So far I have been able to do the initial part of KKT. However, now final optimization of the n(w) and k(w) need to be performed.
Let me brief you the steps we are following (probably this will help you to understand the problem):
1. From experimentally measured Transmission spectrum, estimate k(w) [use Beer-Lambert's Law]
2. From k(w), use KKT to derive n(w) (These are used as initial guess)
3. Use these n(w) and k(w) values to theoretically generate the Transmission spectrum.
4. Compare the "Experimental" and "Theoretical" spectrum. Evidently, there will be mismatch between these two. The reason is that
the experimentally measured spectrum also includes terms of reflectivity (even, multiple times)
5. Now use the current values of n(w) and k(w) as starting points and adjust their values such a way that the resultant "theoretical"
spectrum matches with "Experimental" one. If n and k were single valued or constants then simple curve-fitting would have been
sufficient. however, at present both n and k are function of frequency i.e. n = n(w) and k = k(w)
It is for this particular reason, some optimization routine / approach has to be performed. This is what I am looking for now.
I suggest the following :
select a set of frequency values w1, w2,..., wn, at each such frequency value take your inputs as the guess n(w) and k(w) and outputs as the theoretical n(w) and k(w) values, this constitute your training set, now use a Multi layered Feed forward Neural Network setup with back propagation algorithm or meta heuristics like Genetic algorithm as the algorithm for iterative update of the network parameters, at selected level of threshold for convergence the network parameters would be optimal that describe the input output behavior in terms of the experimental and theoretical spectrum.
my apologies prof. Bagchi,
use the theoretical n(w) and k(w) as inputs and experimental n(w) and k(w) as outputs in constructing the training set.
The above approach is possible, but also completely unnecessary. In performing backprop, your feed forward network is learning to represent a function. Even if it is able to learn the function exactly, this protocol is unnecessary (and possibly error prone - we also don't know that a priori). In your problem the function relating inputs to outputs is already known! Use the adjoint sensitivity method (if needed), or just plain gradient descent - estimating gradients from finite differences in parameter space given the small number of parameters. Using ridge regression should also help. Both of the above are extensions least squares regression. These approaches also completely avoid any generalization error from learning to represent the function via neural nets.
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