Numerical solution to the vector tomography problem using polynomial basis

The problem of a reconstruction of a solenoidal part of a vector field in the circle is considered if its ray transform is known. Two variants of numerical solution of the problem are developed. In the first of them, a polynomial approximation of the vector field that was obtained by means of the least squares method contains an potential part. Thus a further step of solving the problem is to separate from the approximation a potential vector field by finding a solution for a homogeneous boundary value problem for the Poisson equation. Investigation of the structure of finite-dimensional subspaces of solenoidal and potential vector fields of the polynomial type allows to state a problem of determining of coefficients of the polynomial approximation of the potential part as the problem of step-by-step solving of a set of systems of linear equations of increasing dimensions. The second way consists in constructing subspaces of the basis polynomial solenoidal fields. In this case, the least squares method immediately gives a polynomial approximation of a solenoidal part of the vector field. Efficiency of the constructed algorithms is verified by the numerical simulation. The results of comparative test of the algorithms show that the accuracy of both algorithms is good and similar to one another.