The paper is concerned with developing the methods for solving inverse problems of ultrasound tomography in vector elastic models. In the general case, the objective is to reconstruct two Lamé parameters and density of the object as functions of spatial coordinates by measuring scattered ultrasound waves. To construct iterative solution methods, a representation for the gradient of the residual functional between the computed and measured wave fields at the detectors with respect to the sought-for parameters was used. Precise reconstruction of three independent parameters without additional information seems unlikely even in a full-view tomographic scheme. The main result of the work demonstrates the possibility of reconstructing spatially dependent velocities of longitudinal and transverse waves with high accuracy. These physical parameters are important in interpreting tomographic imaging data using ultrasound sources. A GPU platform of the supercomputer was used to reconstruct tomographic images. Comparing to scalar wave models, vector models require approximately four times more computing power.