• SolverIssues

WimBrouw has outlined his ideas about solving strategy in a series of recent emails (which should be copied here, of course). This is just my summary, and a few comments, so that we have at least some record of these things.

A reminder of some background:

• -) The solver is linear if the LM parameter is zero, and non-linear otherwise. -) When non-linear, the solver assumes a parabolic shape for the minimum of
  • the qhisq surface.
  -) The magnitude of the LM parameter determines the step-size in solution-
  • space. Initially, it is set to a 'large' value, and it is decreased (or increased) as the solvers feels that it is getting closer to the minimum.
  -) A non-zero LM parameter assures that the matrix is positive-definite, so
  • it will always invert. In practice, finite machine precision will cause errors. This is avoided by having a non-zero colinearity factor, which will cause a FAIL when the vectors get more parallel than a certain small angle. This happens if we try to solve with too little information.
  -) Using SVD will avoid this condition, because, if necessary, it generates
  • some extra condition equations, that supply the missing information. This may be detected by looking at the matrix rank, which is smaller than full. NB: The extra processing cost of using SVD is negligible.
  -) The alternative to SVD is to supply the extra condition equations
  • explicitly (using extra condeqs), e.g. equations that require the sum of the unkown phases to be zero. According to WNB, supplying a priori information in this way MAY cause a non-optimal path to the solution...
  -) Note that all this explains why the rank gradually decreases as we get
  • closer to the solution: As the maginitude of the LM parameter decreases, the non-zero colinearity factor will cause SVD to kick in with more and more extra equations. (It is now possible to do tests with different values of the colinearity factor, and with or without SVD.)

The recommended strategy is the following:

• 1) Set the colinearity factor to zero, and switch off SVD. Since the LM
  • parameter is non-zero, it will converge non-linearly. Do NOT supply and extra condition equations.
  2) After convergence, set the LM parameter to zero, and do a last linear
  • solution, with either SVD or explicit condition equations (the latter are by definition linear in the unknowns!). NB: Only at this point is it meaningful to ask the WNB fitter for covariance values...

Some remarks:

• -) I have little feeling of what machine inprecesion will do to the solution
  • in the first phase. It seems to me that it is safer (at little cos) to keep SVD switched on, and have a small colinearity factor (i.e. what we are doing now).
  -) In order to include extra condition equations in the last step, the solver
  • needs to know which of its condeq children contain them. This is certainly possible, because the tree builder can set a switch when the condeq is created.
  -) If we decide to go this route, the intelligence has to be built into the

  • MeqSolver node. In this stage, I am not yet convinced of the superiority of this strategy over our current practice, but that could change...

  -) See also the Bugzilla item on estimating the chisq surface

There are two issues here: The accuracy of the solution (especially in the presence of noise), and efficiency (the number of iterations needed). As to the latter, I am much more concerned about using the last solution as starting position for the next, which is impossible in tiled solutions....