Summary: The boundary element method (BEM) is applied to solve numerically some inverse boundary value problem associated to the biharmonic equation which involve over- and under-specified boundary problems of the solution domain. The resulting ill-conditioned system of linear equations is solved using the regularization and the minimal energy methods, followed by a further application of the singular value decomposition method. The regularization method incorporates a smoothing effect into the least squares functional, whilst the minimal energy method is based on minimizing the energy functional for the Laplace equation subject to the linear constraints generated by the BEM discretization of the biharmonic equation. The numerical results are compared with known analytical solutions and the stability of the numerical solution is investigated by introducing noise into the input data.