Tikhonov functional

Therefore, the problem of minimizing the parametric functional, given by equation (5.117), can be treated in a similar way to the minimization of the conventional Tikhonov functional. The only difference is that now we introduce some variable weighting matrix Wg for the model parameters. The minimization problem for the parametric functional introduced by equation (5.117) can be solved using the ideas of traditional gradient-type methods. 

The minimum of the Tikhonov functional is achieved with the optimal value of the a regularisation parameter a = 9A x 10, while the residual norm value was 1.07x1This method allows the variation of the error value in the integral equation core selection process. However, as the model core was the Schulz-Flory function, the error h was assumed to be h 0. 

Skaggs and Kabala [60] extended their study of TR using Monte Carlo simulation to answer the question of how far back one can use the Tikhonov procedure in recovering the release history of the plume, since the procedure always produces the recovered release curve that accurately reproduces the data. A table containing the percentage of test function recovery accuracy was produced. From the table, one can extract the information on how likely is that the recov-... 

Based on Tikhonov s theorem, we know that the operator A is a continuous one on the correctness set C. Therefore, we conclude from (2.22) that the quasi-solution is a continuous function of d. Note that this property holds only in the correctness set C. If one selects a solution, m, from outside the correctness set, it may be no longer a continuous function of the data (see Figure 2-1). 

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. 

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... 

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. 

Note that this stabilizer works similarly to the maximum entropy regularization principles, considered, for example, in Smith et al. (1991), and Wernecke and D Addario (1977). However, in the framework of the Tikhonov regularization, the goal is to minimize a stabilizing functional, which justifies the minimum entropy name for this stabilizer. 

Repeating the considerations described above for s s (m), one can demonstrate that the minimum gradient support functional satisfies the Tikhonov criterion for a stabilizer. 

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. 

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 ... 

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... 

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 ... 

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 ... 

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) ... 

Kuznetsov, P. I., Stratonovich, R, L., and Tikhonov, V. I. Quasi-moment functions in the theory of random processes. Theory Prob. Appl. 5, 80-97 (1960). [English translation series.]... 

This stabilization approach was introduced by Phillips [PHI 62] and Tikhonov [TIK 63, TIK 77], and is often referred to as the Tikhonov stabilization . It consists of a general approach to ill-posed problems, when the signal resulting from a convolution is used to try to determine the pure profile. The main idea is to limit the number of solutions to equation [6.2] by including additional information. This information, suggested by Tikhonov, is that the function is smooth or, in other words, that its second derivative is as close as possible to zero. 

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