The defining stage of the process of solving ill-posed inverse problems is the choice of the regularization parameter

Research methods and received results.
The general form of the model in the problem of interpreting observations is a nonlinear integral equation of the Fredholm type of the 1st kind
(1)
where ‑ kernel; ‑ right side; ‑ required function. Let us introduce some notation and definitions necessary to formulate the method of model experiments.
Let the approximate dependences in (1) be defined as follows
(2)
(3)


Definition 1. Two problems ‑ and , respectively described by the equations


we call them similar if
(4)
For such problems the following equality holds: [7]
. (5)
Remark 1. The number of problems W that are similar to some given original P is infinite due to the possibility of changing the factor g.
Definition 2. Let’s call a set of parameters and functions
(6)
a mode, where , are the steps (generally speaking, non-constant) of the finite-step approximation of equation (1) when solving it numerically, is the relative error of the right-hand side, and is the relative error of the kernel.
Definition 3. An estimation problem with respect to some original P is a problem V that has the same mode as the problem P.
Remark 2. The number of evaluation problems corresponding to a given original is infinite, since given values of and can correspond to various laws of distribution and implementation of errors and .
Definition 4. Let us call a model problem a problem that is intermediate in the amount of a priori information (type of function, information about the nature of the smoothness of the function) about the desired solution between and . The more a priori information included in the problem , the closer to .
Determination of the regularization parameter when solving nonlinear integral equations of the Fredholm type of the 1st kind based on the method of model experiments is as follows:
1. Let some practical problem (original) of the form (1) be given, namely , and the mode
(7)
and , where and — exact values; and — their practical values; and — errors (i.e. errors and are assumed to be known approximately).
2. A model problem is created, namely, a function is specified taking into account a priori information about the desired solution . For example, if the problem of resolving sources is being solved and it is known a priori that the number of sources is two, then should be specified as the sum of two peaks using a priori estimated values of the distance between the peaks, the ratio of their heights, peak widths, etc.
3. is defined from (1) at such , that
. (8)
4. The equation obtained as a result of regularization (1) is solved (for different ) (the form of the regularized equation depends on which regularization method is chosen) relatively in mode (6) with the kernel = = and the right side , such that
. (9)
Moreover, even if the errors and are random in nature, they can be obtained using a random number sensor. If the errors are systematic, for example, it is known a priori that and are functions of the same smoothness, then to fulfill (8) the errors should be obtained by specifying the same smoothness .
5. is defined, i.e. that at which
. (10)
6. is defined — solution (1), for which .
The algorithm for determining the optimal regularization parameter using model experiments is as follows:
1. Let the kernel and the right side of the problem being solved be given on the grids , .
2. The kernel and exact solution of the model problem are defined on the grids , .
3. is defined.
4. The right side , =0, , , of the model problem is calculated.
5. Defined such , that .
6. For (1) is solved by the method of finite sums and differences with , .
7. The is determined such that the function takes the minimum value.
8.  is defined — solution of the original from (1) at ,
, .
As noted above, the choice of a regularizing method plays an important role in solving nonlinear integral equations of the Fredholm type of the first kind. The analysis allows us to conclude that the most suitable for this class of problems are the Tikhonov method and the Lavrentiev-Vasin method [8].
Let a nonlinear Fredholm type integral equation of the first kind be given
(11)
Using the Tikhonov regularization method, we reduce equation (11) to a nonlinear Fredholm type integral equation of the second kind [9].
, (12)
where


— regularization parameter.
(13)
(14)
— error of the kernel, — error of the right side, — error of (11)’s solution, , and — relative errors.
The application of the Tikhonov regularization method in combination with the method of model experiments comes down to the following experimental steps:
1. A practical problem (original) of type (1) is specified by its right side and mode.
(15)
2. A model problem is created, namely, a function is specified taking into account a priori information about the desired solution , kernel , such that
. (16)
3. Using the selection method, is determined from (11), with and , such that the equality
(17)
is satisfied.
4. A nonlinear Fredholm type integral equation of the 2nd kind (12) is solved (for different α>0) numerically for in mode (15) with kernel = = and right side .
5. is defined — such , that
. (18)
As an example of the application of the model experiment method, let us consider the problem of angular resolution of sources.
Example 1. Practical problem has the form
,
where — the desired field (acoustic, radio astronomical) at the antenna input; — measured field at antenna output; — antenna directivity characteristics, and

where ,
exact solution

where is given in degrees дается в градусах. The right side is assumed to be given with a relative error and presented in the form
.
Here — random number, distributed according to the normal law with a unit mean square value; mean of the exact right hand side , which in this case can be determined by calculating the integral on the left side of equation (1).
We formulate the model problem in the following form:
kernel

where

,
exact solution

In both problems , , .
Right side of the model problem will present in the form
,
where , and function is determined by calculating the integral in (1) with the kernel and exact solution .
Note that the values of and in the examples and are taken to be somewhat different, since in practice , and are known with errors.
Then the problem was solved by Tikhonov’s method for a number of values (by the quadrature method) for ; ; and . Next, the dependence on is calculated for the relative error in solving the equation :

On the segment , the relative error reaches a minimum at . This value should be accepted for the original problem . In Fig. 1 and fig. 2 displays the exact solutions of , and solutions of , for and ; ; and .
Consideration of Fig. 1 and fig. 2 shows the following:
— spikes in the solution (sources) are resolved confidently, despite the fact that they do not appear at all in the functions and ; this is due to the fact that the distance between the bursts is small - approximately equal to the width of the DCh (directional characteristics) at a level of 0.7;
— the coordinates of the bursts are also restored with a small error, namely, up to a step;

Fig. 1. Exact (1) and approximate (2) solutions of the model problem Q at α=10^—5

Fig. 2. Exact solution (1) and solution at α=10^—5
of the required function P (2)
— the functions and are directly restored much worse (to increase accuracy, it is necessary to reduce the errors and , as well as the step ).
Let us compare the method of model experiments and the generalized residual principle based on the problem considered. The results of the numerical experiment are shown in Fig. 3 and in table 1
Table 1.