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

Authors:
Gen Nakamura;Shengzhang Wang;Yanbo Wang
Affiliations:
Department of Mathematics, Graduate School of Science, Hokkaido University, Sapporo 060-0810, Japan;Department of Mechanics and Engineering Science, Fudan University, Shanghai 200433, PR China;School of Mathematical Sciences, Fudan University, Shanghai 200433, PR China
Venue:
Journal of Computational and Applied Mathematics
Year:
2008

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Abstract

A regularized optimization problem for computing numerical differentiation for the second order derivatives of functions with two variables from noisy values at scattered points is discussed in this article. We prove the existence and uniqueness of the solution to this problem, provide a constructive scheme for the solution which is based on bi-harmonic Green's function and give a convergence estimate of the regularized solution to the exact solution for the problem under a simple choice of regularization parameter. The efficiency of the constructive scheme is shown by some numerical examples.