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I Tikhonov regularization and SVD to compute condition number

  1. Apr 7, 2016 #1
    I am solving linear least squares problems with generalized Tikhonov regularization, minimizing the function:
    [tex]
    \chi^2 = || b - A x ||^2 + \lambda^2 || L x ||
    [/tex]
    where [itex]L[/itex] is a diagonal regularization matrix and [itex]\lambda[/itex] is the regularization parameter. I am solving this system using the singular value decomposition, and I use the normal trick to convert the system into "Tikhonov standard form":
    [tex]
    \tilde{A} = A L^{-1}, \tilde{x} = L x
    [/tex]
    so that
    [tex]
    \chi^2 = || b - \tilde{A} \tilde{x} ||^2 + \lambda^2 || \tilde{x} ||
    [/tex]
    This system can now be solved easily with the SVD of [itex]\tilde{A}[/itex]. My problem is that I want to compute the condition number of [itex]A[/itex] while solving the system, but I don't see any easy way to compute this since I calculate only the SVD of [itex]\tilde{A}[/itex]. It seems that there is no simple relationship between the singular values of [itex]A[/itex] and the singular values of [itex]\tilde{A}[/itex], even when [itex]L[/itex] is a diagonal matrix.

    Does anyone know of a solution to this problem, without having to separately compute the SVD of [itex]A[/itex] itself?
     
  2. jcsd
  3. Apr 12, 2016 #2
    Thanks for the post! This is an automated courtesy bump. Sorry you aren't generating responses at the moment. Do you have any further information, come to any new conclusions or is it possible to reword the post?
     
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