This is a completely revised and updated translation of the Russian original work Numerical Metho(Js for Solving lll-Posed Problems, Nauka, Moscow © 1990 Translation by R.A.M. Hoksbergen All Rights Reserved

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This study focuses on the development of efficient methods for solving inverse problems of ultrasound tomography as coefficient inverse problems for the wave equation. The inverse problem consists in finding the unknown wave propagation velocity as a function of coordinates in three-dimensional space. Efficient iterative methods are proposed for solving the inverse problem based on a direct computation of the residual functional. One of the most promising directions of ultrasound tomography is the development of ultrasound tomographs for medical research, and primarily for the differential diagnosis of breast cancer. From a medical viewpoint, diagnostic facilities for the differential cancer diagnosis should have a resolution of 3 mm or better. Because of this requirement, inverse problems of ultrasound tomography have to be solved on dense grids with sizes of up to 1000 × 1000 on cross sections of three-dimensional objects studied. Supercomputers are needed to address such inverse problems in terms of the wave model described by second-order hyperbolic equations. The algorithms developed in this study are easily scalable on supercomputers running up to several tens of thousands of processes. The problem of choosing the initial approximation for iterative algorithms when solving the inverse problem has been studied.