On Jun 22, 2012, at 2:54 AM, Alexander Grayver wrote:
> 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.
Ok, then it is an algorithm that is not currently implemented in PETSc.
CGNE is only for people who have A (which is square) and want to solve the
normal equations with CG using the preconditioner of A and its transpose for
the preconditioner. Basically it allows the user to avoid computing A'A
explicitly or making their own shell matrix. It is definitely not a substitute
for LSQR.
Barry
>
>> 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
>
