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After solving the inverse problem*, how to investigate whether the solution is unique or not? When I check theoretical values and published values during the convergence and search for the lowest value for chi-squared, is it sufficient?

*The task when the observed data is given (radial velocity, light curves for an eclipsing binary), and someone needs to determine the set of parameters for which the model yields the best match of the observed curves.

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  • $\begingroup$ I think you mean "when identifying the causes of a set of observations"... "how to I determine if my solution is unique. Is that right? $\endgroup$
    – James K
    Aug 30, 2021 at 7:20
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    $\begingroup$ I mean that the observed data is given (radial velocity, light curves for an eclipsing binary), and someone needs to determine the set of parameters for which the model yields the best match of the observed curves. $\endgroup$
    – Anna-Kat
    Aug 30, 2021 at 7:21
  • $\begingroup$ @JamesK yes it is right $\endgroup$
    – Anna-Kat
    Aug 30, 2021 at 7:24
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    $\begingroup$ In general you can't. An method based on convergence can hope at best to find a local minimum (of chi-squared or however you are judging the goodness of fit) There might be other, better minima that your method doesn't find. $\endgroup$
    – James K
    Aug 30, 2021 at 7:25
  • $\begingroup$ So the only inspection is to try whether slightly different initial values led to a similar solution and check that the resulting values make sence, right? $\endgroup$
    – Anna-Kat
    Aug 30, 2021 at 7:28

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In general you can't prove that your solution is unique. Methods for fitting a model to data that are based on some form of convergence can get caught in a local optimum and not find the global optimum.

For specific models with specific criteria the best model can be found analytically. For a simple example, fitting a linear model with least squares optimisation can be found by the usual linear regression line. No convergence is used and the method provably gives the best fit according to the given measure.

But for non-linear model fitting, there are no general proofs, and it is quite possible for a particular convergence to fail.

However "the best" model may not be required. You may only need a "good enough" model.

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