A local version of a numerical method for solving an inverse problem (Q1358026)

We develop the numerical method for solving inverse problems for hyperbolic equations proposed by the author [\((*)\) ibid. 37, No. 3, 633-655 (1996; Zbl 0879.65098)]. Unlike the article \((*)\), wherein the case was considered in which the information used in the inverse problem is given on the whole boundary of a physical domain, here similar information is given only on part of the boundary. To simplify exposition and make comparison convenient, in the present article we consider the same initial inverse problem as in \((*)\). This enables us to use the local solvability theorems and estimates for a solution to the direct problem which were established in \((*)\). We suppose that the data of the inverse problem are given with some error. We demonstrate that the numerical algorithm based on solving some auxiliary system of algebraic equations is regularizing, provided that the parameters of the system agree in an appropriate manner with the data error. As in \((*)\), the main result is obtained in the case of analytic dependence of the sought functions on part of variables and is based on L. Nirenberg's method and some of its modifications adapted for studying inverse problems.