A stabilized algorithm for multi-dimensional numerical differentiation

We develop a multi-dimensional numerical differentiation method in this paper. To obtain stable numerical derivatives, the Tikhonov regularization method in Hilbert scales is proposed to deal with illposedness of the problem. The penalty term in Tikhonov functional is more general so that we can choose the regularization parameter without the smoothness parameter and the a priori bound of exact solution. Numerical examples are also presented to check the effectiveness of the method.