[petsc-users] newbie questions on preconditioner LU

[Ling Zou]

Thank you Matt.
I guess I got the impression that M is generally the same as A from reading
the manual.(PETSc Users Manual, Reversion 3.3, page 71, under 4.1 Using KSP)

"Typically the preconditioning matrix (i.e., the matrix from which the
preconditioner is to be constructed), Pmat, is the same as the matrix that
defines the linear system, Amat; however, occasionally these matrices
differ (for instance, when a preconditioning matrix is obtained from a
lower order method than that employed to form the linear system matrix)."

Best,

Ling

On Mon, Jan 28, 2013 at 12:29 PM, Matthew Knepley <knepley at gmail.com> wrote:

> On Mon, Jan 28, 2013 at 2:27 PM, Ling Zou <lingzou80 at gmail.com> wrote:
>
>>
>>
>> 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?
>>
>
> Yes.
>
>    Matt
>
>
>>
>>>    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
>>>
>>
>>
>
>
> --
> 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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