Gel'fand-Levitan, Marchenko and Krein equations. Theory, numeric and applications
We consider the method of regularization of 2D inverse acoustic problem based on projection method and the approach of I.M. Gelfand, B.M. Levitan and M.G. Krein (GLK method). In 1951 M. Gel'fand and B.M. Levitan presented a method of reconstructing the Sturm-Liouville operator from a spectral function. It is important to mention results by M.G. Krein (1951, 1954) concerning the inverse problems for the wave equation. The inverse problem for 2D acoustic equation is considered. We propose a method of reconstruction of the density approximating 2D inverse acoustic problem by a finite system of one dimensional inverse acoustic problems. The 2D analogy of the Gel'fand-Levitan and M.G. Krein method is established. The inverse acoustic problem is formulated and the short outline of the history and development in this field are given. We consider the 2D analogy of the Gel'fand-Levitan and M.G. Krein and V.A. Marchenko equations. The N-approximation of the M.G. Krein equation is obtained for inverse acoustic problem. The numerical results are presented.
This work is partially supported by the Ministry of Education and Science of the Russian Federation and under SB RAS interdisciplinary project 14 "Inverse Problems and Applications: Theory, Algorithms, Software".