[petsc-users] Ambiguity of KSPCGNE

[Alexander Grayver]

Hi Barry,

>> In this case the application of the method is quite restricted since all practical least squares problems formulated in form of normal equations are solved with regularization, e.g.:
>>
>> (A'A + \lamba I)x = A'b
> Yes it is restrictive. There is no concept of lambda in CGNE in PETSc

In this case, since there is LSQR in PETSc, there is hardly any reason 
to use CGNE.

>> Assume I have A computed and use matrix free approach to represent (A'A + \lamba I) without ever forming it, so what should I do then to apply KSPCGNE?
>     If you supply a shell matrix that applies (A'A + \lamba I)  why not just use KSPCG?

That is what I do at the moment. However, as far as I understand, CGLS 
is not just about shifting original matrix with some lambda, it has 
other advantages over CG for normal equations.

>      But if you provide this shell matrix, how do you plan to apply a preconditioner?

One can easily compute diag(A'A + \lamba I), thanks to 
MatGetColumnNorms, and thus Jacobi is possible. Since the matrix is 
diagonally dominant, in my case it is enough to converge. Although 
convergence for normal equations does not imply accurate solution, so 
that one needs CGLS or LSQR.

-- 
Regards,
Alexander