Error Estimate of Method Based on Generalized Residual Principle for Problem of Recovering Spectral Density of Crystals

The problem of determining the phonon spectrum from its heat capacity dependent on the temperature is reduced to a new integral equation with respect to the derivative of the desired solution. The resulting integral equation is subjected to finite-dimensional approximation of a special form, which allows reducing the problem to a special system of linear algebraic equations, using the Tikhonov variational regularization method with a choice of the regularization parameter by the generalized residual principle. An a priori estimate of the accuracy of the obtained stable finite-dimensional approximate solution is also carried out taking into account the accuracy of the finite-dimensional approximation of the problem. The use of this approach is given by the example of the problem of determining the phonon spectrum from its temperature-dependent heat capacity, which is known to be reduced to an integral equation of the first kind. In this paper, the generalized residual principle was used to select the regularization parameter and an error of the approximate solution was obtained, taking into account the discretization of the problem. Previously, discretization was disregarded when deriving an estimate. Studying the possibility of detecting the fine structure, beginning with the number, position and peak values of the function n(s), and developing the efficient methods for solving ill-posed problems, which are optimal in accuracy and require a minimum of a priori information, have an important theoretical and practical significance, beyond the considered inverse problem.