[petsc-users] newbie questions on preconditioner LU

[Matthew Knepley]

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.

   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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