Variational interpolation of functionals in transport theory inverse problems

V. V. Uchaikin ; V. A. Litvinov

Sibirskij žurnal vyčislitelʹnoj matematiki, Tome 22 (2019) no. 3, pp. 363-380

Résumé

It is known that the dual representation of problems (through solutions of the main and the conjugate in the Lagrange sense equations) allows one to formulate the perturbation theory serving as basement for the successive approximation method in the inverse problems theory. If, according to preliminary predictions, the solution of an inverse problem (for example, the structure of the medium of interest) belongs to a certain set $A$, then selecting a suitable (trial, reference) element $a_0$ as an unperturbed one and applying the perturbation theory, one can approximately describe the behavior of the solution of the direct problem in this domain and find a subset $A_0$ that best matches the measurement data. However, as the accuracy requirements increase, the domain $A_0$ of the first approximation is rapidly narrowing, expanding it by adding higher terms of the expansion complicates the decision procedure. For this reason, a number of works have been devoted to the search for unperturbed approaches. Among them is the method of variational interpolation (VI-method), in the frame of which not one, but several problems $a_1, a_2,\dots ,a_n$ an are used in order to compose from their solutions the desired one. The functional of interest is represented in the stationary form, and the coefficients of the expansion are determined from the condition of stationarity of the bilinear form. This paper demonstrates the application of VI-method to solving inverse problems in the frame of simple model situations associated with cosmic rays astrophysics.