[petsc-users] Number of levels of multigrid : 2-3 is sufficient ??

[Matthew Knepley]

On Wed, Oct 14, 2015 at 7:34 AM, Timothée Nicolas <
timothee.nicolas at gmail.com> wrote:

> OK, I see. Does it mean that the coarse grid solver is by default set up
> with the options -ksp_type preonly -pc_type lu ? What about the
> multiprocessor case ?
>

Small scale: We use redundant LU

Large Scale: We use GAMG

   Matt


> Thx
>
> Timothee
>
> 2015-10-14 21:22 GMT+09:00 Matthew Knepley <knepley at gmail.com>:
>
>> On Tue, Oct 13, 2015 at 9:23 PM, Timothée Nicolas <
>> timothee.nicolas at gmail.com> wrote:
>>
>>> Dear all,
>>>
>>> I have been playing around with multigrid recently, namely with
>>> /ksp/ksp/examples/tutorials/ex42.c, with /snes/examples/tutorial/ex5.c and
>>> with my own implementation of a laplacian type problem. In all cases, I
>>> have noted no improvement whatsoever in the performance, whether in CPU
>>> time or KSP iteration, by varying the number of levels of the multigrid
>>> solver. As an example, I have attached the log_summary for ex5.c with
>>> nlevels = 2 to 7, launched by
>>>
>>> mpiexec -n 1 ./ex5 -da_grid_x 21 -da_grid_y 21 -ksp_rtol 1.0e-9
>>> -da_refine 6 -pc_type mg -pc_mg_levels # -snes_monitor -ksp_monitor
>>> -log_summary
>>>
>>> where -pc_mg_levels is set to a number between 2 and 7.
>>>
>>> So there is a noticeable CPU time improvement from 2 levels to 3 levels
>>> (30%), and then no improvement whatsoever. I am surprised because with 6
>>> levels of refinement of the DMDA the fine grid has more than 1200 points so
>>> with 3 levels the coarse grid still has more than 300 points which is still
>>> pretty large (I assume the ratio between grids is 2). I am wondering how
>>> the coarse solver efficiently solves the problem on the coarse grid with
>>> such a large number of points ? Given the principle of multigrid which is
>>> to erase the smooth part of the error with relaxation methods, which are
>>> usually efficient only for high frequency, I would expect optimal
>>> performance when the coarse grid is basically just a few points in each
>>> direction. Does anyone know why the performance saturates at low number of
>>> levels ? Basically what happens internally seems to be quite different from
>>> what I would expect...
>>>
>>
>> A performance model that counts only flops is not sophisticated enough to
>> understand this effect. Unfortunately, nearly all MG
>> books/papers use this model. What we need is a model that incorporates
>> memory bandwidth (for pulling down the values), and
>> also maybe memory latency. For instance, your relaxation pulls down all
>> the values and makes a little progress. It does few flops,
>> but lots of memory access. An LU solve does a little memory access, many
>> more flops, but makes a lots more progress. If memory
>> access is more expensive, then we have a tradeoff, and can understand
>> using a coarse grid which is not just a few points.
>>
>>   Thanks,
>>
>>      Matt
>>
>>
>>> Best
>>>
>>> Timothee
>>>
>>
>>
>>
>> --
>> 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
>>
>
>


-- 
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
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