Trading run-time for higher fit accuracy?

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I am using the fit function to fit a curve with quite a few parameters. It turns out that in my problem it is quite hard to predict the parameter ranges that the value should lie in (it would require a lot of pre-analysing of the data). In addition, the accuracy level I need for the fit is quite high. I have tried and I can't really get around these two facts, they are just set by my physics problem at hand.

However one upside is that I don't really have run-time issues. So what I would like to do is sacrifice run-time for a higher fit accuracy. Essentially I would just like the program to run longer and sample more of the parameter space.

I have read the fit options page. The parameters DiffMinChange, DiffMaxChange, MaxIter and TolX seemed promising at first, but from what I have gathered none of these do what I want. Have I missed something? Should I be using a different function maybe?

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