I've been considering using a basis pursuit to determine the parameters. Not done anything more than look closely at the math. It's an underdetermined problem, but basis pursuit can handle that very well using GLPK if one has an accurate dictionary of simulations for the ranges of values spanning the true values.
I was getting less than 1% errors inverting the heat equation for fluid flow in porous media to determine layer thicknesses above and below a horizontal crack. In that case the flow is the sum of two flows, both of which are infinite series of exponentials.
I just ordered an H4, so when that arrives I'll have a good excuse for playing with the Octave solution.
Have Fun!
Reg
