Geophysical Journal | 1999 volume 21 Ή3

On stable methods of solving problems of the definition of star-shaped domains similar to the preset ones

© Chernaya O.A.

Functionals of discrepancy and those of the type of discrepancy of a linear integral equation of the first kind with a non-negative operator in the Gilbert space of input data are studied. It is shown that both functionals are convex, twice continuously differentiable. The sets of points of a minimum of fuctionals over the space are sets of the second category with respect to the set of all accurate (normal) solutions of an integral equation (a set of uniqueness). Therefore not all minimizing functionals of a succession converge to the set of uniqueness, which makes the problem of the search of the minimum of the functionals over the Gilbert space of data incorrect. Also insufficient are also the widely used additional requirements of the minimality of the norm of solution at the minimization of discrepancy functionals. To orient the search of the normal solution of the equation by using the minimization of the functionals of discrepancy and the type of discrepancy special stabilizing functionals defined over sets of uniqueness are propozed. They are constructed by using definite differential operators whose proper functions are closely related with the proper functions of the operators of direct correspondence in the problems of defining star-shaped domains and are non-negative, highly convex, twice continuously differentiable functionals without local minima different from the global minimum.

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