Numerical differentiation for two-dimensional functions by a Fourier extension method. Issue 1 (2nd January 2020)

• Record Type: Journal Article
• Authors: Meng, Zehong; Zhao, Zhenyu; Mei, Duan; Zhou, Yongxiong
• Is Part Of: Inverse problems in science and engineering. Volume 28:Issue 1(2020)
• Page Start: 126
• Page End: 143
• Publication Date: 2020-01-02
• Subjects: Ill-posed problem -- regularization -- numerical differentiation -- Fourier extension; 47A52; Engineering mathematics -- Periodicals; Inverse problems (Differential equations) -- Periodicals; 620.001515357
• DOI: 10.1080/17415977.2019.1661410
• Languages: English
• ISSNs: 1741-5977

Abstract

ABSTRACT: Based on the idea of Fourier extension, we develop a new method for numerical differentiation of two-dimensional functions on an arbitrary domain. The function will be extended to a periodic function on a larger domain. The Tikhonov regularization method in Hilbert scales is presented to deal with the ill-posedness of the problem. The Sobolev norm defined on the larger domain will be used as a stabilizer. An error analysis is provided with a regularization parameter chosen by a discrepancy principle. Numerical experiments are also presented to illustrate the effectiveness of the proposed method.