Tikhonov parametric functional

It has been proved by Tikhonov and Arsenin (1977) that, for a wide class of stabilizing functionals, their minimum is reached on the model such that /r ,(A(m ), d ) = 6. Thus, we can solve the problem of minimization (2.31) under the condition that 

In other words, one should consider the problem of minimization of the stabilizing functional (2.31), when the model m is subject to the constraint (2.39). A common 

Functional is often called a misfit functional. Thus, the parametric 

Proof There is an exact lower bound of the parametric functional inf P — 7 q , because for any m, P 0. Therefore, wo can select the sequence of models m with the property 

Therefore, the sequence of the models m belongs to the subset Me C M, for which s(m) c. According to the definition of the stabilizing functional, the subset Me is a compact. Therefore, we can select from the sequence m a subsequence which converges to some model M. Inasmuch as the operator A is a continuous operator, we obtain 

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Thus, as an approximate solution of the inverse problem (2.14), we take the solution of another problem (2.41) (problem of minimization of the Tikhonov parametric functional F"(m, d, )), close to the initial problem for the small values of the data errors 6. 

Therefore, the problem of the minimization of the parametric functional introduced by equation (2.79) can be treated in a similar way to the minimization of the conventional Tikhonov parametric functional. The only difference is that now we introduce some variable weighting operator My, which depends on the model parameters. We will discuss in Chapter 5 the different practical techniques of minimizing the parametric functional (2.79). 

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. 

The matrix column of the unknown coefficients can now be found based on the minimization of the Tikhonov parametric functional ... 

Thus, the Euler equation has a unique solution, ma, which can be obtained by the minimal residual method, MRM, or by the generalized MRM. We noted in the beginning of this section that the solution of the minimization problem (4.99) is also unique. Thus, we can conclude that it is equal to mo. In other words, we have proved that minimization of the Tikhonov parametric functional (4.99) is equivalent to the solution of the corresponding Euler equation (4.100). 

Theorem 22. Let A be an arbitrary linear continuous operator, acting from a complex Hilbert space M to a complex Hilbert space D, and W be a positively determined linear continuous operator in M. Then the Tikhonov parametric functional... 

We use the method of constrained inversion developed by Zhdanov and Chernyak (1987). A similar approach to 2-D inverse scattering problem was discussed also by Kleinman and van den Berg (1993). It is based on introducing the Tikhonov parametric functional... 

The inverse problem (10.103) is usually ill-posed, i.e., the solution can be non-unique and unstable. The conventional way of solving ill-posed inverse problems, according to regularization theory (Chapter 2), is to minimize the Tikhonov parametric functional ... 

The only problem now is how to find the family of regularizing operators. Tikhonov and Arsenin (1977) suggested the following scheme for constructing regularizing operators. It is based on introducing special stabilizing and parametric functionals. 

Let us consider first the general approach based on the Tikhonov regularization technique (Tikhonov and Arsenin, 1977). The corresponding parametric functional can be introduced in the following form ... 

According to the conventional Tikhonov regularization method, we substitute for the solution of the linear inverse problem (10.16) a minimization of the corresponding parametric functional with, for example, a minimum norm stabilizer ... 

Our goal is to find the parameters inverse problem. The inverse problem is ill-posed. To solve it, we use the Tikhonov regularization method and minimize the parametric functional with the appropriate stabilizer s cr) ... 

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