how to inverse a sparse matrix in Petsc?

Barry Smith bsmith at
Tue Feb 5 20:16:18 CST 2008

On Feb 5, 2008, at 8:04 PM, Ben Tay wrote:

> Hi Lisandro,
> I'm using the fractional step mtd to solve the NS eqns as well. I've  
> tried the direct mtd and also boomerAMG in solving the poisson eqn.  
> Experience shows that for smaller matrix, direct mtd is slightly  
> faster but if the matrix increases in size, boomerAMG  is faster.  
> Btw, if I'm not wrong, the default solver will be GMRES. I've also  
> tried using the "Struct" interface solely under Hypre. It's even  
> faster for big matrix, although the improvement doesn't seem to be a  
> lot. I need to do more tests to confirm though.
> I'm now doing 2D simulation with 1400x2000 grids. It's takes quite a  
> while to solve the eqns. I'm wondering if it'll be faster if I get  
> the inverse and then do matrix multiplication. Or just calling  
> KSPSolve is actually doing something similar and there'll not be any  
> speed difference. Hope someone can enlighten...
> Thanks!

      Forming the inverse explicitly will be a complete failure.  
Because it is dense it will have (1400x2000)^2 values and
each multiply will take 2*(1400x2000)^2 floating point operations,  
while boomerAMG should take only O(1400x2000).

      BTW: if this is a constant coefficient Poisson operator with  
Neumann or Dirchelet boundary conditions then
likely a parallel FFT based algorithm would be fastest. Alas we do not  
yet have this in PETSc. It looks like FFTW finally
has an updated MPI version so we need to do the PETSc interface for  


> Lisandro Dalcin wrote:
>> Ben, some time ago I was doing some testing with PETSc for solving
>> incompressible NS eqs with fractional step method. I've found that in
>> our software and hardware setup, the best way to solve the pressure
>> problem was by using HYPRE BoomerAMG. This preconditioner usually  
>> have
>> some heavy setup, but if your Poison matrix does not change, then the
>> sucessive solves at each time step are really fast.
>> If you still want to use a direct method, you should use the
>> combination '-ksp_type preonly -pc_type lu' (by default, this will
>> only work on sequential mode, unless you build PETSc with an external
>> package like MUMPS). This way, PETSc computes the LU factorization
>> only once, and at each time step, the call to KSPSolve end-up only
>> doing the triangular solvers.
>> The nice thing about PETSc is that, if you next realize the
>> factorization take a long time (as it usually take in big problems),
>> you can switch BoomerAMG by only passing in the command line
>> '-ksp_type cg -pc_type hypre -pc_hypre_type boomeramg'. And that's
>> all, you do not need to change your code. And more, depending on your
>> problem you can choose the direct solvers or algebraic multigrid as
>> you want, by simply pass the appropriate combination options in the
>> command line (or a options file, using the -options_file option).
>> Please, if you ever try HYPRE BoomerAMG preconditioners, I would like
>> to know about your experience.
>> Regards,
>> On 2/5/08, Ben Tay <zonexo at> wrote:
>>> Hi everyone,
>>> I was reading about the topic abt inversing a sparse matrix. I  
>>> have to
>>> solve a poisson eqn for my CFD code. Usually, I form a system of  
>>> linear
>>> eqns and solve Ax=b. The "A" is always the same and only the "b"  
>>> changes
>>> every timestep. Does it mean that if I'm able to get the inverse  
>>> matrix
>>> A^(-1), in order to get x at every timestep, I only need to do a  
>>> simple
>>> matrix multiplication ie x=A^(-1)*b ?
>>> Hi Timothy, if the above is true, can you email me your Fortran code
>>> template? I'm also programming in fortran 90. Thank you very much
>>> Regards.
>>> Timothy Stitt wrote:
>>>> Yes Yujie, I was able to put together a parallel code to invert a
>>>> large sparse matrix with the help of the PETSc developers. If you  
>>>> need
>>>> any help or maybe a Fortran code template just let me know.
>>>> Best,
>>>> Tim.
>>>> Waad Subber wrote:
>>>>> Hi
>>>>> There was a discussion between Tim Stitt and petsc developers  
>>>>> about
>>>>> matrix inversion, and it was really helpful. That was in last Nov.
>>>>> You can check the emails archive
>>>>> Waad
>>>>> */Yujie <recrusader at>/* wrote:
>>>>>    what is the difference between sequantial and parallel AIJ  
>>>>> matrix?
>>>>>    Assuming there is a matrix A, if
>>>>>    I partitaion this matrix into A1, A2, Ai... An.
>>>>>    A is a parallel AIJ matrix at the whole view, Ai
>>>>>    is a sequential AIJ matrix? I want to operate Ai at each node.
>>>>>    In addition, whether is it possible to get general inverse  
>>>>> using
>>>>>    MatMatSolve() if the matrix is not square? Thanks a lot.
>>>>>    Regards,
>>>>>    Yujie
>>>>>    On 2/4/08, *Barry Smith* <bsmith at
>>>>>    <mailto:bsmith at>> wrote:
>>>>>            For sequential AIJ matrices you can fill the B matrix
>>>>> with the
>>>>>        identity and then use
>>>>>        MatMatSolve().
>>>>>            Note since the inverse of a sparse matrix is dense  
>>>>> the B
>>>>>        matrix is
>>>>>        a SeqDense matrix.
>>>>>            Barry
>>>>>        On Feb 4, 2008, at 12:37 AM, Yujie wrote:
>>>>>        > Hi,
>>>>>        > Now, I want to inverse a sparse matrix. I have browsed  
>>>>> the
>>>>>        manual,
>>>>>        > however, I can't find some information. could you give me
>>>>>        some advice?
>>>>>        >
>>>>>        > thanks a lot.
>>>>>        >
>>>>>        > Regards,
>>>>>        > Yujie
>>>>>        >
>>>>> ------------------------------------------------------------------------
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>>>>> Search.
>>>>> <* 
>>>>> >

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