[petsc-users] Natural norm

[Jed Brown]

Barry Smith <bsmith at mcs.anl.gov> writes:

> On Sep 21, 2014, at 12:35 PM, De Groof, Vincent Frans Maria <Vincent.De-Groof at uibk.ac.at> wrote:
>
>> the natural norm for positive definite systems in Petsc uses the
>> preconditioner B, and is defined by r' * B * r. Am I right assuming
>> that this way we want to obtain an estimate for r' * K^-1 * r, which
>> is impossible since we don't have K^-1? But we do know B which is
>> approximately K^-1.
>
>    I think so. The way I look at it is r’ * B * r = e’ *A *B *A e and
>    if B is inv(A) then it = e’*A*e which is the “energy” of the error
>    as measured by A,

Hmm, unpreconditioned CG minimizes the A-norm (energy norm) of the error:
i.e., |e|_A = e' * A * e.  This is in contrast to GMRES which simply
minimizes the 2-norm of the residual: |r|_2 = r' * r = e' * A' * A * e =
|e|_{A'*A}.  Note that CG's norm is stronger.

When you add preconditioning, CG minimizes the B^{T/2} A B^{1/2} norm of
the error as compared to GMRES, which minimizes the B' A' A B norm (or
A' B' B A for left preconditioning).

If the preconditioner B = A^{-1}, then all methods minimize both the
error and residual (in exact arithmetic) because the preconditioned
operator is the identity.
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