First and second order numerical differentiation with Tikhonov regularization

Shuai Lu; Yanbo Wang

doi:10.1007/s11464-006-0014-x

Front. Math. China ›› 2006, Vol. 1 ›› Issue (3) : 354 -367. DOI: 10.1007/s11464-006-0014-x

Research Article

First and second order numerical differentiation with Tikhonov regularization

Shuai Lu ¹^,^a
, Yanbo Wang ²

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Abstract

This work deals with the numerical differentiation for an unknown smooth function whose data on a given set are available. The numerical differentiation is an ill-posed problem. In this work, the first and second derivatives of the smooth function are approximated by using the Tikhonov regularization method. It is proved that the approximate function can be chosen as a minimizer to a cost functional. The existence and uniqueness theory of the minimizer is established. Errors in the derivatives between the smooth unknown function and the approximate function are obtained, which depend on the mesh size of the grid and the noise level in the data. The numerical results are provided to support the theoretical analysis of this work.

Keywords

ill-posed problem / numerical differentiation / error estimate / Tikhonov regularization / 65D25 / 65F22

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Shuai Lu, Yanbo Wang. First and second order numerical differentiation with Tikhonov regularization. Front. Math. China, 2006, 1(3): 354-367 DOI:10.1007/s11464-006-0014-x

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