Barry Smith <[email protected]> writes: > On Sep 21, 2014, at 12:35 PM, De Groof, Vincent Frans Maria > <[email protected]> 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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