[petsc-users] Increasing norm with finer mesh

Matthew Knepley knepley at gmail.com
Mon Oct 8 17:46:59 CDT 2018 

On Mon, Oct 8, 2018 at 6:13 PM Weizhuo Wang <weizhuo2 at illinois.edu> wrote:

> 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.
>

What are you plotting? It looks like only the algebraic error or residual.
There is absolutely no way your discretization error is 1e-14.

  Thanks,

    Matt


> 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
>


-- 
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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