Automatic numerical differentiation by discrete mollification

Numerical Differentiation of 2D Functions by a Mollification Method Based on Legendre Expansion

In this paper, we consider numerical differentiation of bivariate functions when a set of noisy data is given. A mollification method based on spanned by Legendre polynomials is proposed and the mollification parameter is chosen by a discrepancy principle. The theoretical analyses show that the smoother the genuine solution, the higher the convergence rates of the numerical solution. To get a p...

Automatic differentiation of numerical integration algorithms

Automatic differentiation (AD) is a technique for automatically augmenting computer programs with statements for the computation of derivatives. This article discusses the application of automatic differentiation to numerical integration algorithms for ordinary differential equations (ODEs), in particular, the ramifications of the fact that AD is applied not only to the solution of such an algo...

Automatic differentiation and numerical software design

Automatic diierentiation (AD) tools can generate accurate and eecient derivative code for computer programs of arbitrary length. In some cases, however, the developer of the code to be diierentiated may be required to provide additional information to an AD tool to ensure the desired solution. We illustrate these issues with nondiierentiable language intrinsics such as max() in the context of c...

Automatic Differentiation by Program Transformation

Mollification Based onWavelets

The mollification obtained by truncating the expansion in wavelets is studied, where the wavelets are so chosen that noise is reduced and the Gibbs phenomenon does not occur. The estimations of the error of approximation of the mollification are given for the case when the fractional derivative of a function is calculated. Noting that the estimations are applicable even when the orthogonality o...