A numerical method for an inverse biharmonic problem

An inverse boundary value problem associated to the biharmonic equation in two dimensions, which requires the determination of the unspecified boundary values from interior domain measurements, is discretised numerically using the boundary element method (BEM) and the resulting ill-conditioned system of linear equations is solved using a Gaussian elimination and the minimal energy methods. The latter procedure is based on minimizing the energy of the Laplace equation subject to linear constraints generated from the discretisation of the biharmonic problem. Whilst the direct method of solution which is based on the Gaussian elimination is shown, as expected, to be unstable, the minimal energy method provides a stable numerical solution with respect to the errors in the interior domain measurements. The choice of the regularization parameter for this method is based on an L-curve type analysis.