Numerical differentiation for two-dimensional scattered data

Abstract

In this paper, we propose a regularization method for numerical differentiation of two-dimensional mildly scattered input data. A regularized solution is constructed based on the Green's function. The existence and uniqueness of the regularized solution are proved and the convergence estimates are provided under a simple choice of regularization parameter. Numerical results show that our method is quite effective. One of the advantages for our proposed method is that the basis functions are independent of input data.  2005 Elsevier Inc. All rights reserved.

A truncated Legendre spectral method for solving numerical differentiation

“…The main difficulty is that it is an ill-posed problem, which means small errors in the measurement of a function may lead to large errors in its computed derivatives [6,12]. A number of techniques have been developed for numerical differentiation [2,[8][9][10]12,[15][16][17][18][19][21][22][23]. However, most of the results given in these methods are corresponding to the functions with an inferior smoothness.…”

“…However, most of the results given in these methods are corresponding to the functions with an inferior smoothness. In [21,23], the convergence results corresponding to smooth scales of accurate functions in the Sobolev space have been obtained, but the methods are not self-adaptive. In order to obtain the corresponding convergence results, the methods have to be changed to fit the different smooth scales of the accurate functions.…”

A New Mollification Method for Numerical Differentiation of 2D Periodic Functions

“…A number of techniques have been well developed for numerical differentiation in one-dimensional case [6], [7], [9], [10], [14]. Besides, as far as we know, the literatures on noisy data in two dimensions are relatively poor [2], [12], [13]. In [14], we present a mollification method to deal with numerical differentiation in one-dimensional case.…”

Numerical differentiation for the second order derivatives of functions of two variables

“…For higher order derivatives in the one-dimensional case and for first order derivatives in two-dimensional case, numerical differentiation method along the line of this method were given in [13,15], respectively. For the twodimensional case, the new ingredient was that the variational problem for the regularized minimization problem is solved by using Green's function for the Laplacian with Dirichlet boundary condition and a scheme for computing the first order derivative was given in [15].…”

“…For the twodimensional case, the new ingredient was that the variational problem for the regularized minimization problem is solved by using Green's function for the Laplacian with Dirichlet boundary condition and a scheme for computing the first order derivative was given in [15]. The numerical example showed that this method was efficient.…”