>>>>
>>>>>>> > On Oct 1, 2018, at 7:51 PM, Weizhuo Wang <weizhuo2 at illinois.edu>
>>>>>>> wrote:
>>>>>>> >
>>>>>>> > I also tried to divide the norm by m*n , which is the number of
>>>>>>> grids, the trend of norm still increases.
>>>>>>> >
>>>>>>> > Thanks!
>>>>>>> >
>>>>>>> > Weizhuo
>>>>>>> >
>>>>>>> > Matthew Knepley <knepley at gmail.com>
>>>>>>> > On Mon, Oct 1, 2018 at 6:31 PM Weizhuo Wang <weizhuo2 at illinois.edu>
>>>>>>> wrote:
>>>>>>> > Hi!
>>>>>>> >
>>>>>>> > I'm recently trying out the example code provided with the KSP
>>>>>>> solver (ex12.c). I noticed that the mean norm of the grid increases as I
>>>>>>> use finer meshes. For example, the mean norm is 5.72e-8 at m=10 n=10.
>>>>>>> However at m=100, n=100, mean norm increases to 9.55e-6. This seems counter
>>>>>>> intuitive, since most of the time error should decreases when using finer
>>>>>>> grid. Am I doing this wrong?
>>>>>>> >
>>>>>>> > The norm is misleading in that it is the l_2 norm, meaning just
>>>>>>> the sqrt of the sum of the squares of
>>>>>>> > the vector entries. It should be scaled by the volume element to
>>>>>>> approximate a scale-independent
>>>>>>> > norm (like the L_2 norm).
>>>>>>> >
>>>>>>> >   Thanks,
>>>>>>> >
>>>>>>> >      Matt
>>>>>>> >
>>>>>>> > Thanks!
>>>>>>> > --
>>>>>>> > Wang Weizhuo
>>>>>>> >
>>>>>>> >
>>>>>>> > --
>>>>>>> > 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/
>>>>>>> <https://urldefense.proofpoint.com/v2/url?u=https-3A__www.cse.buffalo.edu_-7Eknepley_&d=DwMFaQ&c=OCIEmEwdEq_aNlsP4fF3gFqSN-E3mlr2t9JcDdfOZag&r=hsLktHsuxNfF6zyuWGCN8x-6ghPYxhx4cV62Hya47oo&m=KjmDEsZ6w8LEry7nlv3Bw7-pczqWbKGueFU59VoIWZg&s=tEv9-AHhL2CIlmmVos0gFa5PAY9oMG3aTQlnfi62ivA&e=>
>>>>>>> >
>>>>>>> >
>>>>>>> > --
>>>>>>> > Wang Weizhuo
>>>>>>>
>>>>>>>
>>>>>>
>>>>>> --
>>>>>> Wang Weizhuo
>>>>>>
>>>>>
>>>>
>>>> --
>>>> Wang Weizhuo
>>>>
>>>
>>
>> --
>> Wang Weizhuo
>>
>
>
> --
> 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/>
>

-- 
Wang Weizhuo
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