Optimal regularization parameter selection

Let us assume that data are observed with some noise, d = dj -l-6d, where d( is the true solution of the problem and the level of the errors in the observed data is equal to 6  

Then the regularization parameter can be determined by the misfit condition (2.49) 

To justify this approach we will examine more carefully the properties of all three functionals involved in the regularization method the Tikhonov parametric functional and the stabilizing and misfit functionals. 

Subtracting the left-hand side of inequality (2.86) from the right-hand side of inequality (2.84) and the right-hand side of inequality (2.86) from the left-hand side of inequality (2.84), we obtain 

It can be proved that the functions p a), i a), s a) are continuous functions (if the element rric, is unique). 

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Figure 2-5 Illustration of the principle of optimal regularization parameter selection.

Figure 2-5 helps in understanding of the principle of optimal regularization parameter selection. One can see that because of the monotonic character of the function i a), there is only one point, Qq, where i(ao) = p, (A(maJ,d ) = 6. ... 

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

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