TY - JOUR
T1 - Construction of Nonlinear Approximation Schemes for Piecewise Smooth Data
AU - Yang, Hyoseon
AU - Yoon, Jungho
N1 - Funding Information:
J. Yoon was supported by the National Research Foundation (NRF) of Korea under the grants NRF-2015R1A5A1009350 and NRF-2020R1A2C1A01005894.
Publisher Copyright:
© 2022, The Author(s), under exclusive licence to Springer Science+Business Media, LLC, part of Springer Nature.
PY - 2022
Y1 - 2022
N2 - The objective of this paper is to develop a family of nonlinear approximation schemes for piecewise smooth data on R. The quasi-interpolation method is a very efficient tool for reconstructing functions from a discrete set of their values on R. It has advantages in simplicity and fast computation. However, when approximating near singularities or sharp gradients of the underlying functions, it often suffers from spurious oscillations or blurring edges. Motivated by this observation, we present a nonlinear modification of the quasi-interpolation to prevent such undesirable artifacts near singular points, while achieving high-order accuracy in smooth regions. To this end, we first introduce two important tools, a smoothness indicator and a singularity detector, and then construct new nonlinear kernels. A detailed error analysis of the proposed scheme is provided. We show that, in smooth regions, the proposed scheme achieves the same approximation order as its linear counterpart, while maintaining essentially non-oscillatory behavior near the singularities. Finally, some numerical results are presented to demonstrate the ability of the proposed nonlinear scheme.
AB - The objective of this paper is to develop a family of nonlinear approximation schemes for piecewise smooth data on R. The quasi-interpolation method is a very efficient tool for reconstructing functions from a discrete set of their values on R. It has advantages in simplicity and fast computation. However, when approximating near singularities or sharp gradients of the underlying functions, it often suffers from spurious oscillations or blurring edges. Motivated by this observation, we present a nonlinear modification of the quasi-interpolation to prevent such undesirable artifacts near singular points, while achieving high-order accuracy in smooth regions. To this end, we first introduce two important tools, a smoothness indicator and a singularity detector, and then construct new nonlinear kernels. A detailed error analysis of the proposed scheme is provided. We show that, in smooth regions, the proposed scheme achieves the same approximation order as its linear counterpart, while maintaining essentially non-oscillatory behavior near the singularities. Finally, some numerical results are presented to demonstrate the ability of the proposed nonlinear scheme.
KW - Approximation order
KW - B-spline
KW - Nonlinear approximation
KW - Quasi-interpolation
KW - Singularity detector
KW - Smoothness indicator
UR - http://www.scopus.com/inward/record.url?scp=85125568186&partnerID=8YFLogxK
U2 - 10.1007/s00365-022-09567-4
DO - 10.1007/s00365-022-09567-4
M3 - Article
AN - SCOPUS:85125568186
SN - 0176-4276
JO - Constructive Approximation
JF - Constructive Approximation
ER -