How to scale the linear system Ax = b?

[klausbu]

Hello,

I am working on a stronger preconditioner for symmetric matrices which requires scaling of the linear system Ax = b as follows:

A' = D^-1/2 A D^-1/2

x' = D^1/2 x

b' = D^-1/2 b

Where A'x' = b' is the scaled linear system and D is the Diagonal as defined in the OpenFOAM standard LDU matrix storage format.

Are you aware of functors/transformations I could reuse, implemented in other parts of RapidCFD?

Klaus

[daniel-jasinski]

Hi,

You can take a look at the AINVPreconditioner. Instead of using the divideOperatorSFFunctor<scalar,scalar,scalar>(1.0), apply sqrtUnaryFunctionFunctor<scalar,scalar>().

You can then use the computed scaling field and use it in the actual preconditioner kernel.

The AINV implementation is a bit more complicated than it could be, because of some optimization tricks. Just to make your implementation working, you can remove half of the branches in preconditionImpl by assuming that fastPath is always false.

Daniel