TY - JOUR

T1 - Reconstructing piecewise smooth bivariate functions from scattered data

AU - Amir, Anat

AU - Levin, David

N1 - Publisher Copyright:
© 2021. Universidad de Jaen

PY - 2021

Y1 - 2021

N2 - Given scattered data values of a piecewise smooth function f within a domain,we look for a piecewise adaptive approximation to f. Approximation techniques for scattered data approximation, as Radial Basis Function (RBF) or Moving Least-Squares (MLS), achieve reduced approximation orders near the boundary of the domain and near the unknown curves of jump singularities of the function or its derivatives. The idea used here is that the approximation errors near the boundaries, and near a singularity curve, fully characterize the behavior of the function at these locations. We refer to these approximation error values as the signature of f. In this paper, we aim at using these values in order to define the approximation. Assuming smoothness of the singularity curve, we suggest using a signed-distance approach to construct an approximation of the singularity curve. Now we find approximations to the different smooth segments of f, based upon matching the signatures of the approximant to the signature of f. The resulting approximation captures the singularity of the given data. As a result, the error in this first stage approximation is smooth.

AB - Given scattered data values of a piecewise smooth function f within a domain,we look for a piecewise adaptive approximation to f. Approximation techniques for scattered data approximation, as Radial Basis Function (RBF) or Moving Least-Squares (MLS), achieve reduced approximation orders near the boundary of the domain and near the unknown curves of jump singularities of the function or its derivatives. The idea used here is that the approximation errors near the boundaries, and near a singularity curve, fully characterize the behavior of the function at these locations. We refer to these approximation error values as the signature of f. In this paper, we aim at using these values in order to define the approximation. Assuming smoothness of the singularity curve, we suggest using a signed-distance approach to construct an approximation of the singularity curve. Now we find approximations to the different smooth segments of f, based upon matching the signatures of the approximant to the signature of f. The resulting approximation captures the singularity of the given data. As a result, the error in this first stage approximation is smooth.

KW - Bivariate approximation

KW - Moving least squares

KW - Piecewise smooth functions

UR - http://www.scopus.com/inward/record.url?scp=85129410126&partnerID=8YFLogxK

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AN - SCOPUS:85129410126

SN - 1889-3066

VL - 12

SP - 155

JO - Jaen Journal on Approximation

JF - Jaen Journal on Approximation

IS - 1

ER -