Two-dimensional analogs of the equations of Gelfand, Levitan, Krein, and Marchenko

An algorithm for regularization of two-dimensional inverse coefficient problems is considered on the basis of a projection method and an approach proposed by I.M. Gelfand, B.M. Levitan, M.G. Krein, and V.A. Marchenko. An algorithm to reconstruct the potential, density, and velocity in two-dimensional inverse coefficient problems is described. Two dimensional analogs for the equations of Gelfand, Levitan, and Krein are obtained. A two dimensional analog of the Marchenko equation is considered for the Kadomtsev–Petviashvili equation. This approach can be generalized for corresponding multi-dimensional inverse problems. The results of numerical calculations are presented.