[petsc-users] Linear Least Square with regularization using petsc4py

[Matthew Knepley]

On Tue, Jul 28, 2015 at 5:55 AM, Romain Jolivet <jolivetinsar at gmail.com>
wrote:

> Dear PETSc developers and users,
>
> I am trying to solve a linear least square problem that requires some
> regularization. My plan is to use a Tikhonov scheme as a first try.
> Hence, I would like to minimise the function S(x) defined as:
>
> S(x) = ||Ax - b||2 + alpha || Cx ||
>
> where A is my operator (rectangular matrix with far more lines than
> columns), x is the vector of unknown, b is some data vector, alpha is a
> damping factor and C some operator of my own.
> Ideally C is a gaussian form of a covariance matrix. But at first, I would
> like to check what happens with an identity (good), gradient (better) or
> laplacian (best) operator.
> There is a mention to a MatShift function in some previous post in the
> mailing list concerning Tikhonov regularization, but I don’t see how to
> incorporate that into my python set of tools that I have set up to solve my
> problem (would there be a useful example somewhere?).
>

It would be nice to have an example for this, but I have not made it there
yet. For now, it looks like
what you want is

  A.shift(aC)

where A is the Mat that you are currently using in your linear solve, and
aC is alpha*C. Does this make sense?

  Thanks,

     Matt


> I feel like there is some possibilities using the TAO wrappers of
> petsc4py, but I have no idea how to get this guy to work (how do I create
> the TAO_objective, TAO_gradient, etc?).
>
> To get a better understanding of my problem, my ultimate goal is to
> minimise the function S(x) defined as
>
> S(x) = 1/2. * [(Ax-b)’ inv(D) (Ax-b) + (x-x0)’ inv(M) (x-x0)],
>
> where D is some covariance matrix describing noise distribution on b, M is
> the covariance matrix describing the correlations between the parameters of
> x and x0 is the starting point.
> But that seems a bit complex so far and I will stick to the first problem
> yet.
>
> Cheers,
> Romain
>
>
>
> —————————————————————————————————————
> —————————————————————————————————————
> Romain Jolivet
> Postdoctoral Fellow
>
> University of Cambridge
> Department of Earth Sciences
> Bullard Labs
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> Cambridge CB3 0EZ
> United Kingdom
>
> email: rpj29 at cam.ac.uk
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>
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-- 
What most experimenters take for granted before they begin their
experiments is infinitely more interesting than any results to which their
experiments lead.
-- Norbert Wiener
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