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Linearizing a nonlinear least squares model

  1. Dec 7, 2015 #1
    I have a nonlinear least squares problem with a set of parameters [itex]\bf{g}[/itex], where I need to minimize the function:
    [tex]
    \chi^2 = \sum_i \left( y_i - M(t_i ; {\bf g}) \right)^2
    [/tex]
    The [itex]t_i[/itex] are some independent parameters associated with the observations [itex]y_i[/itex] and the model function has the form
    [tex]
    M(t_i ; {\bf g}) = \sqrt{X(t_i; {\bf g})^2 + Y(t_i;{\bf g})^2}
    [/tex]
    The functions [itex]X(t_i;\bf{g})[/itex] and [itex]Y(t_i;\bf{g})[/itex] are linear in the model parameters [itex]\bf{g}[/itex], ie:
    [tex]
    X(t_i;{\bf g}) = \sum_k g_k X_k(t_i)
    [/tex]
    and
    [tex]
    Y(t_i;{\bf g}) = \sum_k g_k Y_k(t_i)
    [/tex]
    In order to solve the nonlinear least squares problem, I need to construct the matrix [itex]J^T J[/itex] at each iteration, where [itex]J[/itex] is the Jacobian:
    [tex]
    J_{ij} = - \frac{\partial}{\partial g_j} M(t_i;{\bf g})
    [/tex]
    My question is can anyone see a clever way to optimize the computation of the matrix [itex]J^T J[/itex], since all of the [itex]X_k(t_i)[/itex] and [itex]Y_k(t_i)[/itex] can be precomputed? I have millions of observations (rows of the Jacobian) and so its extremely slow to compute each row of the Jacobian and add it into the [itex]J^T J[/itex] matrix. I'm hoping that the linearity of the functions [itex]X[/itex] and [itex]Y[/itex] might allow some way to linearize or precompute large portions of the Jacobian matrix, but I don't see an easy way to do this.
     
  2. jcsd
  3. Dec 12, 2015 #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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