Construction of Nonlinear Approximation Schemes for Piecewise Smooth Data

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.