Finding numerical derivatives for unstructured and noisy data by multiscale kernels

The recently developed multiscale kernel of R. Opfer [Adv. Comput. Math., 25 (2006), pp. 357-380] is applied to approximate numerical derivatives. The proposed method is truly mesh-free and can handle unstructured data with noise in any dimension. The method of Tikhonov and the method of L-curve are employed for regularization; no information about the noise level is required. An error analysis is provided in a general setting for all dimensions. Numerical comparisons are given in two dimensions which show competitive results with recently published thin plate spline methods.