[petsc-users] newbie questions on preconditioner LU

[Jed Brown]

On Mon, Jan 28, 2013 at 1:27 PM, Ling Zou <lingzou80 at gmail.com> wrote:

> 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?
>

Not really, preconditioners are based on inexact algorithms applied to A,
not explicit formation of an M that is easier to factor exactly. Since the
preconditioner P ("=M^{-1}") is non-singular, there _exists_ an M such that
P=M^{-1}, but M is not explicitly computed and it's not used in the solve.
Only P is used, and only in special cases (like incomplete factorization)
is there even a practical algorithm available to compute M if you wanted
to. (For many interesting algorithms, M is dense even though A is sparse.)
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