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 practical approach, we also derive corresponding results for Legendre-Gauss-Lobatto interpolation. Numerical examples are also given to show the efficiency of the method.
Keywords: Ill-posed problem, Numerical differentiation, Legendre spectral method, Discrepancy principle.
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ABOUT THE AUTHORS
Ou Xie
College of Science, Guangdong Ocean University
Zhenyu Zhao
College of Science, Guangdong Ocean University
Ou Xie
College of Science, Guangdong Ocean University
Zhenyu Zhao
College of Science, Guangdong Ocean University