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.
We develop efficient iterative methods for solving inverse problems of wave tomography in models incorporating both diffraction effects and attenuation. In the inverse problem the aim is to reconstruct the velocity structure and the function that characterizes the distribution of attenuation properties in the object studied. We prove mathematically and rigorously the differentiability of the residual functional in normed spaces, and derive the corresponding formula for the Fréchet derivative. The computation of the Fréchet derivative includes solving both the direct problem with the Neumann boundary condition and the reversed-time conjugate problem. We develop efficient methods for numerical computations where the approximate solution is found using the detector measurements of the wave field and its normal derivative. The wave field derivative values at detector locations are found by solving the exterior boundary value problem with the Dirichlet boundary conditions. We illustrate the efficiency of this approach by applying it to model problems. The algorithms developed are highly parallelizable and designed to be run on supercomputers. Among the most promising medical applications of our results is the development of ultrasonic tomographs for differential diagnosis of breast cancer.
Two approaches to solving coefficient inverse problems for wave equations are compared. One approach is based on integral representations obtained with the help of the Green's function for the wave equation. In the other approach, the gradient of the error functional is directly computed in terms of the solution of the adjoint problem for a partial differential equation. The methods developed are intended for finding inhomogeneities in homogeneous media and can be applied in medicine diag nostics, acoustic and seismic near surface exploration, engineering seismics, etc.
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