Numerical Differentiation of Noisy, Nonsmooth Data

In many scientific applications, it is necessary to compute the derivative of functions specified by data. Conventional finite-difference approximations will greatly amplify any noise present in the data. Denoising the data before or after differentiating does not generally give satisfactory results see an example in Section 4 . A method which does give good results is to regularize the differentiation process itself. This guarantees that the computed derivative will have some degree of regularity, to an extent that is often under control by adjusting parameters. A common framework for this is Tikhonov regularization 1 of the corresponding inverse problem. That is, the derivative of a function f , say on 0, L , is the minimizer of the functional