The pre/post equilibrated conditioning methods to solve Cauchy problems

In the present paper, the inverse Cauchy problems of Laplace equation and biharmonic equation are transformed, by using the method of fundamental solutions (MFS) and the Trefftz method (TM), to the systems of linear equations for determining the expansion coefficients. Then, we propose three different conditioners together with the conjugate gradient method (CGM) to solve the resultant ill-posed linear systems. They are the post-conditioning CGM and the pre-conditioning CGM based on the idea of equilibrated norm for the conditioned matrices, as well as a minimum-distance conditioner. These algorithms are convergent fast and accurate by solving the inverse Cauchy problems under random noise.