[petsc-dev] Solving under determined systems

[Mark Adams]

On Wed, Sep 21, 2016 at 10:20 PM, Munson, Todd <tmunson at mcs.anl.gov> wrote:

>
> You can set up TAO to solve such a problem.
>
> However, your problem boils down to solving the
> linear system
>
>   w - V*lambda = 0
>   V'*w         = b
>
> Taking the Schur complement with respect to w, you get
> the system
>
>   V'*V*lambda = b
>
> You then form and invert V'*V, which is a 5x5 matrix
> and recover w = V*lambda.
>

> That will get you the least 2-norm solution for your
> underdetermined system.
>

Fantastic, this is is perfect for us,
Thanks,


> LSQR will solve the underdetermined system and give
> you the least norm solution if you don't want to
> do the matrix-matrix product and inverse.  LSQR
> may also be more stable.  See
>
>   http://web.stanford.edu/group/SOL/software/lsqr/
>
> That site suggests using CRAIG in the underdetermined
> case, but I don't know if CRAIG is implemented
> in PETSc.
>
> Other norms are more difficult to obtain, but can
> be done.  One and infinity norms are recast as
> linear programming problems.  Matrix norms are
> equality constrained quadratic programs.
> p-norms with p >= 1 are convex
> nonlinear programs.
>
> Todd.
>
> > On Sep 21, 2016, at 8:24 PM, Mark Adams <mfadams at lbl.gov> wrote:
> >
> > I thought least squares was for tall skinny (overdetermined) solves? I
> have a short fat (5 x ~100) matrix to solve.
> >
> > On Wed, Sep 21, 2016 at 4:24 PM, Stefano Zampini <
> stefano.zampini at gmail.com> wrote:
> > Mark,
> >
> > You can use KSPLSQR
> >
> > Stefano
> >
> >
> > Il 21 set 2016 11:21 PM, "Mark Adams" <mfadams at lbl.gov> ha scritto:
> > I want to solve for w in V' w = b, where V is tall and skinny. So a
> short fat matrix "solve". This is underdetermined. I would like to minimize
> the two norm (or any norm) of w. This looks like an optimization problem,
> would TAO do this?
> > Mark
> >
>
>
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