[petsc-users] Natural norm

[Barry Smith]

On Sep 21, 2014, at 12:35 PM, De Groof, Vincent Frans Maria <Vincent.De-Groof at uibk.ac.at> wrote:

> Hi all,
> 
> 
> 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, .ie. ||e||_A = sort(e’*A*e) .  Now, for example, if B = I then one gets e’*A*A*e so it is no longer the “natural norm” of the error, just some other norm.


> 
> 
> I'm sorry for this, I guess, basic question, but I just wanted to be sure.
> 
> 
> kind regards,
> Vincent