Solution of Inverse Spectral Problems for Discrete Semi-Bounded Operators Given on Geometric Graphs

Using the numerical method of regularized traces and the Galerkin method, linear formulas were previously obtained for calculating the approximate eigenvalues of discrete semi-bounded operators. These formulas can be used to find approximate eigenvalues of discrete operators with any ordinal number without using the previous eigenvalues. It removes many of the computational difficulties arising in other methods. The comparison of the results of computational experiments showed that the eigenvalues found by both linear formulas and the Galerkin method are in a good agreement. On the basis of linear formulas for calculating the eigenvalues of discrete semi-bounded operators, we describe a numerical method for solving inverse spectral problems given on sequential geometric graphs with a finite number of links. The method allows to recover the values of unknown functions included in the operators at the discretization nodes using the eigenvalues of the operators and the spectral characteristics of the corresponding self-adjoint operators. We construct an algorithm for solving inverse spectral problems given on sequential geometric graphs with a finite number of links, and test the algorithm on a sequential two-edge graph. The results of numerous experiments shown good accuracy and a high computational efficiency of the developed method.

Keywords

eigenvalues and eigenfunctions; discrete and self-adjoint operators; inverse spectral problem; Galerkin method; ill-posed problems; Fredholm integral equation of the first kind; geometric graph.

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