Fast algorithms for boundary integral equations connected with Robin boundary value problem for the Laplace equation in domains with ellipse or close to ellipse boundaries are developed. It is shown that the coefficient matrices of discretisation systems have a special structure. This fact is used to develop a fast algorithm for matrix vector multiplication and to implement it in the numerical methods used. Such an approach is especially helpful in numerical methods for inverse problems, since many methods of their solution repeatedly use forward solvers. The efficiency of the methods is illustrated by numerical examples.