Geodesic
Iterative methods for solving inverse problems of ultrasonic tomography
Numerical methods and programming, Tome 16 (2015) no. 4, pp. 464-475.

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This paper is dedicated to rigorous mathematical substantiation of iterative methods for solving inverse problems of ultrasonic tomography. These inverse problems are considered in the framework of a scalar model for the wave equation. This model takes into account such wave effects as diffraction, refraction, etc. The inverse problem is considered as a coefficient inverse problem. A rigorous mathematical representation is given for the Frechet derivative of the residual functional with respect to the wave velocity $с(r)$ characterizing a nonuniform structure of the object under study. The representation for the Frechet derivative is obtained both for the two-dimensional problems and for the three-dimensional case. It is suggested that the inverse problem can be solved using this representation of the Frechet derivative together with the gradient methods of minimization for the residual functional. The proposed iterative procedure is highly parallelizable and implementable on supercomputers.
Keywords: coefficient inverse problems, wave equation, ultrasonic tomography, Frechet derivative, iterative methods. 
@article{VMP_2015_16_4_a1,
     author = {A. V. Goncharsky and S. Y. Romanov},
     title = {Iterative methods for solving inverse problems of ultrasonic tomography},
     journal = {Numerical methods and programming},
     pages = {464--475},
     publisher = {mathdoc},
     volume = {16},
     number = {4},
     year = {2015},
     language = {ru},
     url = {http://geodesic.mathdoc.fr/item/VMP_2015_16_4_a1/}
}
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A. V. Goncharsky; S. Y. Romanov. Iterative methods for solving inverse problems of ultrasonic tomography. Numerical methods and programming, Tome 16 (2015) no. 4, pp. 464-475. http://geodesic.mathdoc.fr/item/VMP_2015_16_4_a1/