[petsc-users] newbie questions on preconditioner LU

[Ling Zou]

On Mon, Jan 28, 2013 at 12:01 PM, Matthew Knepley <knepley at gmail.com> wrote:

> On Mon, Jan 28, 2013 at 1:39 PM, Ling Zou <lingzou80 at gmail.com> wrote:
>
>> Hi, All
>>
>> I am trying to understand how the preconditioner works when using KSP.
>>
>> For example, when using KSP to solve the linear system problem,
>>
>> Ax = b
>>
>> with the default left preconditioning. We actually solve,
>>
>> M^(-1) * A x = M^(-1) * b
>>
>> where, M is the preconditioning matrix and in many cases, we just use A
>> as the preconditioning matrix.
>>
>>
>> Question:
>> 1), Is the understanding above correct?
>>
>
> This is too simplistic. If you really mean M^{-1}, then no, you (almost)
> never use A as M. If you mean an
> approximate inverse to M, then yes.
>
>
>> 2), If the understanding above is correct, is it correct to state the
>> different methods provided in PETSc (such as PCLU, PCILU, etc) are to
>> calculate the inverse matrix M^(-1) from M?
>>
>
>  An approximate inverse.
>
>
>> 3), How to understand this sentence in the manual (PETSc Users Manual,
>> Reversion 3.3, page 78, under 4.4 Preconditioners)
>> "The direct preconditioner, PCLU, is, in fact, a direct solver for the
>> linear system that uses LU factorization. PCLU is included as a
>> preconditioner so that PETSc has a consistent interface among direct and
>> iterative linear solvers."
>> Does this indicate when using PCLU, we solve Ax = b directly using LU
>> factorization, or, we solve M^(-1) from M using LU factorization?
>>
>
> Same thing, if M = A,
>
>   M^{-1} A x = A^{-1} A x = x = A^{-1} b
>
> which is Gaussian elimination for the original problem.
>

Ahh.. that's true!
In case M is not A (as you pointed out earlier), does PCLU provide the
approximated inverse matrix of M^{-1} using LU factorization on M?


>    Matt
>
>
>> As a beginner to the PETSc, all questions are probably too simple. I'd
>> appreciate it if someone could answer my questions.
>>
>> Best,
>>
>> Ling
>>
>
>
>
> --
> 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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