L-curve method of regularization parameter selection

L-curve analysis (Hansen, 1998) represents a simple graphical tool for qualitative selection of the quasi-optimal regularization parameter. 

It is based on plotting for all possible a the curve of the misfit functional, f(a),versus the. stabilizing functional, s(a) (where we use notations (2.82)). The L-curve illustrates the trade-off between the best fitting (minimizing a misfit) and most reasonable stabilization (minimizing a stabilizer). In a case where a is selected to be too small, the minimization of the parametric functional P (m) is equivalent to the minimization of the misfit functional therefore i a) decreases, while s(q) increases. 

The distinct corner, separating the vertical and the horizontal branches of this curve, corresponds to the quasi-optimal value of the regularization parameter a. 

Analysis of total variation penalty methods Inverse Problems, 10, 1217-1229. 

Constable, S. C., Parker, R. C., and G. G. Constable, 1987, Occam s inversion a practical algorithm for generating smooth models from EM sounding data Geophysics, 52, 289-300. 

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