Fitting sums of exponentials is indeed very tricky and can be
ill-conditioned. Do not blame the least-squares for this, it is
because of the exponentials.
The Pade'-Laplace approximation does a very attractive thing.
Namely, it relies on the Laplace transform which is essentially
an integral. Integrals are lovely because (due to their "averaging"
nature) they 'smooth' the data, rendering them more tractable by
numerical analysis (i.e. parameter estimation in this case).
However, be warned: there are no wonders. IMHO everyone should
refrain from fitting complicated models to noisy data -- and ALL
data are noisy in biology. Try to prune your models first,
do not attempt estimating more than, say, 3 parameters. Perform
sensitivity analysis, hunt down hidden relationships between
parameters, always test your estimated correlation matrix
to see the eigenvalue spectrum... I'm speaking from experience;
I have suffered enough ;-)
Good luck to all fitters,