Forming a sparse approximation of a MatShell

[Jed Brown]

On Wed 2008-04-09 15:10, Barry Smith wrote:
>
>    Jed,
>
>     The Mat coloring code can also be used directly, not through SNES. Once 
> you have
> the coloring for the matrix (you can get that with MatGetColoring(), of 
> course, this assumes
> you have already set a nonzero pattern for your matrix)). Call 
> MatFDColoringCreate()
> then MatFDColoringSetFunction(), MatFDColoringSetFromOptions() and then 
> MatFDColoringApply().

Cool, I tried this and I can confirm that it is generating the correct matrix
(by comparing entries with the output of -snes_fd) but unfortunately the matrix
entries of the spectral operator corresponding to neighbors are actually not a
very good approximation of the full operator.  Bummer.  It looks like I'm stuck
with formulating the problem twice, once for the spectral operators and once for
the FD/FE preconditioner.  Thanks for the help.

Jed

> On Apr 9, 2008, at 2:13 PM, Jed Brown wrote:
>> I'm trying to improve the preconditioning of my spectral collocation 
>> method for
>> non-Newtonian incompressible Stokes flow.  My current algorithm uses 
>> MatShell
>> for the full Jacobian as well as each of its blocks [A B1'; B2 0] and the 
>> Schur
>> complement S = -B2*A*B1'.  I needed a preconditioner for A so I thought 
>> I'd
>> solve the same problem using finite differences on the Chebyshev nodes.  
>> In
>> reality, the stencil is really ugly in 3D so I just used a simpler 
>> elliptic
>> operator.  This works okay, but it's performance decays significantly as I
>> increase the continuation parameter.  Also, dealing with general boundary
>> conditions is rather tricky and it seems to be a much weaker 
>> preconditioner when
>> I have mixed boundary conditions.  To rectify this, I tried a finite 
>> element
>> discretization on the Chebyshev nodes (using Q1 elements).  This must be 
>> scaled
>> by the inverse (lumped) mass matrix due to the collocation nature of the
>> spectral method.  Strangely, even though it captures all the terms in the
>> Jacobian, it is slightly weaker than the finite difference version.  At 
>> least it
>> is less error-prone and boundary conditions are easier to get right.
>> Regardless, forming the explicit matrix separately from the spectral 
>> matrix
>> causes a duplication of concepts that have to be kept in sync.  So I 
>> started
>> thinking, the spectral matrix is pretty cheap to apply a few times, so 
>> perhaps I
>> can use a coloring to compute a sparse approximation.  However, the
>> documentation I found is using the function from the SNES context to form 
>> the
>> matrix.  In my case, the entire Jacobian doesn't help, I just want an
>> approximation of A.  (A itself is full, but implemented via FFT.)  What is 
>> the
>> correct way to do this?  Should I just stick with finite differences or 
>> finite
>> elements?
>>
>> Also, any ideas for preconditioning S?  It's condition number also grows
>> significantly with the continuation parameter.
>>
>> Thanks,
>>
>> Jed
>
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