Global and explicit approximation of piecewise smooth 2D functions from cell-average data

by   Sergio Amat, et al.

Given cell-average data values of a piecewise smooth bivariate function f within a domain Ω, we look for a piecewise adaptive approximation to f. We are interested in an explicit and global (smooth) approach. Bivariate approximation techniques, as trigonometric or splines approximations, achieve reduced approximation orders near the boundary of the domain and near curves of jump singularities of the function or its derivatives. Whereas the boundary of Ω is assumed to be known, the subdivision of Ω to subdomains on which f is smooth is unknown. The first challenge of the proposed approximation algorithm would be to find a good approximation to the curves separating the smooth subdomains of f. In the second stage, we simultaneously look for approximations to the different smooth segments of f, where on each segment we approximate the function by a linear combination of basis functions {p_i}_i=1^M, considering the corresponding cell-averages. A discrete Laplacian operator applied to the given cell-average data intensifies the structure of the singularity of the data across the curves separating the smooth subdomains of f. We refer to these derived values as the signature of the data, and we use it for both approximating the singularity curves separating the different smooth regions of f. The main contributions here are improved convergence rates to both the approximation of the singularity curves and the approximation of f, an explicit and global formula, and, in particular, the derivation of a piecewise smooth high order approximation to the function.


page 1

page 2

page 3

page 4


Corrected approximation strategy for piecewise smooth bivariate functions

Given values of a piecewise smooth function f on a square grid within a ...

Reconstruction of piecewise-smooth multivariate functions from Fourier data

In some applications, one is interested in reconstructing a function f f...

Explicit multivariate approximations from cell-average data

Given gridded cell-average data of a smooth multivariate function, we pr...

Corrected subdivision approximation of piecewise smooth functions

Subdivision schemes are useful mathematical tools for the generation of ...

Numerical Differentiation using local Chebyshev-Approximation

In applied mathematics, especially in optimization, functions are often ...

Approximation of curves with piecewise constant or piecewise linear functions

In this paper we compute the Hausdorff distance between sets of continuo...

Adaptive Padé-Chebyshev Type Approximation of Piecewise Smooth Functions

A piecewise Padé-Chebyshev type (PiPCT) approximation method is proposed...

Please sign up or login with your details

Forgot password? Click here to reset