Three light emitters are placed in a room and one detector in each corner (-1;1), (-1;-1), (1;-1) and (1,1). Also the positions of emitters are given. This system should be represented by a matrix. Source values should be found by LSQR. After that, SVD incl. gaussian 0.1 standard deviation noise has to be performed. Finally, Tikhonov regularization and the L-curve are needed. Anyone know how these two work mathematically and what their results represent?
Tikhonov regularization is a way to control the complexity of models by placing a penalty on the parameter vector (usually such that the parameters are closer to zero, thereby avoiding very large parameters that can easily occur in case of overadaptation to the training data, especially if the problem at hand is ill-posed).
As far as I understand your problem description above, tikhonov regularization is already done in the least squares regression assuming Gaussian noise. Assuming zero-mean i.i.d. Gaussian noise, this is equal to ridge regression.
To add to my predecessor:
Performing the SVD on your coefficient matrix basically equals to a stability analysis.
From the singular values, the spectral condition number can be derived, giving you a tool to measure the noise amplification induced by your inversion.
If one does not use LSQR, SVD can be applied to find a suitable Tikhonov Matrix to stabilize the inverse.
To contribute a bit here. (mainly on the introductory question)
Tikhonov reguarization is, as already mentioned, a way to constrain the solution to small values. You do this by taking the 2nd norm of the solution vector, multiplying with a parameter λ and adding to the LSQ error. Then you try to minimize the whole sum.
The L-curve is a graphical indication of how the solution norm (y-axis) changes to the LSQ error (x-axis) for different values of λ. The curve has an L-shape and thus the name.
Maybe my contribution seems a bit confusing but you can check the attached file for further explanation. There are a couple of great paper on the subject by P.C.Hansen (he has them online on his webpage).
Good luck!
I would suggest the Introduction and chapter 5 (Determination of the Tikhonov factor) for L-curve in the following webpage. It is very easy to read:
http://en.wikipedia.org/wiki/Tikhonov_regularization#Determination_of_the_Tikhonov_factor
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