On stable numerical differentiation

Based on a regularized Volterra equation, two different approaches for numeri- cal differentiation are considered. The first approach consists of solving a regularized Volterra equation while the second approach is based on solving a disretized version of the regularized Volterra equation. Numerical experiments show that these methods are efficient and compete fa- vorably with the variational regularization method for stable calculating the derivatives of noisy functions.

[1]  Alexander G. Ramm,et al.  INEQUALITIES FOR THE DERIVATIVES , 1999 .

[2]  H. E. Salzer,et al.  Some problems in optimally stable Lagrangian differentiation , 1974 .

[3]  V. K. Ivanov,et al.  On numerical differentiation , 1966 .

[4]  A. N. Tikhonov,et al.  Solutions of ill-posed problems , 1977 .

[5]  A. Krall.  Applied Analysis , 1986 .

[6]  R. Qu.  A new approach to numerical differentiation and integration , 1996 .

[7]  J. Cullum.  Numerical Differentiation and Regularization , 1971 .

[8]  Alexander G. Ramm,et al.  Estimates of derivatives of random functions I , 1984 .

[9]  P. Bloomfield,et al.  Numerical differentiation procedures for non-exact data , 1974 .

[10]  Ian Knowles,et al.  A variational method for numerical differentiation , 1995 .

[11]  A. Ramm,et al.  The RADON TRANSFORM and LOCAL TOMOGRAPHY , 1996 .

[12]  V. V. Vasin,et al.  Ill-posed problems with a priori information , 1995 .

[13]  Jean-François Richard,et al.  Methods of Numerical Integration , 2000 .

[14]  Alexander G. Ramm,et al.  Continuous Methods for Solving Nonlinear Ill-Posed Problems , 2000 .

[15]  T. J. Rivlin,et al.  Optimally Stable Lagrangian Numerical Differentiation , 1975 .

[16]  L PhillipsDavid.  A Technique for the Numerical Solution of Certain Integral Equations of the First Kind , 1962 .

[17]  Alexander G. Ramm,et al.  Random fields estimation theory , 1990 .

[18]  Alexander G. Ramm,et al.  On Simultaneous Approximation of a Function and its Derivative by Interpolation Polynomials , 1977 .

[19]  G RammAlexander,et al.  On stable numerical differentiation , 2001 .

[20]  Alexander G. Ramm,et al.  STABLE NUMERICAL DIFFERENTIATION: WHEN IS IT POSSIBLE? , 2003 .

[21]  R. Landau,et al.  Numerical Differentiation , 2019, Numerical Methods.

[22]  Alexander G. Ramm,et al.  Stable solutions of some ill-posed problems , 1981 .

[23]  C. Lanczos.  Applied Analysis , 1961 .