Variational interpolation of functionals in transport theory inverse problems
V.V. Uchaikin1, V.A. Litvinov2
1Ulyanovsk State University, Ulyanovsk, Russia
2Barnaul law Institute of MIA of Russia, Barnaul, Russia
Keywords: теория возмущений, сопряжённые функции, операторы, стационарный функционал, perturbation theory, adjoint functions, operators, stationary functional
Abstract
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 A0 that best matches the measurement data. However, as the accuracy requirements increase, the domain A0 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 a1, a2, …, 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.
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