Summary: The problem of reconstructing the solenoidal part of a vector field in the circle is considered from its ray transform. Two variants of numerical solution of this problem are developed.
In the first of them, a polynomial approximation of the vector field is obtained by means of the least squares method and may contain a potential part. Thus a further step of solving the problem is to separate the potential vector field from the above approximation by finding a solution to a homogeneous boundary value problem for the Poisson equation. Investigation of the structure of finite-dimensional subspaces of solenoidal and potential polynomial vector fields allows us to state the problem of determining the coefficients of the polynomial approximation of the potential part as the problem of step-by-step solution of a set of systems of linear equations of increasing dimensions.
The second variant 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 algorithms constructed is verified by numerical simulation. Comparison of test results obtained by the algorithms shows that the accuracy of both algorithms is good enough.