A three-dimensional multifrequency inverse problem of acoustic sounding of a stationary inhomogeneous medium is considered. This nonlinear inverse problem is reduced to solving an auxiliary three-dimensional linear Fredholm integral equation of the first kind. In the analysis of the uniqueness of the solution to the inverse problem, the connection between the integral equation and determining the source in the Helmholtz equation is indicated. The last problem is ambiguously solvable in the general case. Examples of such ambiguity are given. Questions about detailed data (frequencies, sources) ensuring or not the uniqueness of solutions are considered. A speed-efficient algorithm for solving the inverse problem based on Fourier transforms is proposed. This algorithm makes it possible to calculate uniquely an approximate solution by a stable method under data perturbations. The results of numerical experiments on solving a three-dimensional model inverse problem on fairly detailed grids are presented.
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International Mathematical Schools. Vol. 4. Problems for Partial Differential Equations and Topics in Analysis
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Bakushinsky, A.B., Leonov, A.S. Multifrequency Inverse Problem of Scalar Acoustics: Remarks on Nonuniqueness and Solution Algorithm. J Math Sci 274, 460–474 (2023). https://doi.org/10.1007/s10958-023-06613-9
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DOI: https://doi.org/10.1007/s10958-023-06613-9