An iterated quasi-interpolation approach for derivative approximation

• Published in: Numerical algorithms Vol. 85; no. 1; pp. 255 - 276
• Main Authors: Sun, Zhengjie, Wu, Zongmin, Gao, Wenwu
• Format: Journal Article
• Language: English
• Published: New York Springer US 01.09.2020; Springer Nature B.V
• Subjects: Algebra; Algorithms; Approximation; Computer Science; Derivatives; Discrete functions; Fourier transforms; Interpolation; Mathematical analysis; Numeric Computing; Numerical Analysis; Operators (mathematics); Original Paper; Periodic functions; Theory of Computation; Quasi-interpolation; 41A63; 65D10; 41A65; 65D05; Fourier transform; 65D15; Strang–Fix condition; 41A05; Numerical differentiation
• ISSN: 1017-1398; 1572-9265
• DOI: 10.1007/s11075-019-00812-9

Given discrete function values sampled at uniform centers, the iterated quasi-interpolation approach for approximating the m th derivative consists of two steps. The first step adopts m successive applications of the operator DQ (the quasi-interpolation operator Q first, and then the differentiation operator D ) to get approximated values of the m th derivative at uniform centers. Then, by one further application of the quasi-interpolation operator Q to corresponding approximated derivative values gives the final approximation of the m th derivative. The most salient feature of the approach is that it approximates all derivatives with the same convergence rate. In addition, it is valid for a general multivariate function, compared with the existing iterated interpolation approaches that are only valid for periodic functions, so far. Numerical examples of approximating high-order derivatives using both the iterated and direct approach based on B-spline quasi-interpolation and multiquadric quasi-interpolation are presented at the end of the paper, which demonstrate that the iterated quasi-interpolation approach provides higher approximation orders than the corresponding direct approach.